diff options
Diffstat (limited to 'libquadmath/math')
77 files changed, 13563 insertions, 0 deletions
diff --git a/libquadmath/math/acoshq.c b/libquadmath/math/acoshq.c new file mode 100644 index 000000000..b0a1e4e54 --- /dev/null +++ b/libquadmath/math/acoshq.c @@ -0,0 +1,59 @@ +/* e_acoshl.c -- long double version of e_acosh.c. + * Conversion to long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* __ieee754_acoshl(x) + * Method : + * Based on + * acoshl(x) = logl [ x + sqrtl(x*x-1) ] + * we have + * acoshl(x) := logl(x)+ln2, if x is large; else + * acoshl(x) := logl(2x-1/(sqrtl(x*x-1)+x)) if x>2; else + * acoshl(x) := log1pl(t+sqrtl(2.0*t+t*t)); where t=x-1. + * + * Special cases: + * acoshl(x) is NaN with signal if x<1. + * acoshl(NaN) is NaN without signal. + */ + +#include "quadmath-imp.h" + +static const __float128 +one = 1.0Q, +ln2 = 0.6931471805599453094172321214581766Q; + +__float128 +acoshq (__float128 x) +{ + __float128 t; + uint64_t lx; + int64_t hx; + GET_FLT128_WORDS64(hx,lx,x); + if(hx<0x3fff000000000000LL) { /* x < 1 */ + return (x-x)/(x-x); + } else if(hx >=0x4035000000000000LL) { /* x > 2**54 */ + if(hx >=0x7fff000000000000LL) { /* x is inf of NaN */ + return x+x; + } else + return logq(x)+ln2; /* acoshl(huge)=logl(2x) */ + } else if(((hx-0x3fff000000000000LL)|lx)==0) { + return 0.0Q; /* acosh(1) = 0 */ + } else if (hx > 0x4000000000000000LL) { /* 2**28 > x > 2 */ + t=x*x; + return logq(2.0Q*x-one/(x+sqrtq(t-one))); + } else { /* 1<x<2 */ + t = x-one; + return log1pq(t+sqrtq(2.0Q*t+t*t)); + } +} diff --git a/libquadmath/math/acosq.c b/libquadmath/math/acosq.c new file mode 100644 index 000000000..a8a361d23 --- /dev/null +++ b/libquadmath/math/acosq.c @@ -0,0 +1,317 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* + __float128 expansions are + Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> + and are incorporated herein by permission of the author. The author + reserves the right to distribute this material elsewhere under different + copying permissions. These modifications are distributed here under + the following terms: + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + +/* __ieee754_acosl(x) + * Method : + * acos(x) = pi/2 - asin(x) + * acos(-x) = pi/2 + asin(x) + * For |x| <= 0.375 + * acos(x) = pi/2 - asin(x) + * Between .375 and .5 the approximation is + * acos(0.4375 + x) = acos(0.4375) + x P(x) / Q(x) + * Between .5 and .625 the approximation is + * acos(0.5625 + x) = acos(0.5625) + x rS(x) / sS(x) + * For x > 0.625, + * acos(x) = 2 asin(sqrt((1-x)/2)) + * computed with an extended precision square root in the leading term. + * For x < -0.625 + * acos(x) = pi - 2 asin(sqrt((1-|x|)/2)) + * + * Special cases: + * if x is NaN, return x itself; + * if |x|>1, return NaN with invalid signal. + * + * Functions needed: __ieee754_sqrtl. + */ + +#include "quadmath-imp.h" + +static const __float128 + one = 1.0Q, + pio2_hi = 1.5707963267948966192313216916397514420986Q, + pio2_lo = 4.3359050650618905123985220130216759843812E-35Q, + + /* acos(0.5625 + x) = acos(0.5625) + x rS(x) / sS(x) + -0.0625 <= x <= 0.0625 + peak relative error 3.3e-35 */ + + rS0 = 5.619049346208901520945464704848780243887E0Q, + rS1 = -4.460504162777731472539175700169871920352E1Q, + rS2 = 1.317669505315409261479577040530751477488E2Q, + rS3 = -1.626532582423661989632442410808596009227E2Q, + rS4 = 3.144806644195158614904369445440583873264E1Q, + rS5 = 9.806674443470740708765165604769099559553E1Q, + rS6 = -5.708468492052010816555762842394927806920E1Q, + rS7 = -1.396540499232262112248553357962639431922E1Q, + rS8 = 1.126243289311910363001762058295832610344E1Q, + rS9 = 4.956179821329901954211277873774472383512E-1Q, + rS10 = -3.313227657082367169241333738391762525780E-1Q, + + sS0 = -4.645814742084009935700221277307007679325E0Q, + sS1 = 3.879074822457694323970438316317961918430E1Q, + sS2 = -1.221986588013474694623973554726201001066E2Q, + sS3 = 1.658821150347718105012079876756201905822E2Q, + sS4 = -4.804379630977558197953176474426239748977E1Q, + sS5 = -1.004296417397316948114344573811562952793E2Q, + sS6 = 7.530281592861320234941101403870010111138E1Q, + sS7 = 1.270735595411673647119592092304357226607E1Q, + sS8 = -1.815144839646376500705105967064792930282E1Q, + sS9 = -7.821597334910963922204235247786840828217E-2Q, + /* 1.000000000000000000000000000000000000000E0 */ + + acosr5625 = 9.7338991014954640492751132535550279812151E-1Q, + pimacosr5625 = 2.1682027434402468335351320579240000860757E0Q, + + /* acos(0.4375 + x) = acos(0.4375) + x rS(x) / sS(x) + -0.0625 <= x <= 0.0625 + peak relative error 2.1e-35 */ + + P0 = 2.177690192235413635229046633751390484892E0Q, + P1 = -2.848698225706605746657192566166142909573E1Q, + P2 = 1.040076477655245590871244795403659880304E2Q, + P3 = -1.400087608918906358323551402881238180553E2Q, + P4 = 2.221047917671449176051896400503615543757E1Q, + P5 = 9.643714856395587663736110523917499638702E1Q, + P6 = -5.158406639829833829027457284942389079196E1Q, + P7 = -1.578651828337585944715290382181219741813E1Q, + P8 = 1.093632715903802870546857764647931045906E1Q, + P9 = 5.448925479898460003048760932274085300103E-1Q, + P10 = -3.315886001095605268470690485170092986337E-1Q, + Q0 = -1.958219113487162405143608843774587557016E0Q, + Q1 = 2.614577866876185080678907676023269360520E1Q, + Q2 = -9.990858606464150981009763389881793660938E1Q, + Q3 = 1.443958741356995763628660823395334281596E2Q, + Q4 = -3.206441012484232867657763518369723873129E1Q, + Q5 = -1.048560885341833443564920145642588991492E2Q, + Q6 = 6.745883931909770880159915641984874746358E1Q, + Q7 = 1.806809656342804436118449982647641392951E1Q, + Q8 = -1.770150690652438294290020775359580915464E1Q, + Q9 = -5.659156469628629327045433069052560211164E-1Q, + /* 1.000000000000000000000000000000000000000E0 */ + + acosr4375 = 1.1179797320499710475919903296900511518755E0Q, + pimacosr4375 = 2.0236129215398221908706530535894517323217E0Q, + + /* asin(x) = x + x^3 pS(x^2) / qS(x^2) + 0 <= x <= 0.5 + peak relative error 1.9e-35 */ + pS0 = -8.358099012470680544198472400254596543711E2Q, + pS1 = 3.674973957689619490312782828051860366493E3Q, + pS2 = -6.730729094812979665807581609853656623219E3Q, + pS3 = 6.643843795209060298375552684423454077633E3Q, + pS4 = -3.817341990928606692235481812252049415993E3Q, + pS5 = 1.284635388402653715636722822195716476156E3Q, + pS6 = -2.410736125231549204856567737329112037867E2Q, + pS7 = 2.219191969382402856557594215833622156220E1Q, + pS8 = -7.249056260830627156600112195061001036533E-1Q, + pS9 = 1.055923570937755300061509030361395604448E-3Q, + + qS0 = -5.014859407482408326519083440151745519205E3Q, + qS1 = 2.430653047950480068881028451580393430537E4Q, + qS2 = -4.997904737193653607449250593976069726962E4Q, + qS3 = 5.675712336110456923807959930107347511086E4Q, + qS4 = -3.881523118339661268482937768522572588022E4Q, + qS5 = 1.634202194895541569749717032234510811216E4Q, + qS6 = -4.151452662440709301601820849901296953752E3Q, + qS7 = 5.956050864057192019085175976175695342168E2Q, + qS8 = -4.175375777334867025769346564600396877176E1Q; + /* 1.000000000000000000000000000000000000000E0 */ + +__float128 +acosq (__float128 x) +{ + __float128 z, r, w, p, q, s, t, f2; + int32_t ix, sign; + ieee854_float128 u; + + u.value = x; + sign = u.words32.w0; + ix = sign & 0x7fffffff; + u.words32.w0 = ix; /* |x| */ + if (ix >= 0x3fff0000) /* |x| >= 1 */ + { + if (ix == 0x3fff0000 + && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0) + { /* |x| == 1 */ + if ((sign & 0x80000000) == 0) + return 0.0; /* acos(1) = 0 */ + else + return (2.0 * pio2_hi) + (2.0 * pio2_lo); /* acos(-1)= pi */ + } + return (x - x) / (x - x); /* acos(|x| > 1) is NaN */ + } + else if (ix < 0x3ffe0000) /* |x| < 0.5 */ + { + if (ix < 0x3fc60000) /* |x| < 2**-57 */ + return pio2_hi + pio2_lo; + if (ix < 0x3ffde000) /* |x| < .4375 */ + { + /* Arcsine of x. */ + z = x * x; + p = (((((((((pS9 * z + + pS8) * z + + pS7) * z + + pS6) * z + + pS5) * z + + pS4) * z + + pS3) * z + + pS2) * z + + pS1) * z + + pS0) * z; + q = (((((((( z + + qS8) * z + + qS7) * z + + qS6) * z + + qS5) * z + + qS4) * z + + qS3) * z + + qS2) * z + + qS1) * z + + qS0; + r = x + x * p / q; + z = pio2_hi - (r - pio2_lo); + return z; + } + /* .4375 <= |x| < .5 */ + t = u.value - 0.4375Q; + p = ((((((((((P10 * t + + P9) * t + + P8) * t + + P7) * t + + P6) * t + + P5) * t + + P4) * t + + P3) * t + + P2) * t + + P1) * t + + P0) * t; + + q = (((((((((t + + Q9) * t + + Q8) * t + + Q7) * t + + Q6) * t + + Q5) * t + + Q4) * t + + Q3) * t + + Q2) * t + + Q1) * t + + Q0; + r = p / q; + if (sign & 0x80000000) + r = pimacosr4375 - r; + else + r = acosr4375 + r; + return r; + } + else if (ix < 0x3ffe4000) /* |x| < 0.625 */ + { + t = u.value - 0.5625Q; + p = ((((((((((rS10 * t + + rS9) * t + + rS8) * t + + rS7) * t + + rS6) * t + + rS5) * t + + rS4) * t + + rS3) * t + + rS2) * t + + rS1) * t + + rS0) * t; + + q = (((((((((t + + sS9) * t + + sS8) * t + + sS7) * t + + sS6) * t + + sS5) * t + + sS4) * t + + sS3) * t + + sS2) * t + + sS1) * t + + sS0; + if (sign & 0x80000000) + r = pimacosr5625 - p / q; + else + r = acosr5625 + p / q; + return r; + } + else + { /* |x| >= .625 */ + z = (one - u.value) * 0.5; + s = sqrtq (z); + /* Compute an extended precision square root from + the Newton iteration s -> 0.5 * (s + z / s). + The change w from s to the improved value is + w = 0.5 * (s + z / s) - s = (s^2 + z)/2s - s = (z - s^2)/2s. + Express s = f1 + f2 where f1 * f1 is exactly representable. + w = (z - s^2)/2s = (z - f1^2 - 2 f1 f2 - f2^2)/2s . + s + w has extended precision. */ + u.value = s; + u.words32.w2 = 0; + u.words32.w3 = 0; + f2 = s - u.value; + w = z - u.value * u.value; + w = w - 2.0 * u.value * f2; + w = w - f2 * f2; + w = w / (2.0 * s); + /* Arcsine of s. */ + p = (((((((((pS9 * z + + pS8) * z + + pS7) * z + + pS6) * z + + pS5) * z + + pS4) * z + + pS3) * z + + pS2) * z + + pS1) * z + + pS0) * z; + q = (((((((( z + + qS8) * z + + qS7) * z + + qS6) * z + + qS5) * z + + qS4) * z + + qS3) * z + + qS2) * z + + qS1) * z + + qS0; + r = s + (w + s * p / q); + + if (sign & 0x80000000) + w = pio2_hi + (pio2_lo - r); + else + w = r; + return 2.0 * w; + } +} diff --git a/libquadmath/math/asinhq.c b/libquadmath/math/asinhq.c new file mode 100644 index 000000000..be044dcd8 --- /dev/null +++ b/libquadmath/math/asinhq.c @@ -0,0 +1,71 @@ +/* s_asinhl.c -- long double version of s_asinh.c. + * Conversion to long double by Ulrich Drepper, + * Cygnus Support, drepper@cygnus.com. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* asinhl(x) + * Method : + * Based on + * asinhl(x) = signl(x) * logl [ |x| + sqrtl(x*x+1) ] + * we have + * asinhl(x) := x if 1+x*x=1, + * := signl(x)*(logl(x)+ln2)) for large |x|, else + * := signl(x)*logl(2|x|+1/(|x|+sqrtl(x*x+1))) if|x|>2, else + * := signl(x)*log1pl(|x| + x^2/(1 + sqrtl(1+x^2))) + */ + +#include "quadmath-imp.h" + +static const __float128 + one = 1.0Q, + ln2 = 6.931471805599453094172321214581765681e-1Q, + huge = 1.0e+4900Q; + +__float128 +asinhq (__float128 x) +{ + __float128 t, w; + int32_t ix, sign; + ieee854_float128 u; + + u.value = x; + sign = u.words32.w0; + ix = sign & 0x7fffffff; + if (ix == 0x7fff0000) + return x + x; /* x is inf or NaN */ + if (ix < 0x3fc70000) + { /* |x| < 2^ -56 */ + if (huge + x > one) + return x; /* return x inexact except 0 */ + } + u.words32.w0 = ix; + if (ix > 0x40350000) + { /* |x| > 2 ^ 54 */ + w = logq (u.value) + ln2; + } + else if (ix >0x40000000) + { /* 2^ 54 > |x| > 2.0 */ + t = u.value; + w = logq (2.0 * t + one / (sqrtq (x * x + one) + t)); + } + else + { /* 2.0 > |x| > 2 ^ -56 */ + t = x * x; + w = log1pq (u.value + t / (one + sqrtq (one + t))); + } + if (sign & 0x80000000) + return -w; + else + return w; +} diff --git a/libquadmath/math/asinq.c b/libquadmath/math/asinq.c new file mode 100644 index 000000000..d4321a58e --- /dev/null +++ b/libquadmath/math/asinq.c @@ -0,0 +1,254 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* + __float128 expansions are + Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> + and are incorporated herein by permission of the author. The author + reserves the right to distribute this material elsewhere under different + copying permissions. These modifications are distributed here under the + following terms: + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + +/* __ieee754_asin(x) + * Method : + * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... + * we approximate asin(x) on [0,0.5] by + * asin(x) = x + x*x^2*R(x^2) + * Between .5 and .625 the approximation is + * asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x) + * For x in [0.625,1] + * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) + * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; + * then for x>0.98 + * asin(x) = pi/2 - 2*(s+s*z*R(z)) + * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) + * For x<=0.98, let pio4_hi = pio2_hi/2, then + * f = hi part of s; + * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) + * and + * asin(x) = pi/2 - 2*(s+s*z*R(z)) + * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) + * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) + * + * Special cases: + * if x is NaN, return x itself; + * if |x|>1, return NaN with invalid signal. + * + */ + + +#include "quadmath-imp.h" + +static const __float128 + one = 1.0Q, + huge = 1.0e+4932Q, + pio2_hi = 1.5707963267948966192313216916397514420986Q, + pio2_lo = 4.3359050650618905123985220130216759843812E-35Q, + pio4_hi = 7.8539816339744830961566084581987569936977E-1Q, + + /* coefficient for R(x^2) */ + + /* asin(x) = x + x^3 pS(x^2) / qS(x^2) + 0 <= x <= 0.5 + peak relative error 1.9e-35 */ + pS0 = -8.358099012470680544198472400254596543711E2Q, + pS1 = 3.674973957689619490312782828051860366493E3Q, + pS2 = -6.730729094812979665807581609853656623219E3Q, + pS3 = 6.643843795209060298375552684423454077633E3Q, + pS4 = -3.817341990928606692235481812252049415993E3Q, + pS5 = 1.284635388402653715636722822195716476156E3Q, + pS6 = -2.410736125231549204856567737329112037867E2Q, + pS7 = 2.219191969382402856557594215833622156220E1Q, + pS8 = -7.249056260830627156600112195061001036533E-1Q, + pS9 = 1.055923570937755300061509030361395604448E-3Q, + + qS0 = -5.014859407482408326519083440151745519205E3Q, + qS1 = 2.430653047950480068881028451580393430537E4Q, + qS2 = -4.997904737193653607449250593976069726962E4Q, + qS3 = 5.675712336110456923807959930107347511086E4Q, + qS4 = -3.881523118339661268482937768522572588022E4Q, + qS5 = 1.634202194895541569749717032234510811216E4Q, + qS6 = -4.151452662440709301601820849901296953752E3Q, + qS7 = 5.956050864057192019085175976175695342168E2Q, + qS8 = -4.175375777334867025769346564600396877176E1Q, + /* 1.000000000000000000000000000000000000000E0 */ + + /* asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x) + -0.0625 <= x <= 0.0625 + peak relative error 3.3e-35 */ + rS0 = -5.619049346208901520945464704848780243887E0Q, + rS1 = 4.460504162777731472539175700169871920352E1Q, + rS2 = -1.317669505315409261479577040530751477488E2Q, + rS3 = 1.626532582423661989632442410808596009227E2Q, + rS4 = -3.144806644195158614904369445440583873264E1Q, + rS5 = -9.806674443470740708765165604769099559553E1Q, + rS6 = 5.708468492052010816555762842394927806920E1Q, + rS7 = 1.396540499232262112248553357962639431922E1Q, + rS8 = -1.126243289311910363001762058295832610344E1Q, + rS9 = -4.956179821329901954211277873774472383512E-1Q, + rS10 = 3.313227657082367169241333738391762525780E-1Q, + + sS0 = -4.645814742084009935700221277307007679325E0Q, + sS1 = 3.879074822457694323970438316317961918430E1Q, + sS2 = -1.221986588013474694623973554726201001066E2Q, + sS3 = 1.658821150347718105012079876756201905822E2Q, + sS4 = -4.804379630977558197953176474426239748977E1Q, + sS5 = -1.004296417397316948114344573811562952793E2Q, + sS6 = 7.530281592861320234941101403870010111138E1Q, + sS7 = 1.270735595411673647119592092304357226607E1Q, + sS8 = -1.815144839646376500705105967064792930282E1Q, + sS9 = -7.821597334910963922204235247786840828217E-2Q, + /* 1.000000000000000000000000000000000000000E0 */ + + asinr5625 = 5.9740641664535021430381036628424864397707E-1Q; + + + +__float128 +asinq (__float128 x) +{ + __float128 t = 0; + __float128 w, p, q, c, r, s; + int32_t ix, sign, flag; + ieee854_float128 u; + + flag = 0; + u.value = x; + sign = u.words32.w0; + ix = sign & 0x7fffffff; + u.words32.w0 = ix; /* |x| */ + if (ix >= 0x3fff0000) /* |x|>= 1 */ + { + if (ix == 0x3fff0000 + && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0) + /* asin(1)=+-pi/2 with inexact */ + return x * pio2_hi + x * pio2_lo; + return (x - x) / (x - x); /* asin(|x|>1) is NaN */ + } + else if (ix < 0x3ffe0000) /* |x| < 0.5 */ + { + if (ix < 0x3fc60000) /* |x| < 2**-57 */ + { + if (huge + x > one) + return x; /* return x with inexact if x!=0 */ + } + else + { + t = x * x; + /* Mark to use pS, qS later on. */ + flag = 1; + } + } + else if (ix < 0x3ffe4000) /* 0.625 */ + { + t = u.value - 0.5625; + p = ((((((((((rS10 * t + + rS9) * t + + rS8) * t + + rS7) * t + + rS6) * t + + rS5) * t + + rS4) * t + + rS3) * t + + rS2) * t + + rS1) * t + + rS0) * t; + + q = ((((((((( t + + sS9) * t + + sS8) * t + + sS7) * t + + sS6) * t + + sS5) * t + + sS4) * t + + sS3) * t + + sS2) * t + + sS1) * t + + sS0; + t = asinr5625 + p / q; + if ((sign & 0x80000000) == 0) + return t; + else + return -t; + } + else + { + /* 1 > |x| >= 0.625 */ + w = one - u.value; + t = w * 0.5; + } + + p = (((((((((pS9 * t + + pS8) * t + + pS7) * t + + pS6) * t + + pS5) * t + + pS4) * t + + pS3) * t + + pS2) * t + + pS1) * t + + pS0) * t; + + q = (((((((( t + + qS8) * t + + qS7) * t + + qS6) * t + + qS5) * t + + qS4) * t + + qS3) * t + + qS2) * t + + qS1) * t + + qS0; + + if (flag) /* 2^-57 < |x| < 0.5 */ + { + w = p / q; + return x + x * w; + } + + s = sqrtq (t); + if (ix >= 0x3ffef333) /* |x| > 0.975 */ + { + w = p / q; + t = pio2_hi - (2.0 * (s + s * w) - pio2_lo); + } + else + { + u.value = s; + u.words32.w3 = 0; + u.words32.w2 = 0; + w = u.value; + c = (t - w * w) / (s + w); + r = p / q; + p = 2.0 * s * r - (pio2_lo - 2.0 * c); + q = pio4_hi - 2.0 * w; + t = pio4_hi - (p - q); + } + + if ((sign & 0x80000000) == 0) + return t; + else + return -t; +} diff --git a/libquadmath/math/atan2q.c b/libquadmath/math/atan2q.c new file mode 100644 index 000000000..fbe64d62b --- /dev/null +++ b/libquadmath/math/atan2q.c @@ -0,0 +1,120 @@ +/* e_atan2l.c -- long double version of e_atan2.c. + * Conversion to long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* atan2q(y,x) + * Method : + * 1. Reduce y to positive by atan2q(y,x)=-atan2q(-y,x). + * 2. Reduce x to positive by (if x and y are unexceptional): + * ARG (x+iy) = arctan(y/x) ... if x > 0, + * ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0, + * + * Special cases: + * + * ATAN2((anything), NaN ) is NaN; + * ATAN2(NAN , (anything) ) is NaN; + * ATAN2(+-0, +(anything but NaN)) is +-0 ; + * ATAN2(+-0, -(anything but NaN)) is +-pi ; + * ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2; + * ATAN2(+-(anything but INF and NaN), +INF) is +-0 ; + * ATAN2(+-(anything but INF and NaN), -INF) is +-pi; + * ATAN2(+-INF,+INF ) is +-pi/4 ; + * ATAN2(+-INF,-INF ) is +-3pi/4; + * ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2; + * + * Constants: + * The hexadecimal values are the intended ones for the following + * constants. The decimal values may be used, provided that the + * compiler will convert from decimal to binary accurately enough + * to produce the hexadecimal values shown. + */ + +#include "quadmath-imp.h" + +static const __float128 +tiny = 1.0e-4900Q, +zero = 0.0, +pi_o_4 = 7.85398163397448309615660845819875699e-01Q, /* 3ffe921fb54442d18469898cc51701b8 */ +pi_o_2 = 1.57079632679489661923132169163975140e+00Q, /* 3fff921fb54442d18469898cc51701b8 */ +pi = 3.14159265358979323846264338327950280e+00Q, /* 4000921fb54442d18469898cc51701b8 */ +pi_lo = 8.67181013012378102479704402604335225e-35Q; /* 3f8dcd129024e088a67cc74020bbea64 */ + +__float128 +atan2q (__float128 y, __float128 x) +{ + __float128 z; + int64_t k,m,hx,hy,ix,iy; + uint64_t lx,ly; + + GET_FLT128_WORDS64(hx,lx,x); + ix = hx&0x7fffffffffffffffLL; + GET_FLT128_WORDS64(hy,ly,y); + iy = hy&0x7fffffffffffffffLL; + if(((ix|((lx|-lx)>>63))>0x7fff000000000000LL)|| + ((iy|((ly|-ly)>>63))>0x7fff000000000000LL)) /* x or y is NaN */ + return x+y; + if(((hx-0x3fff000000000000LL)|lx)==0) return atanq(y); /* x=1.0Q */ + m = ((hy>>63)&1)|((hx>>62)&2); /* 2*sign(x)+sign(y) */ + + /* when y = 0 */ + if((iy|ly)==0) { + switch(m) { + case 0: + case 1: return y; /* atan(+-0,+anything)=+-0 */ + case 2: return pi+tiny;/* atan(+0,-anything) = pi */ + case 3: return -pi-tiny;/* atan(-0,-anything) =-pi */ + } + } + /* when x = 0 */ + if((ix|lx)==0) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny; + + /* when x is INF */ + if(ix==0x7fff000000000000LL) { + if(iy==0x7fff000000000000LL) { + switch(m) { + case 0: return pi_o_4+tiny;/* atan(+INF,+INF) */ + case 1: return -pi_o_4-tiny;/* atan(-INF,+INF) */ + case 2: return 3.0Q*pi_o_4+tiny;/*atan(+INF,-INF)*/ + case 3: return -3.0Q*pi_o_4-tiny;/*atan(-INF,-INF)*/ + } + } else { + switch(m) { + case 0: return zero ; /* atan(+...,+INF) */ + case 1: return -zero ; /* atan(-...,+INF) */ + case 2: return pi+tiny ; /* atan(+...,-INF) */ + case 3: return -pi-tiny ; /* atan(-...,-INF) */ + } + } + } + /* when y is INF */ + if(iy==0x7fff000000000000LL) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny; + + /* compute y/x */ + k = (iy-ix)>>48; + if(k > 120) z=pi_o_2+0.5Q*pi_lo; /* |y/x| > 2**120 */ + else if(hx<0&&k<-120) z=0.0Q; /* |y|/x < -2**120 */ + else z=atanq(fabsq(y/x)); /* safe to do y/x */ + switch (m) { + case 0: return z ; /* atan(+,+) */ + case 1: { + uint64_t zh; + GET_FLT128_MSW64(zh,z); + SET_FLT128_MSW64(z,zh ^ 0x8000000000000000ULL); + } + return z ; /* atan(-,+) */ + case 2: return pi-(z-pi_lo);/* atan(+,-) */ + default: /* case 3 */ + return (z-pi_lo)-pi;/* atan(-,-) */ + } +} diff --git a/libquadmath/math/atanhq.c b/libquadmath/math/atanhq.c new file mode 100644 index 000000000..73db957d3 --- /dev/null +++ b/libquadmath/math/atanhq.c @@ -0,0 +1,66 @@ +/* s_atanhl.c -- __float128 version of s_atan.c. + * Conversion to __float128 by Ulrich Drepper, + * Cygnus Support, drepper@cygnus.com. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* __ieee754_atanhl(x) + * Method : + * 1.Reduced x to positive by atanh(-x) = -atanh(x) + * 2.For x>=0.5 + * 1 2x x + * atanhl(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------) + * 2 1 - x 1 - x + * + * For x<0.5 + * atanhl(x) = 0.5*log1pl(2x+2x*x/(1-x)) + * + * Special cases: + * atanhl(x) is NaN if |x| > 1 with signal; + * atanhl(NaN) is that NaN with no signal; + * atanhl(+-1) is +-INF with signal. + * + */ + +#include "quadmath-imp.h" + +static const __float128 one = 1.0Q, huge = 1e4900Q; +static const __float128 zero = 0.0Q; + +__float128 +atanhq (__float128 x) +{ + __float128 t; + uint32_t jx, ix; + ieee854_float128 u; + + u.value = x; + jx = u.words32.w0; + ix = jx & 0x7fffffff; + u.words32.w0 = ix; + if (ix >= 0x3fff0000) /* |x| >= 1.0 or infinity or NaN */ + { + if (u.value == one) + return x/zero; + else + return (x-x)/(x-x); + } + if(ix<0x3fc60000 && (huge+x)>zero) return x; /* x < 2^-57 */ + + if(ix<0x3ffe0000) { /* x < 0.5 */ + t = u.value+u.value; + t = 0.5*log1pq(t+t*u.value/(one-u.value)); + } else + t = 0.5*log1pq((u.value+u.value)/(one-u.value)); + if(jx & 0x80000000) return -t; else return t; +} diff --git a/libquadmath/math/atanq.c b/libquadmath/math/atanq.c new file mode 100644 index 000000000..cb38a340a --- /dev/null +++ b/libquadmath/math/atanq.c @@ -0,0 +1,231 @@ +/* s_atanl.c + * + * Inverse circular tangent for 128-bit __float128 precision + * (arctangent) + * + * + * + * SYNOPSIS: + * + * __float128 x, y, atanl(); + * + * y = atanl( x ); + * + * + * + * DESCRIPTION: + * + * Returns radian angle between -pi/2 and +pi/2 whose tangent is x. + * + * The function uses a rational approximation of the form + * t + t^3 P(t^2)/Q(t^2), optimized for |t| < 0.09375. + * + * The argument is reduced using the identity + * arctan x - arctan u = arctan ((x-u)/(1 + ux)) + * and an 83-entry lookup table for arctan u, with u = 0, 1/8, ..., 10.25. + * Use of the table improves the execution speed of the routine. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -19, 19 4e5 1.7e-34 5.4e-35 + * + * + * WARNING: + * + * This program uses integer operations on bit fields of floating-point + * numbers. It does not work with data structures other than the + * structure assumed. + * + */ + +/* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov> + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + + +#include "quadmath-imp.h" + +/* arctan(k/8), k = 0, ..., 82 */ +static const __float128 atantbl[84] = { + 0.0000000000000000000000000000000000000000E0Q, + 1.2435499454676143503135484916387102557317E-1Q, /* arctan(0.125) */ + 2.4497866312686415417208248121127581091414E-1Q, + 3.5877067027057222039592006392646049977698E-1Q, + 4.6364760900080611621425623146121440202854E-1Q, + 5.5859931534356243597150821640166127034645E-1Q, + 6.4350110879328438680280922871732263804151E-1Q, + 7.1882999962162450541701415152590465395142E-1Q, + 7.8539816339744830961566084581987572104929E-1Q, + 8.4415398611317100251784414827164750652594E-1Q, + 8.9605538457134395617480071802993782702458E-1Q, + 9.4200004037946366473793717053459358607166E-1Q, + 9.8279372324732906798571061101466601449688E-1Q, + 1.0191413442663497346383429170230636487744E0Q, + 1.0516502125483736674598673120862998296302E0Q, + 1.0808390005411683108871567292171998202703E0Q, + 1.1071487177940905030170654601785370400700E0Q, + 1.1309537439791604464709335155363278047493E0Q, + 1.1525719972156675180401498626127513797495E0Q, + 1.1722738811284763866005949441337046149712E0Q, + 1.1902899496825317329277337748293183376012E0Q, + 1.2068173702852525303955115800565576303133E0Q, + 1.2220253232109896370417417439225704908830E0Q, + 1.2360594894780819419094519711090786987027E0Q, + 1.2490457723982544258299170772810901230778E0Q, + 1.2610933822524404193139408812473357720101E0Q, + 1.2722973952087173412961937498224804940684E0Q, + 1.2827408797442707473628852511364955306249E0Q, + 1.2924966677897852679030914214070816845853E0Q, + 1.3016288340091961438047858503666855921414E0Q, + 1.3101939350475556342564376891719053122733E0Q, + 1.3182420510168370498593302023271362531155E0Q, + 1.3258176636680324650592392104284756311844E0Q, + 1.3329603993374458675538498697331558093700E0Q, + 1.3397056595989995393283037525895557411039E0Q, + 1.3460851583802539310489409282517796256512E0Q, + 1.3521273809209546571891479413898128509842E0Q, + 1.3578579772154994751124898859640585287459E0Q, + 1.3633001003596939542892985278250991189943E0Q, + 1.3684746984165928776366381936948529556191E0Q, + 1.3734007669450158608612719264449611486510E0Q, + 1.3780955681325110444536609641291551522494E0Q, + 1.3825748214901258580599674177685685125566E0Q, + 1.3868528702577214543289381097042486034883E0Q, + 1.3909428270024183486427686943836432060856E0Q, + 1.3948567013423687823948122092044222644895E0Q, + 1.3986055122719575950126700816114282335732E0Q, + 1.4021993871854670105330304794336492676944E0Q, + 1.4056476493802697809521934019958079881002E0Q, + 1.4089588955564736949699075250792569287156E0Q, + 1.4121410646084952153676136718584891599630E0Q, + 1.4152014988178669079462550975833894394929E0Q, + 1.4181469983996314594038603039700989523716E0Q, + 1.4209838702219992566633046424614466661176E0Q, + 1.4237179714064941189018190466107297503086E0Q, + 1.4263547484202526397918060597281265695725E0Q, + 1.4288992721907326964184700745371983590908E0Q, + 1.4313562697035588982240194668401779312122E0Q, + 1.4337301524847089866404719096698873648610E0Q, + 1.4360250423171655234964275337155008780675E0Q, + 1.4382447944982225979614042479354815855386E0Q, + 1.4403930189057632173997301031392126865694E0Q, + 1.4424730991091018200252920599377292525125E0Q, + 1.4444882097316563655148453598508037025938E0Q, + 1.4464413322481351841999668424758804165254E0Q, + 1.4483352693775551917970437843145232637695E0Q, + 1.4501726582147939000905940595923466567576E0Q, + 1.4519559822271314199339700039142990228105E0Q, + 1.4536875822280323362423034480994649820285E0Q, + 1.4553696664279718992423082296859928222270E0Q, + 1.4570043196511885530074841089245667532358E0Q, + 1.4585935117976422128825857356750737658039E0Q, + 1.4601391056210009726721818194296893361233E0Q, + 1.4616428638860188872060496086383008594310E0Q, + 1.4631064559620759326975975316301202111560E0Q, + 1.4645314639038178118428450961503371619177E0Q, + 1.4659193880646627234129855241049975398470E0Q, + 1.4672716522843522691530527207287398276197E0Q, + 1.4685896086876430842559640450619880951144E0Q, + 1.4698745421276027686510391411132998919794E0Q, + 1.4711276743037345918528755717617308518553E0Q, + 1.4723501675822635384916444186631899205983E0Q, + 1.4735431285433308455179928682541563973416E0Q, /* arctan(10.25) */ + 1.5707963267948966192313216916397514420986E0Q /* pi/2 */ +}; + + +/* arctan t = t + t^3 p(t^2) / q(t^2) + |t| <= 0.09375 + peak relative error 5.3e-37 */ + +static const __float128 + p0 = -4.283708356338736809269381409828726405572E1Q, + p1 = -8.636132499244548540964557273544599863825E1Q, + p2 = -5.713554848244551350855604111031839613216E1Q, + p3 = -1.371405711877433266573835355036413750118E1Q, + p4 = -8.638214309119210906997318946650189640184E-1Q, + q0 = 1.285112506901621042780814422948906537959E2Q, + q1 = 3.361907253914337187957855834229672347089E2Q, + q2 = 3.180448303864130128268191635189365331680E2Q, + q3 = 1.307244136980865800160844625025280344686E2Q, + q4 = 2.173623741810414221251136181221172551416E1Q; + /* q5 = 1.000000000000000000000000000000000000000E0 */ + + +__float128 +atanq (__float128 x) +{ + int k, sign; + __float128 t, u, p, q; + ieee854_float128 s; + + s.value = x; + k = s.words32.w0; + if (k & 0x80000000) + sign = 1; + else + sign = 0; + + /* Check for IEEE special cases. */ + k &= 0x7fffffff; + if (k >= 0x7fff0000) + { + /* NaN. */ + if ((k & 0xffff) | s.words32.w1 | s.words32.w2 | s.words32.w3) + return (x + x); + + /* Infinity. */ + if (sign) + return -atantbl[83]; + else + return atantbl[83]; + } + + if (sign) + x = -x; + + if (k >= 0x40024800) /* 10.25 */ + { + k = 83; + t = -1.0/x; + } + else + { + /* Index of nearest table element. + Roundoff to integer is asymmetrical to avoid cancellation when t < 0 + (cf. fdlibm). */ + k = 8.0Q * x + 0.25Q; + u = 0.125Q * k; + /* Small arctan argument. */ + t = (x - u) / (1.0 + x * u); + } + + /* Arctan of small argument t. */ + u = t * t; + p = ((((p4 * u) + p3) * u + p2) * u + p1) * u + p0; + q = ((((u + q4) * u + q3) * u + q2) * u + q1) * u + q0; + u = t * u * p / q + t; + + /* arctan x = arctan u + arctan t */ + u = atantbl[k] + u; + if (sign) + return (-u); + else + return u; +} diff --git a/libquadmath/math/cacoshq.c b/libquadmath/math/cacoshq.c new file mode 100644 index 000000000..8acc570de --- /dev/null +++ b/libquadmath/math/cacoshq.c @@ -0,0 +1,89 @@ +/* Return arc hyperbole cosine for __float128 value. + Copyright (C) 1997, 1998, 2006 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + + +__complex128 +cacoshq (__complex128 x) +{ + __complex128 res; + int rcls = fpclassifyq (__real__ x); + int icls = fpclassifyq (__imag__ x); + + if (rcls <= QUADFP_INFINITE || icls <= QUADFP_INFINITE) + { + if (icls == QUADFP_INFINITE) + { + __real__ res = HUGE_VALQ; + + if (rcls == QUADFP_NAN) + __imag__ res = nanq (""); + else + __imag__ res = copysignq ((rcls == QUADFP_INFINITE + ? (__real__ x < 0.0 + ? M_PIq - M_PI_4q : M_PI_4q) + : M_PI_2q), __imag__ x); + } + else if (rcls == QUADFP_INFINITE) + { + __real__ res = HUGE_VALQ; + + if (icls >= QUADFP_ZERO) + __imag__ res = copysignq (signbitq (__real__ x) ? M_PIq : 0.0, + __imag__ x); + else + __imag__ res = nanq (""); + } + else + { + __real__ res = nanq (""); + __imag__ res = nanq (""); + } + } + else if (rcls == QUADFP_ZERO && icls == QUADFP_ZERO) + { + __real__ res = 0.0; + __imag__ res = copysignq (M_PI_2q, __imag__ x); + } + else + { + __complex128 y; + + __real__ y = (__real__ x - __imag__ x) * (__real__ x + __imag__ x) - 1.0; + __imag__ y = 2.0 * __real__ x * __imag__ x; + + y = csqrtq (y); + + if (__real__ x < 0.0) + y = -y; + + __real__ y += __real__ x; + __imag__ y += __imag__ x; + + res = clogq (y); + + /* We have to use the positive branch. */ + if (__real__ res < 0.0) + res = -res; + } + + return res; +} diff --git a/libquadmath/math/cacosq.c b/libquadmath/math/cacosq.c new file mode 100644 index 000000000..3c257b029 --- /dev/null +++ b/libquadmath/math/cacosq.c @@ -0,0 +1,35 @@ +/* Return cosine of complex __float128 value. + Copyright (C) 1997, 1998 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + +__complex128 +cacosq (__complex128 x) +{ + __complex128 y; + __complex128 res; + + y = casinq (x); + + __real__ res = M_PI_2q - __real__ y; + __imag__ res = -__imag__ y; + + return res; +} diff --git a/libquadmath/math/casinhq.c b/libquadmath/math/casinhq.c new file mode 100644 index 000000000..ffa45fa81 --- /dev/null +++ b/libquadmath/math/casinhq.c @@ -0,0 +1,78 @@ +/* Return arc hyperbole sine for __float128 value. + Copyright (C) 1997, 1998 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + + +__complex128 +casinhq (__complex128 x) +{ + __complex128 res; + int rcls = fpclassifyq (__real__ x); + int icls = fpclassifyq (__imag__ x); + + if (rcls <= QUADFP_INFINITE || icls <= QUADFP_INFINITE) + { + if (icls == QUADFP_INFINITE) + { + __real__ res = copysignq (HUGE_VALQ, __real__ x); + + if (rcls == QUADFP_NAN) + __imag__ res = nanq (""); + else + __imag__ res = copysignq (rcls >= QUADFP_ZERO ? M_PI_2q : M_PI_4q, + __imag__ x); + } + else if (rcls <= QUADFP_INFINITE) + { + __real__ res = __real__ x; + if ((rcls == QUADFP_INFINITE && icls >= QUADFP_ZERO) + || (rcls == QUADFP_NAN && icls == QUADFP_ZERO)) + __imag__ res = copysignq (0.0, __imag__ x); + else + __imag__ res = nanq (""); + } + else + { + __real__ res = nanq (""); + __imag__ res = nanq (""); + } + } + else if (rcls == QUADFP_ZERO && icls == QUADFP_ZERO) + { + res = x; + } + else + { + __complex128 y; + + __real__ y = (__real__ x - __imag__ x) * (__real__ x + __imag__ x) + 1.0; + __imag__ y = 2.0 * __real__ x * __imag__ x; + + y = csqrtq (y); + + __real__ y += __real__ x; + __imag__ y += __imag__ x; + + res = clogq (y); + } + + return res; +} diff --git a/libquadmath/math/casinq.c b/libquadmath/math/casinq.c new file mode 100644 index 000000000..122ef5d85 --- /dev/null +++ b/libquadmath/math/casinq.c @@ -0,0 +1,60 @@ +/* Return arc sine of complex __float128 value. + Copyright (C) 1997 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + + +__complex128 +casinq (__complex128 x) +{ + __complex128 res; + + if (isnanq (__real__ x) || isnanq (__imag__ x)) + { + if (__real__ x == 0.0) + { + res = x; + } + else if (isinfq (__real__ x) || isinfq (__imag__ x)) + { + __real__ res = nanq (""); + __imag__ res = copysignq (HUGE_VALQ, __imag__ x); + } + else + { + __real__ res = nanq (""); + __imag__ res = nanq (""); + } + } + else + { + __complex128 y; + + __real__ y = -__imag__ x; + __imag__ y = __real__ x; + + y = casinhq (y); + + __real__ res = __imag__ y; + __imag__ res = -__real__ y; + } + + return res; +} diff --git a/libquadmath/math/catanhq.c b/libquadmath/math/catanhq.c new file mode 100644 index 000000000..6a86e2d02 --- /dev/null +++ b/libquadmath/math/catanhq.c @@ -0,0 +1,76 @@ +/* Return arc hyperbole tangent for __float128 value. + Copyright (C) 1997, 1998 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + + +__complex128 +catanhq (__complex128 x) +{ + __complex128 res; + int rcls = fpclassifyq (__real__ x); + int icls = fpclassifyq (__imag__ x); + + if (rcls <= QUADFP_INFINITE || icls <= QUADFP_INFINITE) + { + if (icls == QUADFP_INFINITE) + { + __real__ res = copysignq (0.0, __real__ x); + __imag__ res = copysignq (M_PI_2q, __imag__ x); + } + else if (rcls == QUADFP_INFINITE || rcls == QUADFP_ZERO) + { + __real__ res = copysignq (0.0, __real__ x); + if (icls >= QUADFP_ZERO) + __imag__ res = copysignq (M_PI_2q, __imag__ x); + else + __imag__ res = nanq (""); + } + else + { + __real__ res = nanq (""); + __imag__ res = nanq (""); + } + } + else if (rcls == QUADFP_ZERO && icls == QUADFP_ZERO) + { + res = x; + } + else + { + __float128 i2, num, den; + + i2 = __imag__ x * __imag__ x; + + num = 1.0 + __real__ x; + num = i2 + num * num; + + den = 1.0 - __real__ x; + den = i2 + den * den; + + __real__ res = 0.25 * (logq (num) - logq (den)); + + den = 1 - __real__ x * __real__ x - i2; + + __imag__ res = 0.5 * atan2q (2.0 * __imag__ x, den); + } + + return res; +} diff --git a/libquadmath/math/catanq.c b/libquadmath/math/catanq.c new file mode 100644 index 000000000..a9c925797 --- /dev/null +++ b/libquadmath/math/catanq.c @@ -0,0 +1,81 @@ +/* Return arc tangent of complex __float128 value. + Copyright (C) 1997, 1998 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + + +__complex128 +catanq (__complex128 x) +{ + __complex128 res; + int rcls = fpclassifyq (__real__ x); + int icls = fpclassifyq (__imag__ x); + + if (rcls <= QUADFP_INFINITE || icls <= QUADFP_INFINITE) + { + if (rcls == QUADFP_INFINITE) + { + __real__ res = copysignq (M_PI_2q, __real__ x); + __imag__ res = copysignq (0.0, __imag__ x); + } + else if (icls == QUADFP_INFINITE) + { + if (rcls >= QUADFP_ZERO) + __real__ res = copysignq (M_PI_2q, __real__ x); + else + __real__ res = nanq (""); + __imag__ res = copysignq (0.0, __imag__ x); + } + else if (icls == QUADFP_ZERO || icls == QUADFP_INFINITE) + { + __real__ res = nanq (""); + __imag__ res = copysignq (0.0, __imag__ x); + } + else + { + __real__ res = nanq (""); + __imag__ res = nanq (""); + } + } + else if (rcls == QUADFP_ZERO && icls == QUADFP_ZERO) + { + res = x; + } + else + { + __float128 r2, num, den; + + r2 = __real__ x * __real__ x; + + den = 1 - r2 - __imag__ x * __imag__ x; + + __real__ res = 0.5 * atan2q (2.0 * __real__ x, den); + + num = __imag__ x + 1.0; + num = r2 + num * num; + + den = __imag__ x - 1.0; + den = r2 + den * den; + + __imag__ res = 0.25 * logq (num / den); + } + + return res; +} diff --git a/libquadmath/math/cbrtq.c b/libquadmath/math/cbrtq.c new file mode 100644 index 000000000..f61f32513 --- /dev/null +++ b/libquadmath/math/cbrtq.c @@ -0,0 +1,64 @@ +#include "quadmath-imp.h" +#include <math.h> +#include <float.h> + +__float128 +cbrtq (const __float128 x) +{ + __float128 y; + int exp, i; + + if (x == 0) + return x; + + if (isnanq (x)) + return x; + + if (x <= DBL_MAX && x >= DBL_MIN) + { + /* Use double result as starting point. */ + y = cbrt ((double) x); + + /* Two Newton iterations. */ + y -= 0.333333333333333333333333333333333333333333333333333Q + * (y - x / (y * y)); + y -= 0.333333333333333333333333333333333333333333333333333Q + * (y - x / (y * y)); + return y; + } + +#ifdef HAVE_CBRTL + if (x <= LDBL_MAX && x >= LDBL_MIN) + { + /* Use long double result as starting point. */ + y = cbrtl ((long double) x); + + /* One Newton iteration. */ + y -= 0.333333333333333333333333333333333333333333333333333Q + * (y - x / (y * y)); + return y; + } +#endif + + /* If we're outside of the range of C types, we have to compute + the initial guess the hard way. */ + y = frexpq (x, &exp); + + i = exp % 3; + y = (i >= 0 ? i : -i); + if (i == 1) + y *= 2, exp--; + else if (i == 2) + y *= 4, exp -= 2; + + y = cbrt (y); + y = scalbnq (y, exp / 3); + + /* Two Newton iterations. */ + y -= 0.333333333333333333333333333333333333333333333333333Q + * (y - x / (y * y)); + y -= 0.333333333333333333333333333333333333333333333333333Q + * (y - x / (y * y)); + return y; +} + diff --git a/libquadmath/math/ceilq.c b/libquadmath/math/ceilq.c new file mode 100644 index 000000000..577d8cd98 --- /dev/null +++ b/libquadmath/math/ceilq.c @@ -0,0 +1,61 @@ +/* s_ceill.c -- long double version of s_ceil.c. + * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include "quadmath-imp.h" + +static const __float128 huge = 1.0e4930Q; + +__float128 +ceilq (__float128 x) +{ + int64_t i0,i1,j0; + uint64_t i,j; + GET_FLT128_WORDS64(i0,i1,x); + j0 = ((i0>>48)&0x7fff)-0x3fff; + if(j0<48) { + if(j0<0) { /* raise inexact if x != 0 */ + if(huge+x>0.0) {/* return 0*sign(x) if |x|<1 */ + if(i0<0) {i0=0x8000000000000000ULL;i1=0;} + else if((i0|i1)!=0) { i0=0x3fff000000000000ULL;i1=0;} + } + } else { + i = (0x0000ffffffffffffULL)>>j0; + if(((i0&i)|i1)==0) return x; /* x is integral */ + if(huge+x>0.0) { /* raise inexact flag */ + if(i0>0) i0 += (0x0001000000000000LL)>>j0; + i0 &= (~i); i1=0; + } + } + } else if (j0>111) { + if(j0==0x4000) return x+x; /* inf or NaN */ + else return x; /* x is integral */ + } else { + i = -1ULL>>(j0-48); + if((i1&i)==0) return x; /* x is integral */ + if(huge+x>0.0) { /* raise inexact flag */ + if(i0>0) { + if(j0==48) i0+=1; + else { + j = i1+(1LL<<(112-j0)); + if(j<i1) i0 +=1 ; /* got a carry */ + i1=j; + } + } + i1 &= (~i); + } + } + SET_FLT128_WORDS64(x,i0,i1); + return x; +} diff --git a/libquadmath/math/cimagq.c b/libquadmath/math/cimagq.c new file mode 100644 index 000000000..9b3bf70a1 --- /dev/null +++ b/libquadmath/math/cimagq.c @@ -0,0 +1,27 @@ +/* Return imaginary part of complex __float128 value. + Copyright (C) 1997, 1998 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + +__float128 +cimagq (__complex128 z) +{ + return __imag__ z; +} diff --git a/libquadmath/math/complex.c b/libquadmath/math/complex.c new file mode 100644 index 000000000..f67448a2c --- /dev/null +++ b/libquadmath/math/complex.c @@ -0,0 +1,210 @@ +#include "quadmath-imp.h" + + +#define REALPART(z) (__real__(z)) +#define IMAGPART(z) (__imag__(z)) +#define COMPLEX_ASSIGN(z_, r_, i_) {__real__(z_) = (r_); __imag__(z_) = (i_);} + + +// Horrible... GCC doesn't know how to multiply or divide these +// __complex128 things. We have to do it on our own. +// Protect it around macros so, some day, we can switch it on + +#if 0 + +# define C128_MULT(x,y) ((x)*(y)) +# define C128_DIV(x,y) ((x)/(y)) + +#else + +#define C128_MULT(x,y) mult_c128(x,y) +#define C128_DIV(x,y) div_c128(x,y) + +static inline __complex128 mult_c128 (__complex128 x, __complex128 y) +{ + __float128 r1 = REALPART(x), i1 = IMAGPART(x); + __float128 r2 = REALPART(y), i2 = IMAGPART(y); + __complex128 res; + COMPLEX_ASSIGN(res, r1*r2 - i1*i2, i2*r1 + i1*r2); + return res; +} + + +// Careful: the algorithm for the division sucks. A lot. +static inline __complex128 div_c128 (__complex128 x, __complex128 y) +{ + __float128 n = hypotq (REALPART (y), IMAGPART (y)); + __float128 r1 = REALPART(x), i1 = IMAGPART(x); + __float128 r2 = REALPART(y), i2 = IMAGPART(y); + __complex128 res; + COMPLEX_ASSIGN(res, r1*r2 + i1*i2, i1*r2 - i2*r1); + return res / n; +} + +#endif + + + +__float128 +cabsq (__complex128 z) +{ + return hypotq (REALPART (z), IMAGPART (z)); +} + + +__complex128 +cexpq (__complex128 z) +{ + __float128 a, b; + __complex128 v; + + a = REALPART (z); + b = IMAGPART (z); + COMPLEX_ASSIGN (v, cosq (b), sinq (b)); + return expq (a) * v; +} + + +__complex128 +cexpiq (__float128 x) +{ + __complex128 v; + COMPLEX_ASSIGN (v, cosq (x), sinq (x)); + return v; +} + + +__float128 +cargq (__complex128 z) +{ + return atan2q (IMAGPART (z), REALPART (z)); +} + + +__complex128 +clogq (__complex128 z) +{ + __complex128 v; + COMPLEX_ASSIGN (v, logq (cabsq (z)), cargq (z)); + return v; +} + + +__complex128 +clog10q (__complex128 z) +{ + __complex128 v; + COMPLEX_ASSIGN (v, log10q (cabsq (z)), cargq (z)); + return v; +} + + +__complex128 +cpowq (__complex128 base, __complex128 power) +{ + return cexpq (C128_MULT(power, clogq (base))); +} + + +__complex128 +csinq (__complex128 a) +{ + __float128 r = REALPART (a), i = IMAGPART (a); + __complex128 v; + COMPLEX_ASSIGN (v, sinq (r) * coshq (i), cosq (r) * sinhq (i)); + return v; +} + + +__complex128 +csinhq (__complex128 a) +{ + __float128 r = REALPART (a), i = IMAGPART (a); + __complex128 v; + COMPLEX_ASSIGN (v, sinhq (r) * cosq (i), coshq (r) * sinq (i)); + return v; +} + + +__complex128 +ccosq (__complex128 a) +{ + __float128 r = REALPART (a), i = IMAGPART (a); + __complex128 v; + COMPLEX_ASSIGN (v, cosq (r) * coshq (i), - (sinq (r) * sinhq (i))); + return v; +} + + +__complex128 +ccoshq (__complex128 a) +{ + __float128 r = REALPART (a), i = IMAGPART (a); + __complex128 v; + COMPLEX_ASSIGN (v, coshq (r) * cosq (i), sinhq (r) * sinq (i)); + return v; +} + + +__complex128 +ctanq (__complex128 a) +{ + __float128 rt = tanq (REALPART (a)), it = tanhq (IMAGPART (a)); + __complex128 n, d; + COMPLEX_ASSIGN (n, rt, it); + COMPLEX_ASSIGN (d, 1, - (rt * it)); + return C128_DIV(n,d); +} + + +__complex128 +ctanhq (__complex128 a) +{ + __float128 rt = tanhq (REALPART (a)), it = tanq (IMAGPART (a)); + __complex128 n, d; + COMPLEX_ASSIGN (n, rt, it); + COMPLEX_ASSIGN (d, 1, rt * it); + return C128_DIV(n,d); +} + + +/* Square root algorithm from glibc. */ +__complex128 +csqrtq (__complex128 z) +{ + __float128 re = REALPART(z), im = IMAGPART(z); + __complex128 v; + + if (im == 0) + { + if (re < 0) + { + COMPLEX_ASSIGN (v, 0, copysignq (sqrtq (-re), im)); + } + else + { + COMPLEX_ASSIGN (v, fabsq (sqrtq (re)), copysignq (0, im)); + } + } + else if (re == 0) + { + __float128 r = sqrtq (0.5 * fabsq (im)); + COMPLEX_ASSIGN (v, r, copysignq (r, im)); + } + else + { + __float128 d = hypotq (re, im); + __float128 r, s; + + /* Use the identity 2 Re res Im res = Im x + to avoid cancellation error in d +/- Re x. */ + if (re > 0) + r = sqrtq (0.5 * d + 0.5 * re), s = (0.5 * im) / r; + else + s = sqrtq (0.5 * d - 0.5 * re), r = fabsq ((0.5 * im) / s); + + COMPLEX_ASSIGN (v, r, copysignq (s, im)); + } + return v; +} + diff --git a/libquadmath/math/conjq.c b/libquadmath/math/conjq.c new file mode 100644 index 000000000..8587d1219 --- /dev/null +++ b/libquadmath/math/conjq.c @@ -0,0 +1,27 @@ +/* Return complex conjugate of complex __float128 value. + Copyright (C) 1997, 1998 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + +__complex128 +conjq (__complex128 z) +{ + return ~z; +} diff --git a/libquadmath/math/copysignq.c b/libquadmath/math/copysignq.c new file mode 100644 index 000000000..b59fcc549 --- /dev/null +++ b/libquadmath/math/copysignq.c @@ -0,0 +1,26 @@ +/* s_copysignl.c -- long double version of s_copysign.c. + * Conversion to long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include "quadmath-imp.h" + +__float128 +copysignq (__float128 x, __float128 y) +{ + uint64_t hx,hy; + GET_FLT128_MSW64(hx,x); + GET_FLT128_MSW64(hy,y); + SET_FLT128_MSW64(x,(hx&0x7fffffffffffffffULL)|(hy&0x8000000000000000ULL)); + return x; +} diff --git a/libquadmath/math/coshq.c b/libquadmath/math/coshq.c new file mode 100644 index 000000000..a6e0eb5f1 --- /dev/null +++ b/libquadmath/math/coshq.c @@ -0,0 +1,108 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* Changes for 128-bit __float128 are + Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> + and are incorporated herein by permission of the author. The author + reserves the right to distribute this material elsewhere under different + copying permissions. These modifications are distributed here under + the following terms: + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + +/* __ieee754_coshl(x) + * Method : + * mathematically coshl(x) if defined to be (exp(x)+exp(-x))/2 + * 1. Replace x by |x| (coshl(x) = coshl(-x)). + * 2. + * [ exp(x) - 1 ]^2 + * 0 <= x <= ln2/2 : coshl(x) := 1 + ------------------- + * 2*exp(x) + * + * exp(x) + 1/exp(x) + * ln2/2 <= x <= 22 : coshl(x) := ------------------- + * 2 + * 22 <= x <= lnovft : coshl(x) := expl(x)/2 + * lnovft <= x <= ln2ovft: coshl(x) := expl(x/2)/2 * expl(x/2) + * ln2ovft < x : coshl(x) := huge*huge (overflow) + * + * Special cases: + * coshl(x) is |x| if x is +INF, -INF, or NaN. + * only coshl(0)=1 is exact for finite x. + */ + +#include "quadmath-imp.h" + +static const __float128 one = 1.0Q, half = 0.5Q, huge = 1.0e4900Q, + ovf_thresh = 1.1357216553474703894801348310092223067821E4Q; + +__float128 +coshq (__float128 x) +{ + __float128 t, w; + int32_t ex; + ieee854_float128 u; + + u.value = x; + ex = u.words32.w0 & 0x7fffffff; + + /* Absolute value of x. */ + u.words32.w0 = ex; + + /* x is INF or NaN */ + if (ex >= 0x7fff0000) + return x * x; + + /* |x| in [0,0.5*ln2], return 1+expm1l(|x|)^2/(2*expl(|x|)) */ + if (ex < 0x3ffd62e4) /* 0.3465728759765625 */ + { + t = expm1q (u.value); + w = one + t; + if (ex < 0x3fb80000) /* |x| < 2^-116 */ + return w; /* cosh(tiny) = 1 */ + + return one + (t * t) / (w + w); + } + + /* |x| in [0.5*ln2,40], return (exp(|x|)+1/exp(|x|)/2; */ + if (ex < 0x40044000) + { + t = expq (u.value); + return half * t + half / t; + } + + /* |x| in [22, ln(maxdouble)] return half*exp(|x|) */ + if (ex <= 0x400c62e3) /* 11356.375 */ + return half * expq (u.value); + + /* |x| in [log(maxdouble), overflowthresold] */ + if (u.value <= ovf_thresh) + { + w = expq (half * u.value); + t = half * w; + return t * w; + } + + /* |x| > overflowthresold, cosh(x) overflow */ + return huge * huge; +} diff --git a/libquadmath/math/cosq.c b/libquadmath/math/cosq.c new file mode 100644 index 000000000..dc321a27d --- /dev/null +++ b/libquadmath/math/cosq.c @@ -0,0 +1,82 @@ +/* s_cosl.c -- long double version of s_cos.c. + * Conversion to long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* cosl(x) + * Return cosine function of x. + * + * kernel function: + * __kernel_sinl ... sine function on [-pi/4,pi/4] + * __kernel_cosl ... cosine function on [-pi/4,pi/4] + * __ieee754_rem_pio2l ... argument reduction routine + * + * Method. + * Let S,C and T denote the sin, cos and tan respectively on + * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 + * in [-pi/4 , +pi/4], and let n = k mod 4. + * We have + * + * n sin(x) cos(x) tan(x) + * ---------------------------------------------------------- + * 0 S C T + * 1 C -S -1/T + * 2 -S -C T + * 3 -C S -1/T + * ---------------------------------------------------------- + * + * Special cases: + * Let trig be any of sin, cos, or tan. + * trig(+-INF) is NaN, with signals; + * trig(NaN) is that NaN; + * + * Accuracy: + * TRIG(x) returns trig(x) nearly rounded + */ + +#include "quadmath-imp.h" + +__float128 +cosq (__float128 x) +{ + __float128 y[2],z=0.0Q; + int64_t n, ix; + + /* High word of x. */ + GET_FLT128_MSW64(ix,x); + + /* |x| ~< pi/4 */ + ix &= 0x7fffffffffffffffLL; + if(ix <= 0x3ffe921fb54442d1LL) + return __quadmath_kernel_cosq(x,z); + + /* cos(Inf or NaN) is NaN */ + else if (ix>=0x7fff000000000000LL) { + if (ix == 0x7fff000000000000LL) { + GET_FLT128_LSW64(n,x); + } + return x-x; + } + + /* argument reduction needed */ + else { + n = __quadmath_rem_pio2q(x,y); + switch(n&3) { + case 0: return __quadmath_kernel_cosq(y[0],y[1]); + case 1: return -__quadmath_kernel_sinq(y[0],y[1],1); + case 2: return -__quadmath_kernel_cosq(y[0],y[1]); + default: + return __quadmath_kernel_sinq(y[0],y[1],1); + } + } +} diff --git a/libquadmath/math/cosq_kernel.c b/libquadmath/math/cosq_kernel.c new file mode 100644 index 000000000..86f39551c --- /dev/null +++ b/libquadmath/math/cosq_kernel.c @@ -0,0 +1,127 @@ +/* Quad-precision floating point cosine on <-pi/4,pi/4>. + Copyright (C) 1999 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Jakub Jelinek <jj@ultra.linux.cz> + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + +static const __float128 c[] = { +#define ONE c[0] + 1.00000000000000000000000000000000000E+00Q, /* 3fff0000000000000000000000000000 */ + +/* cos x ~ ONE + x^2 ( SCOS1 + SCOS2 * x^2 + ... + SCOS4 * x^6 + SCOS5 * x^8 ) + x in <0,1/256> */ +#define SCOS1 c[1] +#define SCOS2 c[2] +#define SCOS3 c[3] +#define SCOS4 c[4] +#define SCOS5 c[5] +-5.00000000000000000000000000000000000E-01Q, /* bffe0000000000000000000000000000 */ + 4.16666666666666666666666666556146073E-02Q, /* 3ffa5555555555555555555555395023 */ +-1.38888888888888888888309442601939728E-03Q, /* bff56c16c16c16c16c16a566e42c0375 */ + 2.48015873015862382987049502531095061E-05Q, /* 3fefa01a01a019ee02dcf7da2d6d5444 */ +-2.75573112601362126593516899592158083E-07Q, /* bfe927e4f5dce637cb0b54908754bde0 */ + +/* cos x ~ ONE + x^2 ( COS1 + COS2 * x^2 + ... + COS7 * x^12 + COS8 * x^14 ) + x in <0,0.1484375> */ +#define COS1 c[6] +#define COS2 c[7] +#define COS3 c[8] +#define COS4 c[9] +#define COS5 c[10] +#define COS6 c[11] +#define COS7 c[12] +#define COS8 c[13] +-4.99999999999999999999999999999999759E-01Q, /* bffdfffffffffffffffffffffffffffb */ + 4.16666666666666666666666666651287795E-02Q, /* 3ffa5555555555555555555555516f30 */ +-1.38888888888888888888888742314300284E-03Q, /* bff56c16c16c16c16c16c16a463dfd0d */ + 2.48015873015873015867694002851118210E-05Q, /* 3fefa01a01a01a01a0195cebe6f3d3a5 */ +-2.75573192239858811636614709689300351E-07Q, /* bfe927e4fb7789f5aa8142a22044b51f */ + 2.08767569877762248667431926878073669E-09Q, /* 3fe21eed8eff881d1e9262d7adff4373 */ +-1.14707451049343817400420280514614892E-11Q, /* bfda9397496922a9601ed3d4ca48944b */ + 4.77810092804389587579843296923533297E-14Q, /* 3fd2ae5f8197cbcdcaf7c3fb4523414c */ + +/* sin x ~ ONE * x + x^3 ( SSIN1 + SSIN2 * x^2 + ... + SSIN4 * x^6 + SSIN5 * x^8 ) + x in <0,1/256> */ +#define SSIN1 c[14] +#define SSIN2 c[15] +#define SSIN3 c[16] +#define SSIN4 c[17] +#define SSIN5 c[18] +-1.66666666666666666666666666666666659E-01Q, /* bffc5555555555555555555555555555 */ + 8.33333333333333333333333333146298442E-03Q, /* 3ff81111111111111111111110fe195d */ +-1.98412698412698412697726277416810661E-04Q, /* bff2a01a01a01a01a019e7121e080d88 */ + 2.75573192239848624174178393552189149E-06Q, /* 3fec71de3a556c640c6aaa51aa02ab41 */ +-2.50521016467996193495359189395805639E-08Q, /* bfe5ae644ee90c47dc71839de75b2787 */ +}; + +#define SINCOSQ_COS_HI 0 +#define SINCOSQ_COS_LO 1 +#define SINCOSQ_SIN_HI 2 +#define SINCOSQ_SIN_LO 3 +extern const __float128 __sincosq_table[]; + +__float128 +__quadmath_kernel_cosq (__float128 x, __float128 y) +{ + __float128 h, l, z, sin_l, cos_l_m1; + int64_t ix; + uint32_t tix, hix, index; + GET_FLT128_MSW64 (ix, x); + tix = ((uint64_t)ix) >> 32; + tix &= ~0x80000000; /* tix = |x|'s high 32 bits */ + if (tix < 0x3ffc3000) /* |x| < 0.1484375 */ + { + /* Argument is small enough to approximate it by a Chebyshev + polynomial of degree 16. */ + if (tix < 0x3fc60000) /* |x| < 2^-57 */ + if (!((int)x)) return ONE; /* generate inexact */ + z = x * x; + return ONE + (z*(COS1+z*(COS2+z*(COS3+z*(COS4+ + z*(COS5+z*(COS6+z*(COS7+z*COS8)))))))); + } + else + { + /* So that we don't have to use too large polynomial, we find + l and h such that x = l + h, where fabsl(l) <= 1.0/256 with 83 + possible values for h. We look up cosl(h) and sinl(h) in + pre-computed tables, compute cosl(l) and sinl(l) using a + Chebyshev polynomial of degree 10(11) and compute + cosl(h+l) = cosl(h)cosl(l) - sinl(h)sinl(l). */ + index = 0x3ffe - (tix >> 16); + hix = (tix + (0x200 << index)) & (0xfffffc00 << index); + x = fabsq (x); + switch (index) + { + case 0: index = ((45 << 10) + hix - 0x3ffe0000) >> 8; break; + case 1: index = ((13 << 11) + hix - 0x3ffd0000) >> 9; break; + default: + case 2: index = (hix - 0x3ffc3000) >> 10; break; + } + + SET_FLT128_WORDS64(h, ((uint64_t)hix) << 32, 0); + l = y - (h - x); + z = l * l; + sin_l = l*(ONE+z*(SSIN1+z*(SSIN2+z*(SSIN3+z*(SSIN4+z*SSIN5))))); + cos_l_m1 = z*(SCOS1+z*(SCOS2+z*(SCOS3+z*(SCOS4+z*SCOS5)))); + return __sincosq_table [index + SINCOSQ_COS_HI] + + (__sincosq_table [index + SINCOSQ_COS_LO] + - (__sincosq_table [index + SINCOSQ_SIN_HI] * sin_l + - __sincosq_table [index + SINCOSQ_COS_HI] * cos_l_m1)); + } +} diff --git a/libquadmath/math/cprojq.c b/libquadmath/math/cprojq.c new file mode 100644 index 000000000..6092c7325 --- /dev/null +++ b/libquadmath/math/cprojq.c @@ -0,0 +1,40 @@ +/* Compute projection of complex __float128 value to Riemann sphere. + Copyright (C) 1997, 1999, 2010 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + + +__complex128 +cprojq (__complex128 x) +{ + if (isnanq (__real__ x) && isnanq (__imag__ x)) + return x; + else if (!finiteq (__real__ x) || !finiteq (__imag__ x)) + { + __complex128 res; + + __real__ res = __builtin_inf (); + __imag__ res = copysignq (0.0, __imag__ x); + + return res; + } + + return x; +} diff --git a/libquadmath/math/crealq.c b/libquadmath/math/crealq.c new file mode 100644 index 000000000..71f4a4405 --- /dev/null +++ b/libquadmath/math/crealq.c @@ -0,0 +1,27 @@ +/* Return real part of complex __float128 value. + Copyright (C) 1997, 1998 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + +__float128 +crealq (__complex128 z) +{ + return __real__ z; +} diff --git a/libquadmath/math/erfq.c b/libquadmath/math/erfq.c new file mode 100644 index 000000000..50db88ae8 --- /dev/null +++ b/libquadmath/math/erfq.c @@ -0,0 +1,935 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* Modifications and expansions for 128-bit long double are + Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> + and are incorporated herein by permission of the author. The author + reserves the right to distribute this material elsewhere under different + copying permissions. These modifications are distributed here under + the following terms: + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + +/* double erf(double x) + * double erfc(double x) + * x + * 2 |\ + * erf(x) = --------- | exp(-t*t)dt + * sqrt(pi) \| + * 0 + * + * erfc(x) = 1-erf(x) + * Note that + * erf(-x) = -erf(x) + * erfc(-x) = 2 - erfc(x) + * + * Method: + * 1. erf(x) = x + x*R(x^2) for |x| in [0, 7/8] + * Remark. The formula is derived by noting + * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) + * and that + * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 + * is close to one. + * + * 1a. erf(x) = 1 - erfc(x), for |x| > 1.0 + * erfc(x) = 1 - erf(x) if |x| < 1/4 + * + * 2. For |x| in [7/8, 1], let s = |x| - 1, and + * c = 0.84506291151 rounded to single (24 bits) + * erf(s + c) = sign(x) * (c + P1(s)/Q1(s)) + * Remark: here we use the taylor series expansion at x=1. + * erf(1+s) = erf(1) + s*Poly(s) + * = 0.845.. + P1(s)/Q1(s) + * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] + * + * 3. For x in [1/4, 5/4], + * erfc(s + const) = erfc(const) + s P1(s)/Q1(s) + * for const = 1/4, 3/8, ..., 9/8 + * and 0 <= s <= 1/8 . + * + * 4. For x in [5/4, 107], + * erfc(x) = (1/x)*exp(-x*x-0.5625 + R(z)) + * z=1/x^2 + * The interval is partitioned into several segments + * of width 1/8 in 1/x. + * + * Note1: + * To compute exp(-x*x-0.5625+R/S), let s be a single + * precision number and s := x; then + * -x*x = -s*s + (s-x)*(s+x) + * exp(-x*x-0.5626+R/S) = + * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); + * Note2: + * Here 4 and 5 make use of the asymptotic series + * exp(-x*x) + * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) + * x*sqrt(pi) + * + * 5. For inf > x >= 107 + * erf(x) = sign(x) *(1 - tiny) (raise inexact) + * erfc(x) = tiny*tiny (raise underflow) if x > 0 + * = 2 - tiny if x<0 + * + * 7. Special case: + * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, + * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, + * erfc/erf(NaN) is NaN + */ + +#include "quadmath-imp.h" + + + +__float128 erfcq (__float128); + + +/* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ + +static __float128 +neval (__float128 x, const __float128 *p, int n) +{ + __float128 y; + + p += n; + y = *p--; + do + { + y = y * x + *p--; + } + while (--n > 0); + return y; +} + + +/* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */ + +static __float128 +deval (__float128 x, const __float128 *p, int n) +{ + __float128 y; + + p += n; + y = x + *p--; + do + { + y = y * x + *p--; + } + while (--n > 0); + return y; +} + + + +static const __float128 +tiny = 1e-4931Q, + half = 0.5Q, + one = 1.0Q, + two = 2.0Q, + /* 2/sqrt(pi) - 1 */ + efx = 1.2837916709551257389615890312154517168810E-1Q, + /* 8 * (2/sqrt(pi) - 1) */ + efx8 = 1.0270333367641005911692712249723613735048E0Q; + + +/* erf(x) = x + x R(x^2) + 0 <= x <= 7/8 + Peak relative error 1.8e-35 */ +#define NTN1 8 +static const __float128 TN1[NTN1 + 1] = +{ + -3.858252324254637124543172907442106422373E10Q, + 9.580319248590464682316366876952214879858E10Q, + 1.302170519734879977595901236693040544854E10Q, + 2.922956950426397417800321486727032845006E9Q, + 1.764317520783319397868923218385468729799E8Q, + 1.573436014601118630105796794840834145120E7Q, + 4.028077380105721388745632295157816229289E5Q, + 1.644056806467289066852135096352853491530E4Q, + 3.390868480059991640235675479463287886081E1Q +}; +#define NTD1 8 +static const __float128 TD1[NTD1 + 1] = +{ + -3.005357030696532927149885530689529032152E11Q, + -1.342602283126282827411658673839982164042E11Q, + -2.777153893355340961288511024443668743399E10Q, + -3.483826391033531996955620074072768276974E9Q, + -2.906321047071299585682722511260895227921E8Q, + -1.653347985722154162439387878512427542691E7Q, + -6.245520581562848778466500301865173123136E5Q, + -1.402124304177498828590239373389110545142E4Q, + -1.209368072473510674493129989468348633579E2Q +/* 1.0E0 */ +}; + + +/* erf(z+1) = erf_const + P(z)/Q(z) + -.125 <= z <= 0 + Peak relative error 7.3e-36 */ +static const __float128 erf_const = 0.845062911510467529296875Q; +#define NTN2 8 +static const __float128 TN2[NTN2 + 1] = +{ + -4.088889697077485301010486931817357000235E1Q, + 7.157046430681808553842307502826960051036E3Q, + -2.191561912574409865550015485451373731780E3Q, + 2.180174916555316874988981177654057337219E3Q, + 2.848578658049670668231333682379720943455E2Q, + 1.630362490952512836762810462174798925274E2Q, + 6.317712353961866974143739396865293596895E0Q, + 2.450441034183492434655586496522857578066E1Q, + 5.127662277706787664956025545897050896203E-1Q +}; +#define NTD2 8 +static const __float128 TD2[NTD2 + 1] = +{ + 1.731026445926834008273768924015161048885E4Q, + 1.209682239007990370796112604286048173750E4Q, + 1.160950290217993641320602282462976163857E4Q, + 5.394294645127126577825507169061355698157E3Q, + 2.791239340533632669442158497532521776093E3Q, + 8.989365571337319032943005387378993827684E2Q, + 2.974016493766349409725385710897298069677E2Q, + 6.148192754590376378740261072533527271947E1Q, + 1.178502892490738445655468927408440847480E1Q + /* 1.0E0 */ +}; + + +/* erfc(x + 0.25) = erfc(0.25) + x R(x) + 0 <= x < 0.125 + Peak relative error 1.4e-35 */ +#define NRNr13 8 +static const __float128 RNr13[NRNr13 + 1] = +{ + -2.353707097641280550282633036456457014829E3Q, + 3.871159656228743599994116143079870279866E2Q, + -3.888105134258266192210485617504098426679E2Q, + -2.129998539120061668038806696199343094971E1Q, + -8.125462263594034672468446317145384108734E1Q, + 8.151549093983505810118308635926270319660E0Q, + -5.033362032729207310462422357772568553670E0Q, + -4.253956621135136090295893547735851168471E-2Q, + -8.098602878463854789780108161581050357814E-2Q +}; +#define NRDr13 7 +static const __float128 RDr13[NRDr13 + 1] = +{ + 2.220448796306693503549505450626652881752E3Q, + 1.899133258779578688791041599040951431383E2Q, + 1.061906712284961110196427571557149268454E3Q, + 7.497086072306967965180978101974566760042E1Q, + 2.146796115662672795876463568170441327274E2Q, + 1.120156008362573736664338015952284925592E1Q, + 2.211014952075052616409845051695042741074E1Q, + 6.469655675326150785692908453094054988938E-1Q + /* 1.0E0 */ +}; +/* erfc(0.25) = C13a + C13b to extra precision. */ +static const __float128 C13a = 0.723663330078125Q; +static const __float128 C13b = 1.0279753638067014931732235184287934646022E-5Q; + + +/* erfc(x + 0.375) = erfc(0.375) + x R(x) + 0 <= x < 0.125 + Peak relative error 1.2e-35 */ +#define NRNr14 8 +static const __float128 RNr14[NRNr14 + 1] = +{ + -2.446164016404426277577283038988918202456E3Q, + 6.718753324496563913392217011618096698140E2Q, + -4.581631138049836157425391886957389240794E2Q, + -2.382844088987092233033215402335026078208E1Q, + -7.119237852400600507927038680970936336458E1Q, + 1.313609646108420136332418282286454287146E1Q, + -6.188608702082264389155862490056401365834E0Q, + -2.787116601106678287277373011101132659279E-2Q, + -2.230395570574153963203348263549700967918E-2Q +}; +#define NRDr14 7 +static const __float128 RDr14[NRDr14 + 1] = +{ + 2.495187439241869732696223349840963702875E3Q, + 2.503549449872925580011284635695738412162E2Q, + 1.159033560988895481698051531263861842461E3Q, + 9.493751466542304491261487998684383688622E1Q, + 2.276214929562354328261422263078480321204E2Q, + 1.367697521219069280358984081407807931847E1Q, + 2.276988395995528495055594829206582732682E1Q, + 7.647745753648996559837591812375456641163E-1Q + /* 1.0E0 */ +}; +/* erfc(0.375) = C14a + C14b to extra precision. */ +static const __float128 C14a = 0.5958709716796875Q; +static const __float128 C14b = 1.2118885490201676174914080878232469565953E-5Q; + +/* erfc(x + 0.5) = erfc(0.5) + x R(x) + 0 <= x < 0.125 + Peak relative error 4.7e-36 */ +#define NRNr15 8 +static const __float128 RNr15[NRNr15 + 1] = +{ + -2.624212418011181487924855581955853461925E3Q, + 8.473828904647825181073831556439301342756E2Q, + -5.286207458628380765099405359607331669027E2Q, + -3.895781234155315729088407259045269652318E1Q, + -6.200857908065163618041240848728398496256E1Q, + 1.469324610346924001393137895116129204737E1Q, + -6.961356525370658572800674953305625578903E0Q, + 5.145724386641163809595512876629030548495E-3Q, + 1.990253655948179713415957791776180406812E-2Q +}; +#define NRDr15 7 +static const __float128 RDr15[NRDr15 + 1] = +{ + 2.986190760847974943034021764693341524962E3Q, + 5.288262758961073066335410218650047725985E2Q, + 1.363649178071006978355113026427856008978E3Q, + 1.921707975649915894241864988942255320833E2Q, + 2.588651100651029023069013885900085533226E2Q, + 2.628752920321455606558942309396855629459E1Q, + 2.455649035885114308978333741080991380610E1Q, + 1.378826653595128464383127836412100939126E0Q + /* 1.0E0 */ +}; +/* erfc(0.5) = C15a + C15b to extra precision. */ +static const __float128 C15a = 0.4794921875Q; +static const __float128 C15b = 7.9346869534623172533461080354712635484242E-6Q; + +/* erfc(x + 0.625) = erfc(0.625) + x R(x) + 0 <= x < 0.125 + Peak relative error 5.1e-36 */ +#define NRNr16 8 +static const __float128 RNr16[NRNr16 + 1] = +{ + -2.347887943200680563784690094002722906820E3Q, + 8.008590660692105004780722726421020136482E2Q, + -5.257363310384119728760181252132311447963E2Q, + -4.471737717857801230450290232600243795637E1Q, + -4.849540386452573306708795324759300320304E1Q, + 1.140885264677134679275986782978655952843E1Q, + -6.731591085460269447926746876983786152300E0Q, + 1.370831653033047440345050025876085121231E-1Q, + 2.022958279982138755020825717073966576670E-2Q, +}; +#define NRDr16 7 +static const __float128 RDr16[NRDr16 + 1] = +{ + 3.075166170024837215399323264868308087281E3Q, + 8.730468942160798031608053127270430036627E2Q, + 1.458472799166340479742581949088453244767E3Q, + 3.230423687568019709453130785873540386217E2Q, + 2.804009872719893612081109617983169474655E2Q, + 4.465334221323222943418085830026979293091E1Q, + 2.612723259683205928103787842214809134746E1Q, + 2.341526751185244109722204018543276124997E0Q, + /* 1.0E0 */ +}; +/* erfc(0.625) = C16a + C16b to extra precision. */ +static const __float128 C16a = 0.3767547607421875Q; +static const __float128 C16b = 4.3570693945275513594941232097252997287766E-6Q; + +/* erfc(x + 0.75) = erfc(0.75) + x R(x) + 0 <= x < 0.125 + Peak relative error 1.7e-35 */ +#define NRNr17 8 +static const __float128 RNr17[NRNr17 + 1] = +{ + -1.767068734220277728233364375724380366826E3Q, + 6.693746645665242832426891888805363898707E2Q, + -4.746224241837275958126060307406616817753E2Q, + -2.274160637728782675145666064841883803196E1Q, + -3.541232266140939050094370552538987982637E1Q, + 6.988950514747052676394491563585179503865E0Q, + -5.807687216836540830881352383529281215100E0Q, + 3.631915988567346438830283503729569443642E-1Q, + -1.488945487149634820537348176770282391202E-2Q +}; +#define NRDr17 7 +static const __float128 RDr17[NRDr17 + 1] = +{ + 2.748457523498150741964464942246913394647E3Q, + 1.020213390713477686776037331757871252652E3Q, + 1.388857635935432621972601695296561952738E3Q, + 3.903363681143817750895999579637315491087E2Q, + 2.784568344378139499217928969529219886578E2Q, + 5.555800830216764702779238020065345401144E1Q, + 2.646215470959050279430447295801291168941E1Q, + 2.984905282103517497081766758550112011265E0Q, + /* 1.0E0 */ +}; +/* erfc(0.75) = C17a + C17b to extra precision. */ +static const __float128 C17a = 0.2888336181640625Q; +static const __float128 C17b = 1.0748182422368401062165408589222625794046E-5Q; + + +/* erfc(x + 0.875) = erfc(0.875) + x R(x) + 0 <= x < 0.125 + Peak relative error 2.2e-35 */ +#define NRNr18 8 +static const __float128 RNr18[NRNr18 + 1] = +{ + -1.342044899087593397419622771847219619588E3Q, + 6.127221294229172997509252330961641850598E2Q, + -4.519821356522291185621206350470820610727E2Q, + 1.223275177825128732497510264197915160235E1Q, + -2.730789571382971355625020710543532867692E1Q, + 4.045181204921538886880171727755445395862E0Q, + -4.925146477876592723401384464691452700539E0Q, + 5.933878036611279244654299924101068088582E-1Q, + -5.557645435858916025452563379795159124753E-2Q +}; +#define NRDr18 7 +static const __float128 RDr18[NRDr18 + 1] = +{ + 2.557518000661700588758505116291983092951E3Q, + 1.070171433382888994954602511991940418588E3Q, + 1.344842834423493081054489613250688918709E3Q, + 4.161144478449381901208660598266288188426E2Q, + 2.763670252219855198052378138756906980422E2Q, + 5.998153487868943708236273854747564557632E1Q, + 2.657695108438628847733050476209037025318E1Q, + 3.252140524394421868923289114410336976512E0Q, + /* 1.0E0 */ +}; +/* erfc(0.875) = C18a + C18b to extra precision. */ +static const __float128 C18a = 0.215911865234375Q; +static const __float128 C18b = 1.3073705765341685464282101150637224028267E-5Q; + +/* erfc(x + 1.0) = erfc(1.0) + x R(x) + 0 <= x < 0.125 + Peak relative error 1.6e-35 */ +#define NRNr19 8 +static const __float128 RNr19[NRNr19 + 1] = +{ + -1.139180936454157193495882956565663294826E3Q, + 6.134903129086899737514712477207945973616E2Q, + -4.628909024715329562325555164720732868263E2Q, + 4.165702387210732352564932347500364010833E1Q, + -2.286979913515229747204101330405771801610E1Q, + 1.870695256449872743066783202326943667722E0Q, + -4.177486601273105752879868187237000032364E0Q, + 7.533980372789646140112424811291782526263E-1Q, + -8.629945436917752003058064731308767664446E-2Q +}; +#define NRDr19 7 +static const __float128 RDr19[NRDr19 + 1] = +{ + 2.744303447981132701432716278363418643778E3Q, + 1.266396359526187065222528050591302171471E3Q, + 1.466739461422073351497972255511919814273E3Q, + 4.868710570759693955597496520298058147162E2Q, + 2.993694301559756046478189634131722579643E2Q, + 6.868976819510254139741559102693828237440E1Q, + 2.801505816247677193480190483913753613630E1Q, + 3.604439909194350263552750347742663954481E0Q, + /* 1.0E0 */ +}; +/* erfc(1.0) = C19a + C19b to extra precision. */ +static const __float128 C19a = 0.15728759765625Q; +static const __float128 C19b = 1.1609394035130658779364917390740703933002E-5Q; + +/* erfc(x + 1.125) = erfc(1.125) + x R(x) + 0 <= x < 0.125 + Peak relative error 3.6e-36 */ +#define NRNr20 8 +static const __float128 RNr20[NRNr20 + 1] = +{ + -9.652706916457973956366721379612508047640E2Q, + 5.577066396050932776683469951773643880634E2Q, + -4.406335508848496713572223098693575485978E2Q, + 5.202893466490242733570232680736966655434E1Q, + -1.931311847665757913322495948705563937159E1Q, + -9.364318268748287664267341457164918090611E-2Q, + -3.306390351286352764891355375882586201069E0Q, + 7.573806045289044647727613003096916516475E-1Q, + -9.611744011489092894027478899545635991213E-2Q +}; +#define NRDr20 7 +static const __float128 RDr20[NRDr20 + 1] = +{ + 3.032829629520142564106649167182428189014E3Q, + 1.659648470721967719961167083684972196891E3Q, + 1.703545128657284619402511356932569292535E3Q, + 6.393465677731598872500200253155257708763E2Q, + 3.489131397281030947405287112726059221934E2Q, + 8.848641738570783406484348434387611713070E1Q, + 3.132269062552392974833215844236160958502E1Q, + 4.430131663290563523933419966185230513168E0Q + /* 1.0E0 */ +}; +/* erfc(1.125) = C20a + C20b to extra precision. */ +static const __float128 C20a = 0.111602783203125Q; +static const __float128 C20b = 8.9850951672359304215530728365232161564636E-6Q; + +/* erfc(1/x) = 1/x exp (-1/x^2 - 0.5625 + R(1/x^2)) + 7/8 <= 1/x < 1 + Peak relative error 1.4e-35 */ +#define NRNr8 9 +static const __float128 RNr8[NRNr8 + 1] = +{ + 3.587451489255356250759834295199296936784E1Q, + 5.406249749087340431871378009874875889602E2Q, + 2.931301290625250886238822286506381194157E3Q, + 7.359254185241795584113047248898753470923E3Q, + 9.201031849810636104112101947312492532314E3Q, + 5.749697096193191467751650366613289284777E3Q, + 1.710415234419860825710780802678697889231E3Q, + 2.150753982543378580859546706243022719599E2Q, + 8.740953582272147335100537849981160931197E0Q, + 4.876422978828717219629814794707963640913E-2Q +}; +#define NRDr8 8 +static const __float128 RDr8[NRDr8 + 1] = +{ + 6.358593134096908350929496535931630140282E1Q, + 9.900253816552450073757174323424051765523E2Q, + 5.642928777856801020545245437089490805186E3Q, + 1.524195375199570868195152698617273739609E4Q, + 2.113829644500006749947332935305800887345E4Q, + 1.526438562626465706267943737310282977138E4Q, + 5.561370922149241457131421914140039411782E3Q, + 9.394035530179705051609070428036834496942E2Q, + 6.147019596150394577984175188032707343615E1Q + /* 1.0E0 */ +}; + +/* erfc(1/x) = 1/x exp (-1/x^2 - 0.5625 + R(1/x^2)) + 0.75 <= 1/x <= 0.875 + Peak relative error 2.0e-36 */ +#define NRNr7 9 +static const __float128 RNr7[NRNr7 + 1] = +{ + 1.686222193385987690785945787708644476545E1Q, + 1.178224543567604215602418571310612066594E3Q, + 1.764550584290149466653899886088166091093E4Q, + 1.073758321890334822002849369898232811561E5Q, + 3.132840749205943137619839114451290324371E5Q, + 4.607864939974100224615527007793867585915E5Q, + 3.389781820105852303125270837910972384510E5Q, + 1.174042187110565202875011358512564753399E5Q, + 1.660013606011167144046604892622504338313E4Q, + 6.700393957480661937695573729183733234400E2Q +}; +#define NRDr7 9 +static const __float128 RDr7[NRDr7 + 1] = +{ +-1.709305024718358874701575813642933561169E3Q, +-3.280033887481333199580464617020514788369E4Q, +-2.345284228022521885093072363418750835214E5Q, +-8.086758123097763971926711729242327554917E5Q, +-1.456900414510108718402423999575992450138E6Q, +-1.391654264881255068392389037292702041855E6Q, +-6.842360801869939983674527468509852583855E5Q, +-1.597430214446573566179675395199807533371E5Q, +-1.488876130609876681421645314851760773480E4Q, +-3.511762950935060301403599443436465645703E2Q + /* 1.0E0 */ +}; + +/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2)) + 5/8 <= 1/x < 3/4 + Peak relative error 1.9e-35 */ +#define NRNr6 9 +static const __float128 RNr6[NRNr6 + 1] = +{ + 1.642076876176834390623842732352935761108E0Q, + 1.207150003611117689000664385596211076662E2Q, + 2.119260779316389904742873816462800103939E3Q, + 1.562942227734663441801452930916044224174E4Q, + 5.656779189549710079988084081145693580479E4Q, + 1.052166241021481691922831746350942786299E5Q, + 9.949798524786000595621602790068349165758E4Q, + 4.491790734080265043407035220188849562856E4Q, + 8.377074098301530326270432059434791287601E3Q, + 4.506934806567986810091824791963991057083E2Q +}; +#define NRDr6 9 +static const __float128 RDr6[NRDr6 + 1] = +{ +-1.664557643928263091879301304019826629067E2Q, +-3.800035902507656624590531122291160668452E3Q, +-3.277028191591734928360050685359277076056E4Q, +-1.381359471502885446400589109566587443987E5Q, +-3.082204287382581873532528989283748656546E5Q, +-3.691071488256738343008271448234631037095E5Q, +-2.300482443038349815750714219117566715043E5Q, +-6.873955300927636236692803579555752171530E4Q, +-8.262158817978334142081581542749986845399E3Q, +-2.517122254384430859629423488157361983661E2Q + /* 1.00 */ +}; + +/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2)) + 1/2 <= 1/x < 5/8 + Peak relative error 4.6e-36 */ +#define NRNr5 10 +static const __float128 RNr5[NRNr5 + 1] = +{ +-3.332258927455285458355550878136506961608E-3Q, +-2.697100758900280402659586595884478660721E-1Q, +-6.083328551139621521416618424949137195536E0Q, +-6.119863528983308012970821226810162441263E1Q, +-3.176535282475593173248810678636522589861E2Q, +-8.933395175080560925809992467187963260693E2Q, +-1.360019508488475978060917477620199499560E3Q, +-1.075075579828188621541398761300910213280E3Q, +-4.017346561586014822824459436695197089916E2Q, +-5.857581368145266249509589726077645791341E1Q, +-2.077715925587834606379119585995758954399E0Q +}; +#define NRDr5 9 +static const __float128 RDr5[NRDr5 + 1] = +{ + 3.377879570417399341550710467744693125385E-1Q, + 1.021963322742390735430008860602594456187E1Q, + 1.200847646592942095192766255154827011939E2Q, + 7.118915528142927104078182863387116942836E2Q, + 2.318159380062066469386544552429625026238E3Q, + 4.238729853534009221025582008928765281620E3Q, + 4.279114907284825886266493994833515580782E3Q, + 2.257277186663261531053293222591851737504E3Q, + 5.570475501285054293371908382916063822957E2Q, + 5.142189243856288981145786492585432443560E1Q + /* 1.0E0 */ +}; + +/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2)) + 3/8 <= 1/x < 1/2 + Peak relative error 2.0e-36 */ +#define NRNr4 10 +static const __float128 RNr4[NRNr4 + 1] = +{ + 3.258530712024527835089319075288494524465E-3Q, + 2.987056016877277929720231688689431056567E-1Q, + 8.738729089340199750734409156830371528862E0Q, + 1.207211160148647782396337792426311125923E2Q, + 8.997558632489032902250523945248208224445E2Q, + 3.798025197699757225978410230530640879762E3Q, + 9.113203668683080975637043118209210146846E3Q, + 1.203285891339933238608683715194034900149E4Q, + 8.100647057919140328536743641735339740855E3Q, + 2.383888249907144945837976899822927411769E3Q, + 2.127493573166454249221983582495245662319E2Q +}; +#define NRDr4 10 +static const __float128 RDr4[NRDr4 + 1] = +{ +-3.303141981514540274165450687270180479586E-1Q, +-1.353768629363605300707949368917687066724E1Q, +-2.206127630303621521950193783894598987033E2Q, +-1.861800338758066696514480386180875607204E3Q, +-8.889048775872605708249140016201753255599E3Q, +-2.465888106627948210478692168261494857089E4Q, +-3.934642211710774494879042116768390014289E4Q, +-3.455077258242252974937480623730228841003E4Q, +-1.524083977439690284820586063729912653196E4Q, +-2.810541887397984804237552337349093953857E3Q, +-1.343929553541159933824901621702567066156E2Q + /* 1.0E0 */ +}; + +/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2)) + 1/4 <= 1/x < 3/8 + Peak relative error 8.4e-37 */ +#define NRNr3 11 +static const __float128 RNr3[NRNr3 + 1] = +{ +-1.952401126551202208698629992497306292987E-6Q, +-2.130881743066372952515162564941682716125E-4Q, +-8.376493958090190943737529486107282224387E-3Q, +-1.650592646560987700661598877522831234791E-1Q, +-1.839290818933317338111364667708678163199E0Q, +-1.216278715570882422410442318517814388470E1Q, +-4.818759344462360427612133632533779091386E1Q, +-1.120994661297476876804405329172164436784E2Q, +-1.452850765662319264191141091859300126931E2Q, +-9.485207851128957108648038238656777241333E1Q, +-2.563663855025796641216191848818620020073E1Q, +-1.787995944187565676837847610706317833247E0Q +}; +#define NRDr3 10 +static const __float128 RDr3[NRDr3 + 1] = +{ + 1.979130686770349481460559711878399476903E-4Q, + 1.156941716128488266238105813374635099057E-2Q, + 2.752657634309886336431266395637285974292E-1Q, + 3.482245457248318787349778336603569327521E0Q, + 2.569347069372696358578399521203959253162E1Q, + 1.142279000180457419740314694631879921561E2Q, + 3.056503977190564294341422623108332700840E2Q, + 4.780844020923794821656358157128719184422E2Q, + 4.105972727212554277496256802312730410518E2Q, + 1.724072188063746970865027817017067646246E2Q, + 2.815939183464818198705278118326590370435E1Q + /* 1.0E0 */ +}; + +/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2)) + 1/8 <= 1/x < 1/4 + Peak relative error 1.5e-36 */ +#define NRNr2 11 +static const __float128 RNr2[NRNr2 + 1] = +{ +-2.638914383420287212401687401284326363787E-8Q, +-3.479198370260633977258201271399116766619E-6Q, +-1.783985295335697686382487087502222519983E-4Q, +-4.777876933122576014266349277217559356276E-3Q, +-7.450634738987325004070761301045014986520E-2Q, +-7.068318854874733315971973707247467326619E-1Q, +-4.113919921935944795764071670806867038732E0Q, +-1.440447573226906222417767283691888875082E1Q, +-2.883484031530718428417168042141288943905E1Q, +-2.990886974328476387277797361464279931446E1Q, +-1.325283914915104866248279787536128997331E1Q, +-1.572436106228070195510230310658206154374E0Q +}; +#define NRDr2 10 +static const __float128 RDr2[NRDr2 + 1] = +{ + 2.675042728136731923554119302571867799673E-6Q, + 2.170997868451812708585443282998329996268E-4Q, + 7.249969752687540289422684951196241427445E-3Q, + 1.302040375859768674620410563307838448508E-1Q, + 1.380202483082910888897654537144485285549E0Q, + 8.926594113174165352623847870299170069350E0Q, + 3.521089584782616472372909095331572607185E1Q, + 8.233547427533181375185259050330809105570E1Q, + 1.072971579885803033079469639073292840135E2Q, + 6.943803113337964469736022094105143158033E1Q, + 1.775695341031607738233608307835017282662E1Q + /* 1.0E0 */ +}; + +/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2)) + 1/128 <= 1/x < 1/8 + Peak relative error 2.2e-36 */ +#define NRNr1 9 +static const __float128 RNr1[NRNr1 + 1] = +{ +-4.250780883202361946697751475473042685782E-8Q, +-5.375777053288612282487696975623206383019E-6Q, +-2.573645949220896816208565944117382460452E-4Q, +-6.199032928113542080263152610799113086319E-3Q, +-8.262721198693404060380104048479916247786E-2Q, +-6.242615227257324746371284637695778043982E-1Q, +-2.609874739199595400225113299437099626386E0Q, +-5.581967563336676737146358534602770006970E0Q, +-5.124398923356022609707490956634280573882E0Q, +-1.290865243944292370661544030414667556649E0Q +}; +#define NRDr1 8 +static const __float128 RDr1[NRDr1 + 1] = +{ + 4.308976661749509034845251315983612976224E-6Q, + 3.265390126432780184125233455960049294580E-4Q, + 9.811328839187040701901866531796570418691E-3Q, + 1.511222515036021033410078631914783519649E-1Q, + 1.289264341917429958858379585970225092274E0Q, + 6.147640356182230769548007536914983522270E0Q, + 1.573966871337739784518246317003956180750E1Q, + 1.955534123435095067199574045529218238263E1Q, + 9.472613121363135472247929109615785855865E0Q + /* 1.0E0 */ +}; + + +__float128 +erfq (__float128 x) +{ + __float128 a, y, z; + int32_t i, ix, sign; + ieee854_float128 u; + + u.value = x; + sign = u.words32.w0; + ix = sign & 0x7fffffff; + + if (ix >= 0x7fff0000) + { /* erf(nan)=nan */ + i = ((sign & 0xffff0000) >> 31) << 1; + return (__float128) (1 - i) + one / x; /* erf(+-inf)=+-1 */ + } + + if (ix >= 0x3fff0000) /* |x| >= 1.0 */ + { + y = erfcq (x); + return (one - y); + /* return (one - erfcq (x)); */ + } + u.words32.w0 = ix; + a = u.value; + z = x * x; + if (ix < 0x3ffec000) /* a < 0.875 */ + { + if (ix < 0x3fc60000) /* |x|<2**-57 */ + { + if (ix < 0x00080000) + return 0.125 * (8.0 * x + efx8 * x); /*avoid underflow */ + return x + efx * x; + } + y = a + a * neval (z, TN1, NTN1) / deval (z, TD1, NTD1); + } + else + { + a = a - one; + y = erf_const + neval (a, TN2, NTN2) / deval (a, TD2, NTD2); + } + + if (sign & 0x80000000) /* x < 0 */ + y = -y; + return( y ); +} + + +__float128 +erfcq (__float128 x) +{ + __float128 y = 0.0Q, z, p, r; + int32_t i, ix, sign; + ieee854_float128 u; + + u.value = x; + sign = u.words32.w0; + ix = sign & 0x7fffffff; + u.words32.w0 = ix; + + if (ix >= 0x7fff0000) + { /* erfc(nan)=nan */ + /* erfc(+-inf)=0,2 */ + return (__float128) (((uint32_t) sign >> 31) << 1) + one / x; + } + + if (ix < 0x3ffd0000) /* |x| <1/4 */ + { + if (ix < 0x3f8d0000) /* |x|<2**-114 */ + return one - x; + return one - erfq (x); + } + if (ix < 0x3fff4000) /* 1.25 */ + { + x = u.value; + i = 8.0 * x; + switch (i) + { + case 2: + z = x - 0.25Q; + y = C13b + z * neval (z, RNr13, NRNr13) / deval (z, RDr13, NRDr13); + y += C13a; + break; + case 3: + z = x - 0.375Q; + y = C14b + z * neval (z, RNr14, NRNr14) / deval (z, RDr14, NRDr14); + y += C14a; + break; + case 4: + z = x - 0.5Q; + y = C15b + z * neval (z, RNr15, NRNr15) / deval (z, RDr15, NRDr15); + y += C15a; + break; + case 5: + z = x - 0.625Q; + y = C16b + z * neval (z, RNr16, NRNr16) / deval (z, RDr16, NRDr16); + y += C16a; + break; + case 6: + z = x - 0.75Q; + y = C17b + z * neval (z, RNr17, NRNr17) / deval (z, RDr17, NRDr17); + y += C17a; + break; + case 7: + z = x - 0.875Q; + y = C18b + z * neval (z, RNr18, NRNr18) / deval (z, RDr18, NRDr18); + y += C18a; + break; + case 8: + z = x - 1.0Q; + y = C19b + z * neval (z, RNr19, NRNr19) / deval (z, RDr19, NRDr19); + y += C19a; + break; + case 9: + z = x - 1.125Q; + y = C20b + z * neval (z, RNr20, NRNr20) / deval (z, RDr20, NRDr20); + y += C20a; + break; + } + if (sign & 0x80000000) + y = 2.0Q - y; + return y; + } + /* 1.25 < |x| < 107 */ + if (ix < 0x4005ac00) + { + /* x < -9 */ + if ((ix >= 0x40022000) && (sign & 0x80000000)) + return two - tiny; + + x = fabsq (x); + z = one / (x * x); + i = 8.0 / x; + switch (i) + { + default: + case 0: + p = neval (z, RNr1, NRNr1) / deval (z, RDr1, NRDr1); + break; + case 1: + p = neval (z, RNr2, NRNr2) / deval (z, RDr2, NRDr2); + break; + case 2: + p = neval (z, RNr3, NRNr3) / deval (z, RDr3, NRDr3); + break; + case 3: + p = neval (z, RNr4, NRNr4) / deval (z, RDr4, NRDr4); + break; + case 4: + p = neval (z, RNr5, NRNr5) / deval (z, RDr5, NRDr5); + break; + case 5: + p = neval (z, RNr6, NRNr6) / deval (z, RDr6, NRDr6); + break; + case 6: + p = neval (z, RNr7, NRNr7) / deval (z, RDr7, NRDr7); + break; + case 7: + p = neval (z, RNr8, NRNr8) / deval (z, RDr8, NRDr8); + break; + } + u.value = x; + u.words32.w3 = 0; + u.words32.w2 &= 0xfe000000; + z = u.value; + r = expq (-z * z - 0.5625) * expq ((z - x) * (z + x) + p); + if ((sign & 0x80000000) == 0) + return r / x; + else + return two - r / x; + } + else + { + if ((sign & 0x80000000) == 0) + return tiny * tiny; + else + return two - tiny; + } +} diff --git a/libquadmath/math/expm1q.c b/libquadmath/math/expm1q.c new file mode 100644 index 000000000..510c98fe4 --- /dev/null +++ b/libquadmath/math/expm1q.c @@ -0,0 +1,158 @@ +/* expm1l.c + * + * Exponential function, minus 1 + * 128-bit __float128 precision + * + * + * + * SYNOPSIS: + * + * __float128 x, y, expm1l(); + * + * y = expm1l( x ); + * + * + * + * DESCRIPTION: + * + * Returns e (2.71828...) raised to the x power, minus one. + * + * Range reduction is accomplished by separating the argument + * into an integer k and fraction f such that + * + * x k f + * e = 2 e. + * + * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1 + * in the basic range [-0.5 ln 2, 0.5 ln 2]. + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -79,+MAXLOG 100,000 1.7e-34 4.5e-35 + * + */ + +/* Copyright 2001 by Stephen L. Moshier + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + + + +#include "quadmath-imp.h" + +/* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x) + -.5 ln 2 < x < .5 ln 2 + Theoretical peak relative error = 8.1e-36 */ + +static const __float128 + P0 = 2.943520915569954073888921213330863757240E8Q, + P1 = -5.722847283900608941516165725053359168840E7Q, + P2 = 8.944630806357575461578107295909719817253E6Q, + P3 = -7.212432713558031519943281748462837065308E5Q, + P4 = 4.578962475841642634225390068461943438441E4Q, + P5 = -1.716772506388927649032068540558788106762E3Q, + P6 = 4.401308817383362136048032038528753151144E1Q, + P7 = -4.888737542888633647784737721812546636240E-1Q, + Q0 = 1.766112549341972444333352727998584753865E9Q, + Q1 = -7.848989743695296475743081255027098295771E8Q, + Q2 = 1.615869009634292424463780387327037251069E8Q, + Q3 = -2.019684072836541751428967854947019415698E7Q, + Q4 = 1.682912729190313538934190635536631941751E6Q, + Q5 = -9.615511549171441430850103489315371768998E4Q, + Q6 = 3.697714952261803935521187272204485251835E3Q, + Q7 = -8.802340681794263968892934703309274564037E1Q, + /* Q8 = 1.000000000000000000000000000000000000000E0 */ +/* C1 + C2 = ln 2 */ + + C1 = 6.93145751953125E-1Q, + C2 = 1.428606820309417232121458176568075500134E-6Q, +/* ln (2^16384 * (1 - 2^-113)) */ + maxlog = 1.1356523406294143949491931077970764891253E4Q, +/* ln 2^-114 */ + minarg = -7.9018778583833765273564461846232128760607E1Q; + + +__float128 +expm1q (__float128 x) +{ + __float128 px, qx, xx; + int32_t ix, sign; + ieee854_float128 u; + int k; + + /* Detect infinity and NaN. */ + u.value = x; + ix = u.words32.w0; + sign = ix & 0x80000000; + ix &= 0x7fffffff; + if (ix >= 0x7fff0000) + { + /* Infinity. */ + if (((ix & 0xffff) | u.words32.w1 | u.words32.w2 | u.words32.w3) == 0) + { + if (sign) + return -1.0Q; + else + return x; + } + /* NaN. No invalid exception. */ + return x; + } + + /* expm1(+- 0) = +- 0. */ + if ((ix == 0) && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0) + return x; + + /* Overflow. */ + if (x > maxlog) + return (HUGE_VALQ * HUGE_VALQ); + + /* Minimum value. */ + if (x < minarg) + return (4.0/HUGE_VALQ - 1.0Q); + + /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */ + xx = C1 + C2; /* ln 2. */ + px = floorq (0.5 + x / xx); + k = px; + /* remainder times ln 2 */ + x -= px * C1; + x -= px * C2; + + /* Approximate exp(remainder ln 2). */ + px = (((((((P7 * x + + P6) * x + + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x; + + qx = (((((((x + + Q7) * x + + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0; + + xx = x * x; + qx = x + (0.5 * xx + xx * px / qx); + + /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2). + + We have qx = exp(remainder ln 2) - 1, so + exp(x) - 1 = 2^k (qx + 1) - 1 + = 2^k qx + 2^k - 1. */ + + px = ldexpq (1.0Q, k); + x = px * qx + (px - 1.0); + return x; +} diff --git a/libquadmath/math/expq.c b/libquadmath/math/expq.c new file mode 100644 index 000000000..2740b4e2c --- /dev/null +++ b/libquadmath/math/expq.c @@ -0,0 +1,1214 @@ +/* Quad-precision floating point e^x. + Copyright (C) 1999 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Jakub Jelinek <jj@ultra.linux.cz> + Partly based on double-precision code + by Geoffrey Keating <geoffk@ozemail.com.au> + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" +#ifdef HAVE_FENV_H +# include <fenv.h> +# if defined HAVE_FEHOLDEXCEPT && defined HAVE_FESETROUND \ + && defined HAVE_FESETENV && defined FE_TONEAREST +# define USE_FENV_H +# endif +#endif + + +/* __expl_table basically consists of four tables, T_EXPL_ARG{1,2} and + T_EXPL_RES{1,2}. All tables use positive and negative indexes, the 0 points + are marked by T_EXPL_* defines. + For ARG1 and RES1 tables lets B be 89 and S 256.0, for ARG2 and RES2 B is 65 + and S 32768.0. + These table have the property that, for all integers -B <= i <= B + expl(__expl_table[T_EXPL_ARGN+2*i]+__expl_table[T_EXPL_ARGN+2*i+1]+r) == + __expl_table[T_EXPL_RESN+i], __expl_table[T_EXPL_RESN+i] is some exact number + with the low 58 bits of the mantissa 0, + __expl_table[T_EXPL_ARGN+2*i] == i/S+s + where absl(s) <= 2^-54 and absl(r) <= 2^-212. */ + +static const __float128 __expl_table [] = { + -3.47656250000000000584188889839535373E-01Q, /* bffd640000000000002b1b04213cf000 */ + 6.90417668990715641167244540876988960E-32Q, /* 3f97667c3fdb588a6ae1af8748357a17 */ + -3.43749999999999981853132895957607418E-01Q, /* bffd5ffffffffffffac4ff5f4050b000 */ + -7.16021898043268093462818380603370350E-33Q, /* bf94296c8219427edc1431ac2498583e */ + -3.39843750000000013418643523138766329E-01Q, /* bffd5c000000000003de1f027a30e000 */ + 8.16920774283317801641347327589583265E-32Q, /* 3f97a82b65774bdca1b4440d749ed8d3 */ + -3.35937500000000014998092453039303051E-01Q, /* bffd5800000000000452a9f4d8857000 */ + -6.55865578425428447938248396879359670E-32Q, /* bf97548b7d240f3d034b395e6eecfac8 */ + -3.32031250000000000981984049529998541E-01Q, /* bffd540000000000004875277cda5000 */ + 6.91213046334032232108944519541512737E-32Q, /* 3f9766e5f925338a19045c94443b66e1 */ + -3.28124999999999986646017645350399708E-01Q, /* bffd4ffffffffffffc26a667bf44d000 */ + -6.16281060996110316602421505683742661E-32Q, /* bf973ffdcdcffb6fbffc86b2b8d42f5d */ + -3.24218749999999991645717430645867963E-01Q, /* bffd4bfffffffffffd97901063e48000 */ + -7.90797211087760527593856542417304137E-32Q, /* bf979a9afaaca1ada6a8ed1c80584d60 */ + -3.20312499999999998918211610690789652E-01Q, /* bffd47ffffffffffffb02d9856d71000 */ + 8.64024799457616856987630373786503376E-32Q, /* 3f97c0a098623f95579d5d9b2b67342d */ + -3.16406249999999998153974811017181883E-01Q, /* bffd43ffffffffffff77c991f1076000 */ + -2.73176610180696076418536105483668404E-32Q, /* bf961baeccb32f9b1fcbb8e60468e95a */ + -3.12500000000000011420976192575972779E-01Q, /* bffd400000000000034ab8240483d000 */ + 7.16573502812389453744433792609989420E-32Q, /* 3f977410f4c2cfc4335f28446c0fb363 */ + -3.08593750000000001735496343854851414E-01Q, /* bffd3c000000000000800e995c176000 */ + -1.56292999645122272621237565671593071E-32Q, /* bf95449b9cbdaff6ac1246adb2c826ac */ + -3.04687499999999982592401295899221626E-01Q, /* bffd37fffffffffffafb8bc1e061a000 */ + 6.48993208584888904958594509625158417E-32Q, /* 3f9750f9fe8366d82d77afa0031a92e1 */ + -3.00781249999999999230616898937763959E-01Q, /* bffd33ffffffffffffc73ac39da54000 */ + 6.57082437496961397305801409357792029E-32Q, /* 3f97552d3cb598ea80135cf3feb27ec4 */ + -2.96874999999999998788769281703245722E-01Q, /* bffd2fffffffffffffa6a07fa5021000 */ + -3.26588297198283968096426564544269170E-32Q, /* bf9653260fc1802f46b629aee171809b */ + -2.92968750000000015318089182805941695E-01Q, /* bffd2c0000000000046a468614bd6000 */ + -1.73291974845198589684358727559290718E-32Q, /* bf9567e9d158f52e483c8d8dcb5961dd */ + -2.89062500000000007736778942676309681E-01Q, /* bffd280000000000023adf9f4c3d3000 */ + -6.83629745986675744404029225571026236E-32Q, /* bf9762f5face6281c1daf1c6aedbdb45 */ + -2.85156250000000001367091555763661937E-01Q, /* bffd2400000000000064dfa11e3fb000 */ + -5.44898442619766878281110054067026237E-32Q, /* bf971aed6d2db9f542986a785edae072 */ + -2.81249999999999986958718100227029406E-01Q, /* bffd1ffffffffffffc3db9265ca9d000 */ + 1.13007318374506125723591889451107046E-32Q, /* 3f94d569fe387f456a97902907ac3856 */ + -2.77343750000000000356078829380495179E-01Q, /* bffd1c0000000000001a462390083000 */ + -4.98979365468978332358409063436543102E-32Q, /* bf970315bbf3e0d14b5c94c900702d4c */ + -2.73437499999999990276993957508540484E-01Q, /* bffd17fffffffffffd32919bcdc94000 */ + -8.79390484115892344533724650295100871E-32Q, /* bf97c89b0b89cc19c3ab2b60da9bbbc3 */ + -2.69531250000000002434203866460082225E-01Q, /* bffd14000000000000b39ccf9e130000 */ + 9.44060754687026590886751809927191596E-32Q, /* 3f97ea2f32cfecca5c64a26137a9210f */ + -2.65624999999999997296320716986257179E-01Q, /* bffd0fffffffffffff3880f13a2bc000 */ + 2.07142664067265697791007875348396921E-32Q, /* 3f95ae37ee685b9122fbe377bd205ee4 */ + -2.61718750000000010237478733739017956E-01Q, /* bffd0c000000000002f3648179d40000 */ + -6.10552936159265665298996309192680256E-32Q, /* bf973d0467d31e407515a3cca0f3b4e2 */ + -2.57812500000000011948220522778370303E-01Q, /* bffd08000000000003719f81275bd000 */ + 6.72477169058908902499239631466443836E-32Q, /* 3f975d2b8c475d3160cf72d227d8e6f9 */ + -2.53906249999999991822993360536596860E-01Q, /* bffd03fffffffffffda4a4b62f818000 */ + -2.44868296623215865054704392917190994E-32Q, /* bf95fc92516c6d057d29fc2528855976 */ + -2.49999999999999986862019457428548084E-01Q, /* bffcfffffffffffff86d2d20d5ff4000 */ + -3.85302898949105073614122724961613078E-32Q, /* bf96901f147cb7d643af71b6129ce929 */ + -2.46093750000000000237554160737318435E-01Q, /* bffcf8000000000000230e8ade26b000 */ + -1.52823675242678363494345369284988589E-32Q, /* bf953d6700c5f3fc303f79d0ec8c680a */ + -2.42187500000000003023380963205457065E-01Q, /* bffcf0000000000001be2c1a78bb0000 */ + -7.78402037952209709489481182714311699E-34Q, /* bf9102ab1f3998e887f0ee4cf940faa5 */ + -2.38281249999999995309623303145485725E-01Q, /* bffce7fffffffffffd4bd2940f43f000 */ + -3.54307216794236899443913216397197696E-32Q, /* bf966fef03ab69c3f289436205b21d02 */ + -2.34374999999999998425804947623207526E-01Q, /* bffcdfffffffffffff17b097a6092000 */ + -2.86038428948386602859761879407549696E-32Q, /* bf96290a0eba0131efe3a05fe188f2e3 */ + -2.30468749999999993822207406785200832E-01Q, /* bffcd7fffffffffffc70519834eae000 */ + -2.54339521031747516806893838749365762E-32Q, /* bf96081f0ad7f9107ae6cddb32c178ab */ + -2.26562499999999997823524030344489884E-01Q, /* bffccffffffffffffebecf10093df000 */ + 4.31904611473158635644635628922959401E-32Q, /* 3f96c083f0b1faa7c4c686193e38d67c */ + -2.22656250000000004835132405125162742E-01Q, /* bffcc8000000000002c98a233f19f000 */ + 2.54709791629335691650310168420597566E-33Q, /* 3f92a735903f5eed07a716ab931e20d9 */ + -2.18749999999999988969454021829236626E-01Q, /* bffcbffffffffffff9a42dc14ce36000 */ + -3.77236096429336082213752014054909454E-32Q, /* bf9687be8e5b2fca54d3e81157eac660 */ + -2.14843750000000010613256919115758495E-01Q, /* bffcb80000000000061e3d828ecac000 */ + -4.55194148712216691177097854305964738E-32Q, /* bf96d8b35c776aa3e1a4768271380503 */ + -2.10937499999999993204656148110447201E-01Q, /* bffcaffffffffffffc152f2aea118000 */ + -2.95044199165561453749332254271716417E-32Q, /* bf96326433b00b2439094d9bef22ddd1 */ + -2.07031250000000012233944895423355677E-01Q, /* bffca80000000000070d695ee0e94000 */ + 1.93146788688385419095981415411012357E-32Q, /* 3f959126729135a5e390d4bb802a0bde */ + -2.03125000000000008030983633336321863E-01Q, /* bffca0000000000004a129fbc51af000 */ + 2.37361904671826193563212931215900137E-32Q, /* 3f95ecfb3c4ba1b97ea3ad45cbb1e68a */ + -1.99218750000000001763815712796132779E-01Q, /* bffc98000000000001044b12d9950000 */ + -3.63171243370923753295192486732883239E-33Q, /* bf932db5fb3f27c38e0fa7bbcfc64f55 */ + -1.95312500000000004883660234506677272E-01Q, /* bffc90000000000002d0b3779d1f9000 */ + -3.19989507343607877747980892249711601E-33Q, /* bf9309d63de96bb3ef744c865f22f1bd */ + -1.91406250000000013720152363227519348E-01Q, /* bffc88000000000007e8bcb387121000 */ + -1.89295754093147174148371614722178860E-32Q, /* bf958926e2e67dfe812c508290add2e7 */ + -1.87500000000000000182342082774432620E-01Q, /* bffc800000000000001ae8b06a39f000 */ + -2.96812835183184815200854214892983927E-32Q, /* bf96343a62d156bbe71f55d14ca4b6e5 */ + -1.83593750000000012410147185883290345E-01Q, /* bffc78000000000007276a1adda8d000 */ + -2.02191931237489669058466239995304587E-32Q, /* bf95a3efab92d26ec2df90df036a117f */ + -1.79687499999999997439177363346082917E-01Q, /* bffc6ffffffffffffe8616db2927d000 */ + -9.92752326937775530007399526834009465E-33Q, /* bf949c5f88ed17041e1a3f1829d543cd */ + -1.75781249999999995824373974504785174E-01Q, /* bffc67fffffffffffd97c94f13ea3000 */ + 1.44184772065335613487885714828816178E-32Q, /* 3f952b75c63476e7fcc2f5841c27bcce */ + -1.71874999999999986685050259043077809E-01Q, /* bffc5ffffffffffff8530f6bc531a000 */ + -3.49007014971241147689894940544402482E-32Q, /* bf966a6dfaa012aea8ffe6d90b02330f */ + -1.67968749999999997316058782350439701E-01Q, /* bffc57fffffffffffe73eb914f2aa000 */ + 3.34025733574205019081305778794376391E-32Q, /* 3f965adf4572561fd5456a6c13d8babf */ + -1.64062499999999993322730602128318480E-01Q, /* bffc4ffffffffffffc269be4f68f3000 */ + -1.83345916769684984022099095506340635E-32Q, /* bf957ccb69026cb2f6024c211576d5f4 */ + -1.60156249999999992419000744447607979E-01Q, /* bffc47fffffffffffba13df21784a000 */ + 2.73442789798110494773517431626534726E-32Q, /* 3f961bf58ff22c9b30f1e2b39f26d7d5 */ + -1.56249999999999987665010524130393080E-01Q, /* bffc3ffffffffffff8e3ad45e7508000 */ + 2.02695576464836145806428118889332191E-32Q, /* 3f95a4fb7435a4a2f71de81eb8ae75d1 */ + -1.52343749999999989905291167951491803E-01Q, /* bffc37fffffffffffa2e48aecfc24000 */ + -3.61436631548815190395331054871041524E-32Q, /* bf967756567ebd108075ae527cc2e7f0 */ + -1.48437500000000006686107754967759751E-01Q, /* bffc30000000000003dab20261b3c000 */ + -2.15524270159131591469319477922198390E-32Q, /* bf95bfa05b82ef3a708c4f0395e9fcf6 */ + -1.44531250000000005132889939177166485E-01Q, /* bffc28000000000002f57b1969e7b000 */ + 2.74741116529653547935086189244019604E-32Q, /* 3f961d4eb77c1185d34fe1b04a3f3cf5 */ + -1.40625000000000000707469094533647325E-01Q, /* bffc2000000000000068676d3d5c4000 */ + 4.40607097220049957013547629906723266E-33Q, /* 3f936e0ac425daf795b42913cf0ef881 */ + -1.36718749999999995713752139187543306E-01Q, /* bffc17fffffffffffd87762255991000 */ + -3.73751317180116492404578048203389108E-32Q, /* bf9684202491e9cbb7ceb67d9ff7e0c9 */ + -1.32812500000000007198453630478482191E-01Q, /* bffc10000000000004264de3a4379000 */ + -3.97050085179660203884930593717220728E-32Q, /* bf969c52048de14be3c9c1971e50869c */ + -1.28906250000000006070486371645733082E-01Q, /* bffc080000000000037fd87db2cb0000 */ + 3.59610068058504988294019521946586131E-32Q, /* 3f967570c10687cb8e9ebd0b280abf5a */ + -1.25000000000000003700729208608337966E-01Q, /* bffc00000000000002222198bbc74000 */ + 3.23464851393124362331846965931995969E-33Q, /* 3f930cb95da3bfc847e593716c91d57a */ + -1.21093750000000013729038501177102555E-01Q, /* bffbf000000000000fd418d1f5fda000 */ + 2.45242487730722066611358741283977619E-32Q, /* 3f95fd5945ad86a464292e26ac192a84 */ + -1.17187499999999999765305306880205578E-01Q, /* bffbdfffffffffffffbabaf869845000 */ + -1.14557520298960389903199646350205537E-32Q, /* bf94dbda735322179d9bcf392e1dd06d */ + -1.13281250000000009579647893740755690E-01Q, /* bffbd000000000000b0b69bae7ab9000 */ + 2.37873962873837390105423621772752350E-32Q, /* 3f95ee0b7e0bd5ac1f6fab1e2a71abc3 */ + -1.09375000000000008981153004560108539E-01Q, /* bffbc000000000000a5ac4bc1d2c3000 */ + 1.53152444860014076105003555837231015E-32Q, /* 3f953e15ce931e12ef9a152522e32bdd */ + -1.05468749999999992399063850363228723E-01Q, /* bffbaffffffffffff73c998091408000 */ + -8.75920903597804862471749360196688834E-33Q, /* bf946bd7e310a01bae5687ebdc47fcc5 */ + -1.01562500000000007685885179918350550E-01Q, /* bffba0000000000008dc7910a648c000 */ + -4.63820993797174451904075397785059501E-33Q, /* bf938153d0e54001a472da180fb5e8aa */ + -9.76562499999999887262211517861331814E-02Q, /* bffb8ffffffffffff300915aa6fd6000 */ + -2.63767025974952608658936466715705903E-33Q, /* bf92b64215bb8d520be5404620d38088 */ + -9.37499999999999939650246024457439795E-02Q, /* bffb7ffffffffffff90aca26bd0fc000 */ + -1.72047822349322956713582039121348377E-32Q, /* bf9565545015c5b9b56d02cfefca2c7d */ + -8.98437500000000033088896383977486369E-02Q, /* bffb70000000000003d09ca1e3cbe000 */ + 3.04831994420989436248526129869697270E-33Q, /* 3f92fa7d30d2ed90e7ebbd6231fd08b1 */ + -8.59374999999999947312400115121319225E-02Q, /* bffb5ffffffffffff9ecefc03376e000 */ + 1.50416954438393392150792422537312281E-32Q, /* 3f9538675ee99bd722fad0023c09c915 */ + -8.20312500000000054182280847004695514E-02Q, /* bffb500000000000063f2dbd40200000 */ + 2.68399664523430004488075638997207289E-33Q, /* 3f92bdf49766629882c49a3da88928ed */ + -7.81250000000000114767533968079748798E-02Q, /* bffb4000000000000d3b56f81ba70000 */ + 1.72318124201659121296305402819694281E-32Q, /* 3f9565e407aaabfb359e8a567d760de3 */ + -7.42187500000000035531829472486812869E-02Q, /* bffb3000000000000418b6e9b5388000 */ + 2.09401756478514117051383998628099655E-32Q, /* 3f95b2e91221fcd74be0a86d8ad658d2 */ + -7.03124999999999987474933134860732535E-02Q, /* bffb1ffffffffffffe8e53453d2ac000 */ + 2.28515798224350800271565551341211666E-32Q, /* 3f95da9bd6adf00894f05b5cc5530125 */ + -6.64062500000000042267533361089054159E-02Q, /* bffb10000000000004df8473dbcf2000 */ + 1.97576478800281368377376002585430031E-32Q, /* 3f959a59acbddb2f53bd3096b66370e9 */ + -6.25000000000000066329769382774201686E-02Q, /* bffb00000000000007a5b5914e336000 */ + -1.46422615813786836245343723048221678E-33Q, /* bf91e69295f069fc0c4a9db181ea25a3 */ + -5.85937500000000002823707957982406053E-02Q, /* bffae0000000000000a6aeab10592000 */ + 9.25637741701318872896718218457555829E-33Q, /* 3f94807eb021f1f40a37d4015b1eb76b */ + -5.46875000000000081586888005226044448E-02Q, /* bffac0000000000012d00a3171e3a000 */ + -4.87144542459404765480424673678105050E-33Q, /* bf9394b42faba6b7036fe7b36269daf3 */ + -5.07812499999999927720348253140567013E-02Q, /* bffa9fffffffffffef555cc8dd914000 */ + -3.01901021987395945826043649523451725E-33Q, /* bf92f59e7e3025691f290f8f67277faf */ + -4.68749999999999935349476738962633103E-02Q, /* bffa7ffffffffffff117b4ea2b876000 */ + 1.21521638219189777347767475937119750E-32Q, /* 3f94f8c7f88c5b56674b94d984ac8ecb */ + -4.29687500000000056305562847814228219E-02Q, /* bffa6000000000000cfbb19be30c0000 */ + -1.18643699217679276275559592978275214E-32Q, /* bf94ecd39f0833a876550e83eb012b99 */ + -3.90624999999999962692914526031373542E-02Q, /* bffa3ffffffffffff765c743922f9000 */ + -4.91277156857520035712509544689973679E-33Q, /* bf939823189996193872e58ac0dececb */ + -3.51562500000000108152468207687602886E-02Q, /* bffa20000000000018f031e41177f000 */ + 1.18599806302656253755207072755609820E-32Q, /* 3f94eca4f23e787fab73ce8f6b9b8d64 */ + -3.12500000000000077376981036742289578E-02Q, /* bffa00000000000011d787e0b386f000 */ + 9.97730386477005171963635210799577079E-33Q, /* 3f949e70e498c46a0173ac0d46c699fc */ + -2.73437500000000139436129596418623235E-02Q, /* bff9c00000000000404db66e70a08000 */ + 2.25755321633070123579875157841633859E-33Q, /* 3f927719b1a93074bdf9f3c2cb784785 */ + -2.34375000000000088003629211828324876E-02Q, /* bff98000000000002895a27d45feb000 */ + 2.84374279216848803102126617873942975E-33Q, /* 3f92d87f70e749d6da6c260b68dc210b */ + -1.95312500000000107408831063404855424E-02Q, /* bff9400000000000318898ba69f71000 */ + 2.47348089686935458989103979140011912E-33Q, /* 3f929afa3de45086fe909fdddb41edce */ + -1.56250000000000081443917555362290635E-02Q, /* bff9000000000000258f335e9cdd6000 */ + -2.43379314483517422161458863218426254E-33Q, /* bf9294621c8a9ccacf2b020ec19cad27 */ + -1.17187500000000051490597418161403184E-02Q, /* bff88000000000002f7ddfa26221f000 */ + 1.83405297208145390679150568810924707E-33Q, /* 3f9230bbfc5d5fe1b534fbcda0465bb9 */ + -7.81249999999999715861805208310174953E-03Q, /* bff7ffffffffffffcb95f3fff157d000 */ + 3.51548384878710915171654413641872451E-34Q, /* 3f8fd349b76c22966f77a39fc37ed704 */ + -3.90625000000000309326013918295097128E-03Q, /* bff7000000000000390f820c8e153000 */ + 6.38058004651791109324060099097251911E-36Q, /* 3f8a0f665d3ac25a1ac94d688273dbcd */ +#define T_EXPL_ARG1 (2*89) + 0.00000000000000000000000000000000000E+00Q, /* 00000000000000000000000000000000 */ + 0.00000000000000000000000000000000000E+00Q, /* 00000000000000000000000000000000 */ + 3.90625000000000245479958859972588985E-03Q, /* 3ff70000000000002d48769ac9874000 */ + -6.58439598384342854976169982902779828E-36Q, /* bf8a1811b923e6c626b07ef29761482a */ + 7.81250000000001311374391093664996358E-03Q, /* 3ff800000000000078f3f3cd89111000 */ + 2.60265650555493781464273319671555602E-33Q, /* 3f92b070c3b635b87af426735a71fc87 */ + 1.17187500000000269581156218247101912E-02Q, /* 3ff8800000000000f8a50d02fe20d000 */ + 1.00961747974945520631836275894919326E-33Q, /* 3f914f80c1a4f8042044fe3b757b030b */ + 1.56249999999999797878275270751825475E-02Q, /* 3ff8ffffffffffff45935b69da62e000 */ + 2.03174577741375590087897353146748580E-33Q, /* 3f925194e863496e0f6e91cbf6b22e26 */ + 1.95312499999999760319884511789111533E-02Q, /* 3ff93fffffffffff917790ff9a8f4000 */ + 4.62788519658803722282100289809515007E-33Q, /* 3f9380783ba81295feeb3e4879d7d52d */ + 2.34374999999999822953909016349145918E-02Q, /* 3ff97fffffffffffae5a163bd3cd5000 */ + -3.19499956304699705390404384504876533E-33Q, /* bf93096e2037ced8194cf344c692f8d6 */ + 2.73437500000000137220327275871555682E-02Q, /* 3ff9c000000000003f481dea5dd51000 */ + -2.25757776523031994464630107442723424E-33Q, /* bf92771abcf988a02b414bf2614e3734 */ + 3.12499999999999790857640618332718621E-02Q, /* 3ff9ffffffffffff9f8cd40b51509000 */ + -4.22479470489989916319395454536511458E-33Q, /* bf935efb7245612f371deca17cb7b30c */ + 3.51562499999999840753382405747597346E-02Q, /* 3ffa1fffffffffffdb47bd275f722000 */ + 1.08459658374118041980976756063083500E-34Q, /* 3f8e2055d18b7117c9db1c318b1e889b */ + 3.90624999999999989384433621470426757E-02Q, /* 3ffa3ffffffffffffd8d5e18b042e000 */ + -7.41674226146122000759491297811091830E-33Q, /* bf94341454e48029e5b0205d91baffdc */ + 4.29687500000000107505739500500200462E-02Q, /* 3ffa60000000000018ca04cd9085c000 */ + -4.74689012756713017494437969420919847E-34Q, /* bf903b7c268103c6f7fbaaa24142e287 */ + 4.68749999999999978700749928325717352E-02Q, /* 3ffa7ffffffffffffb16b6d5479e3000 */ + -1.06208165308448830117773486334902917E-32Q, /* bf94b92be4b3b5b5a596a0a5187cc955 */ + 5.07812499999999815072625435955786253E-02Q, /* 3ffa9fffffffffffd55bd086d5cbc000 */ + -9.37038897148383660401929567549111394E-33Q, /* bf94853b111b0175b491c80d00419416 */ + 5.46874999999999809511553152189867394E-02Q, /* 3ffabfffffffffffd4138bfa74a61000 */ + 1.06642963074562437340498606682822123E-32Q, /* 3f94bafa3fe991b39255d563dfa05d89 */ + 5.85937500000000184331996330905145551E-02Q, /* 3ffae000000000002a810a5f2f8bf000 */ + -1.76639977694797200820296641773791945E-34Q, /* bf8ed596f07ce4408f1705c8ec16864c */ + 6.25000000000000021544696744852045001E-02Q, /* 3ffb000000000000027be32045e2b000 */ + 1.68616371995798354366633034788947149E-32Q, /* 3f955e33d7440794d8a1b25233d086ab */ + 6.64062499999999965563110718495802889E-02Q, /* 3ffb0ffffffffffffc079a38a3fed000 */ + -1.82463217667830160048872113565316215E-32Q, /* bf957af6163bcdb97cefab44a942482a */ + 7.03124999999999759989183341261898222E-02Q, /* 3ffb1fffffffffffe454218acea05000 */ + -1.07843770101525495515646940862541503E-32Q, /* bf94bff72aada26d94e76e71c07e0580 */ + 7.42187499999999898968873730710101412E-02Q, /* 3ffb2ffffffffffff45a166496dc1000 */ + 1.28629441689592874462780757154138223E-32Q, /* 3f950b2724597b8b93ce1e9d1cf4d035 */ + 7.81249999999999957198938523510804668E-02Q, /* 3ffb3ffffffffffffb10bc52adbc5000 */ + 1.13297573459968118467100063135856856E-33Q, /* 3f91787eea895b3c245899cf34ad0abd */ + 8.20312500000000199911640621145851159E-02Q, /* 3ffb500000000000170c59a661a89000 */ + -1.51161335208135146756554123073528707E-32Q, /* bf9539f326c5ca84e7db5401566f3775 */ + 8.59375000000000134175373433347670743E-02Q, /* 3ffb6000000000000f78287547af0000 */ + 1.09763629458404270323909815379924900E-32Q, /* 3f94c7f0b61b6e3e27d44b9f5bbc7e9d */ + 8.98437500000000036533922600308306335E-02Q, /* 3ffb70000000000004364a83b7a14000 */ + 3.11459653680110433194288029777718358E-33Q, /* 3f9302c0248136d65cebeab69488d949 */ + 9.37500000000000184977946245216914691E-02Q, /* 3ffb800000000000155395d870b17000 */ + -4.66656154468277949130395786965043927E-33Q, /* bf9383aec9b993b6db492b1ede786d8a */ + 9.76562500000000237839723100419376084E-02Q, /* 3ffb9000000000001b6bca237f6c4000 */ + -1.03028043424658760249140747856831301E-32Q, /* bf94abf6352e3d2bb398e47919a343fb */ + 1.01562500000000012345545575236836572E-01Q, /* 3ffba000000000000e3bc30cd9a1f000 */ + 2.15755372310795701322789783729456319E-32Q, /* 3f95c01b3b819edd9d07548fafd61550 */ + 1.05468749999999976493840484471911438E-01Q, /* 3ffbafffffffffffe4e634cd77985000 */ + 1.78771847038773333029677216592309083E-32Q, /* 3f95734b6ae650f33dd43c49a1df9fc0 */ + 1.09375000000000002267015055992785402E-01Q, /* 3ffbc00000000000029d1ad08de7b000 */ + 6.23263106693943817730045115112427717E-33Q, /* 3f9402e4b39ce2198a45e1d045868cd6 */ + 1.13281250000000022354208618429577398E-01Q, /* 3ffbd0000000000019c5cc3f9d2b5000 */ + 5.40514416644786448581426756221178868E-33Q, /* 3f93c10ab4021472c662f69435de9269 */ + 1.17187500000000013252367133076817603E-01Q, /* 3ffbe000000000000f47688cc561b000 */ + -7.12412585457324989451327215568641325E-33Q, /* bf9427ecb343a8d1758990565fcfbf45 */ + 1.21093750000000020759863992944300792E-01Q, /* 3ffbf0000000000017ef3af97bf04000 */ + 6.26591408357572503875647872077266444E-33Q, /* 3f940446a09a2da771b45fc075514d12 */ + 1.25000000000000004739659392396765618E-01Q, /* 3ffc00000000000002bb7344ecd89000 */ + -1.55611398459729463981000080101758830E-32Q, /* bf95433135febefa9e6aa4db39e263d2 */ + 1.28906249999999982360888081057894783E-01Q, /* 3ffc07fffffffffff5d4ed3154361000 */ + -1.77531518652835570781208599686606474E-32Q, /* bf9570b7f225ea076f97f418d11359c1 */ + 1.32812500000000010568583998727400436E-01Q, /* 3ffc1000000000000617a5d09526a000 */ + 2.12104021624990594668286391598300893E-32Q, /* 3f95b885d767a1048d93055927a27adc */ + 1.36718749999999998434125157367005292E-01Q, /* 3ffc17ffffffffffff18eaebc7970000 */ + 2.50454798592543203967309921276955297E-32Q, /* 3f9604164e5598528a76faff26cd1c97 */ + 1.40625000000000015550032422969330356E-01Q, /* 3ffc20000000000008f6c79d8928c000 */ + 7.80972982879849783680252962992639832E-33Q, /* 3f9444674acf2b3225c7647e0d95edf3 */ + 1.44531250000000012402535562111122522E-01Q, /* 3ffc28000000000007264a8bc1ff1000 */ + 2.79662468716455159585514763921671876E-32Q, /* 3f96226b095bd78aa650faf95a221993 */ + 1.48437500000000007761020440087419948E-01Q, /* 3ffc3000000000000479530ff8fe3000 */ + 2.15518492972728435680556239996258527E-32Q, /* 3f95bf9d49295e73a957906a029768cb */ + 1.52343750000000001733189947520484032E-01Q, /* 3ffc38000000000000ffc6109f71f000 */ + 8.34032236093545825619420380704500188E-33Q, /* 3f945a71851226a1d0ce5e656693153e */ + 1.56249999999999988073295321246958484E-01Q, /* 3ffc3ffffffffffff91fedd62ae0f000 */ + 2.44119337150624789345260194989620908E-32Q, /* 3f95fb041a57bc1c1280680ac1620bea */ + 1.60156250000000002076894210913572460E-01Q, /* 3ffc48000000000001327ed84a199000 */ + -7.36124501128859978061216696286151753E-33Q, /* bf9431c62f01e59d2c1e00f195a0037f */ + 1.64062500000000000950861276373482172E-01Q, /* 3ffc500000000000008c5285fba85000 */ + -4.80566184447001164583855800470217373E-33Q, /* bf938f3d1fcafd390f22f80e6c19421f */ + 1.67968749999999989878071706155265999E-01Q, /* 3ffc57fffffffffffa2a445c548c5000 */ + -4.42154428718618459799673088733365064E-32Q, /* bf96cb28cf1c1b28006d53ffe633b22a */ + 1.71874999999999999459734108403218175E-01Q, /* 3ffc5fffffffffffffb04554e9dd4000 */ + -3.29736288190321377985697972236270628E-32Q, /* bf96566af0ebc852e84be12859b24a31 */ + 1.75781249999999997987525759778901845E-01Q, /* 3ffc67fffffffffffed702df6ffff000 */ + -1.28800728638468399687523924685844352E-32Q, /* bf950b8236b88ca0c1b739dc91a7e3fc */ + 1.79687500000000004929565820437175783E-01Q, /* 3ffc70000000000002d779bb32d2e000 */ + 1.60624461317978482424582320675174225E-32Q, /* 3f954d9a9cc0c963fd081f3dc922d04e */ + 1.83593750000000016873727045739708856E-01Q, /* 3ffc78000000000009ba1f6263c9a000 */ + -3.83390389582056606880506003118452558E-32Q, /* bf968e22a5d826f77f19ee788474df22 */ + 1.87500000000000013443068740761666872E-01Q, /* 3ffc80000000000007bfd8c72a1bf000 */ + -2.74141662712926256150154726565203091E-32Q, /* bf961caf5ac59c7f941f928e324c2cc1 */ + 1.91406249999999981494101786848611970E-01Q, /* 3ffc87fffffffffff55502eeae001000 */ + 3.68992437075565165346469517256118001E-32Q, /* 3f967f2f03f9096793372a27b92ad79d */ + 1.95312499999999989069921848800501648E-01Q, /* 3ffc8ffffffffffff9b3015280394000 */ + 3.69712249337856518452988332367785220E-32Q, /* 3f967fee5fdb5bd501ff93516999faa0 */ + 1.99218750000000021148042946919300804E-01Q, /* 3ffc9800000000000c30e67939095000 */ + 2.50142536781142175091322844848566649E-32Q, /* 3f9603c34ae58e10b300b07137ee618a */ + 2.03124999999999977732559198825437141E-01Q, /* 3ffc9ffffffffffff329e7df079e4000 */ + -2.41951877287895024779300892731537816E-32Q, /* bf95f683aefe6965f080df8f59dd34a1 */ + 2.07031249999999996744030653771913124E-01Q, /* 3ffca7fffffffffffe1f80f4b73ca000 */ + -1.94346475904454000031592792989765585E-32Q, /* bf9593a44f87870a3d100d498501ecc7 */ + 2.10937500000000000251399259834392298E-01Q, /* 3ffcb000000000000025199873310000 */ + -1.33528748788094249098998693871759411E-33Q, /* bf91bbb9b25c813668d6103d08acac35 */ + 2.14843749999999993936323609611875097E-01Q, /* 3ffcb7fffffffffffc8128c866236000 */ + 1.14839877977014974625242788556545292E-32Q, /* 3f94dd06b4655c9b83a1305b240e7a42 */ + 2.18750000000000015181732784749663837E-01Q, /* 3ffcc0000000000008c06da5fff24000 */ + 1.42689085313142539755499441881408391E-32Q, /* 3f95285a87dfa7ea7dad5b3be8c669f4 */ + 2.22656249999999992172647770539596569E-01Q, /* 3ffcc7fffffffffffb7ce2fe531f6000 */ + -3.34421462850496887359128610229650547E-32Q, /* bf965b487962b5c2d9056ca6ac0c2e5c */ + 2.26562499999999989595607223847082419E-01Q, /* 3ffccffffffffffffa0095277be5c000 */ + -3.08983588107248752517344356508205569E-32Q, /* bf9640dded57157f8eded311213bdbcd */ + 2.30468749999999979130462438434567117E-01Q, /* 3ffcd7fffffffffff3f8332996560000 */ + -3.01407539802851697849105682795217019E-32Q, /* bf9638ffde35dbdfe1a1ffe45185de5d */ + 2.34375000000000012194252337217891971E-01Q, /* 3ffce0000000000007078dd402c86000 */ + -8.46879710915628592284714319904522657E-33Q, /* bf945fc7b29a2ac6c9eff9eb258a510f */ + 2.38281249999999982991877076137149870E-01Q, /* 3ffce7fffffffffff6320b486eece000 */ + -2.93563878880439245627127095245798544E-32Q, /* bf9630daaa4f40ff05caf29ace2ea7d4 */ + 2.42187499999999981447559841442773990E-01Q, /* 3ffceffffffffffff54e24a09a8d5000 */ + -4.56766746558806021264215486909850481E-32Q, /* bf96da556dee11f3113e5a3467b908e6 */ + 2.46093749999999991067720539980207318E-01Q, /* 3ffcf7fffffffffffad9d405dcb5d000 */ + 2.14033004219908074003010247652128251E-32Q, /* 3f95bc8776e8f9ae098884aa664cc3df */ + 2.50000000000000016613825838126835953E-01Q, /* 3ffd00000000000004c9e24c12bb3000 */ + 2.57617532593749185996714235009382870E-32Q, /* 3f960b867cc01178c0ec68226c6cb47d */ + 2.53906250000000013372004437827044321E-01Q, /* 3ffd04000000000003daae05b3168000 */ + 7.20177123439204414298152646284640101E-32Q, /* 3f9775eff59ddad7e7530b83934af87f */ + 2.57812499999999995765234725413886085E-01Q, /* 3ffd07fffffffffffec7878bad9d5000 */ + 6.51253187532920882777046064603770602E-32Q, /* 3f975226659ca241402e71c2011583b0 */ + 2.61718750000000007647689994011222248E-01Q, /* 3ffd0c000000000002344cc793a0f000 */ + 3.02370610028725823590045201871491395E-32Q, /* 3f9639ffe55fa2fa011674448b4e5b96 */ + 2.65624999999999986893899042596554269E-01Q, /* 3ffd0ffffffffffffc38f0c0a1e9f000 */ + -2.07683715950724761146070082510569258E-32Q, /* bf95af579a92e872fef81abfdf06bae8 */ + 2.69531249999999979842788204900639327E-01Q, /* 3ffd13fffffffffffa30a908d67db000 */ + 8.71465252506557329027658736641075706E-32Q, /* 3f97c47d99e19830447a42b1c0ffac61 */ + 2.73437500000000006712165837793818271E-01Q, /* 3ffd18000000000001ef453a58edb000 */ + -6.62704045767568912140550474455810301E-32Q, /* bf9758187a204dcb06ece46588aeeaba */ + 2.77343749999999994411329302988535617E-01Q, /* 3ffd1bfffffffffffe63a0fec9c9e000 */ + -4.87273466291944117406493607771338767E-32Q, /* bf96fa0381b0844a0be46bac2d673f0c */ + 2.81250000000000012677892447379453135E-01Q, /* 3ffd20000000000003a7769e125d6000 */ + -8.55871796664700790726282049552906783E-32Q, /* bf97bc64e01332cf7616b0091b8dff2c */ + 2.85156249999999998558643013736363981E-01Q, /* 3ffd23ffffffffffff95a5894bccf000 */ + -1.33068334720606220176455289635046875E-32Q, /* bf95145f43290ecf5b7adcb24697bc73 */ + 2.89062500000000008831431235621753924E-01Q, /* 3ffd280000000000028ba504fac59000 */ + -9.34157398616814623985483776710704237E-32Q, /* bf97e50ad1115b941fcb5f0c88a428f7 */ + 2.92968750000000019840235286110877063E-01Q, /* 3ffd2c000000000005b7f372d184f000 */ + 4.99302093775173155906059132992249671E-33Q, /* 3f939ecdcfb97bad3f8dbec5df5ec67d */ + 2.96875000000000015867911730971630513E-01Q, /* 3ffd3000000000000492d860c79db000 */ + 7.86107787827057767235127454590866211E-33Q, /* 3f944689517ee8f16cdb97d6a6938f32 */ + 3.00781250000000015814100002286124758E-01Q, /* 3ffd340000000000048edfe73a17d000 */ + -1.65419431293024229981937172317171504E-32Q, /* bf9557900e3efca16c89646b57f68dc0 */ + 3.04687499999999985213157159965287195E-01Q, /* 3ffd37fffffffffffbbcec6f99b36000 */ + 9.68753602893894024018934325652944198E-32Q, /* 3f97f70170e5458660c33a7e8d43d049 */ + 3.08593749999999989969324338045156215E-01Q, /* 3ffd3bfffffffffffd1bdde4d0fb1000 */ + 7.10268609610294706092252562643261106E-32Q, /* 3f9770cae45cdf615010401a4b37d8d4 */ + 3.12500000000000002971606591018488854E-01Q, /* 3ffd40000000000000db440fbc06b000 */ + 6.38924218802905979887732294952782964E-32Q, /* 3f974bbf988bb5622bd8fbaa46e8b811 */ + 3.16406250000000006594921047402056305E-01Q, /* 3ffd44000000000001e69e8954814000 */ + 3.96079878754651470094149874444850097E-32Q, /* 3f969b5017b9fa7a1e86975258c73d3d */ + 3.20312500000000006713799366908329147E-01Q, /* 3ffd48000000000001ef64159c065000 */ + -1.86401314975634286055150437995880517E-32Q, /* bf958323f0434911794e5fb8bfe136ba */ + 3.24218749999999987061246567584951210E-01Q, /* 3ffd4bfffffffffffc4549db9b928000 */ + -3.18643523744758601387071062700407431E-32Q, /* bf964ae5fa7e26c2c3981bed12e14372 */ + 3.28124999999999991782776266707412953E-01Q, /* 3ffd4ffffffffffffda1ad0840ca8000 */ + -4.46964199751314296839915534813144652E-32Q, /* bf96d0277729ffd74727150df6d15547 */ + 3.32031250000000000393816557756032682E-01Q, /* 3ffd540000000000001d0efc04fad000 */ + -9.03246333902065439930373230002688649E-33Q, /* bf947731a008748cc6dee948839ef7ae */ + 3.35937499999999983810482995064392173E-01Q, /* 3ffd57fffffffffffb556cab8ae61000 */ + 5.27742727066129518825981597650621794E-32Q, /* 3f9712050a6ddbf1cabf1b971f4b5d0b */ + 3.39843750000000004310441349760912471E-01Q, /* 3ffd5c0000000000013e0def5ddc4000 */ + -3.85927263474732591932884416445586106E-32Q, /* bf9690c51088ef3db9ca000829c450c2 */ + 3.43749999999999990248130003997484364E-01Q, /* 3ffd5ffffffffffffd3070624a0af000 */ + 9.62005170171527308106468341512327487E-34Q, /* 3f913fae595cea84432eb01430817fca */ + 3.47656250000000004085726414568625697E-01Q, /* 3ffd640000000000012d79309e291000 */ + -6.59664093705705297250259434519072507E-32Q, /* bf97568465eafb0e662e64a5dbfaf35f */ + + -1.98364257812501251077851763965418372E-03Q, /* bff6040000000001cd90f658cf0b1000 */ + -3.71984513103117734260309047540278737E-34Q, /* bf8fee73c54483194782aac4a6154d11 */ + -1.95312500000000378520649630233891879E-03Q, /* bff60000000000008ba643bb5e2e8000 */ + -1.12194202736719050440745599339855038E-34Q, /* bf8e2a436aeff7bc529873354f47a3f5 */ + -1.92260742187499397430259771221991482E-03Q, /* bff5f7fffffffffe4361cb51170da000 */ + -2.30068299876822157331268484824540848E-34Q, /* bf8f31d02f85cfe8c0cc02276ce0f437 */ + -1.89208984375001137424603270262074989E-03Q, /* bff5f0000000000347456ed490c23000 */ + -1.15012507244426243338260435466985403E-34Q, /* bf8e31c174d5677a937a34ad8d2a70b4 */ + -1.86157226562500172319250342061336738E-03Q, /* bff5e800000000007f262fa3617b4000 */ + -3.12438344643346437509767736937785561E-34Q, /* bf8f9f4d426a2457c273d34ef7d9bde9 */ + -1.83105468749999505256246872355430379E-03Q, /* bff5dffffffffffe92f18c1c2b6fa000 */ + -5.91130415288336591179087455220308942E-35Q, /* bf8d3a4c80b42dc036bae446c9807f78 */ + -1.80053710937499445182387245573120522E-03Q, /* bff5d7fffffffffe669dea82b4a4c000 */ + -1.92396289352411531324908916321392100E-34Q, /* bf8eff7a2123fb573ba9778550d669bd */ + -1.77001953125000387737631542516323906E-03Q, /* bff5d000000000011e19915c3ddb7000 */ + 7.91101758977203355387806553469731354E-36Q, /* 3f8a507f5a70faaccf469e3461873dea */ + -1.73950195312500034854670281415554486E-03Q, /* bff5c8000000000019b7dc6ef97bd000 */ + 1.55906551582436824067407021178835755E-34Q, /* 3f8e9e7880333e34955aebcde3cfb053 */ + -1.70898437499998955782591472611429852E-03Q, /* bff5bffffffffffcfd80e88aa6b96000 */ + 8.22951661962611381718215899498500357E-35Q, /* 3f8db58e6031a779b59f6ece191de7cc */ + -1.67846679687500586652037711131708544E-03Q, /* bff5b80000000001b0df6fd21c133000 */ + -8.96642618848426299713145894522897419E-35Q, /* bf8ddcbcab46d531801bfae4121f2f8a */ + -1.64794921875000109499161354039904782E-03Q, /* bff5b0000000000050cbce8915575000 */ + -2.88077905394253859590587789680486639E-34Q, /* bf8f7eebd4dd860ef73b674d5e707959 */ + -1.61743164062501133830507079150388351E-03Q, /* bff5a80000000003449e8700c3e82000 */ + -3.68271725851639066312899986829350273E-34Q, /* bf8fe9845fe20a5fe74059e0cae185d6 */ + -1.58691406249999015546015764131101956E-03Q, /* bff59ffffffffffd2999e668cdd28000 */ + 8.48197657099957029953716507898788812E-35Q, /* 3f8dc2faaebb97392e451b07b28c4b12 */ + -1.55639648437500317366570219290722587E-03Q, /* bff5980000000000ea2cd9a40d256000 */ + -3.45156704719737676412949957712570373E-36Q, /* bf8925a079505516c8e317ac1ff53255 */ + -1.52587890625000568759013197767046039E-03Q, /* bff5900000000001a3ab8a3f6b698000 */ + -1.01902948542497496574967177677556729E-34Q, /* bf8e0ee78d94d9b5ad3d63ae35c9b554 */ + -1.49536132812500945889014955936485340E-03Q, /* bff5880000000002b9f1621b57743000 */ + -3.32264697086631598830366079048117140E-34Q, /* bf8fb9a7d14c32289204fbb0c9eb20e0 */ + -1.46484374999999931883259902869504725E-03Q, /* bff57fffffffffffcdbd1c90e1b4a000 */ + -1.76487524793892929381101031660811433E-34Q, /* bf8ed52f2f724bc1ae870b18356337b4 */ + -1.43432617187498876325946983333888768E-03Q, /* bff577fffffffffcc2dff8faa5570000 */ + -3.54550084538495708816233114576143814E-34Q, /* bf8fd74724576915868c1e8ce9f430f1 */ + -1.40380859374999215367421282192718062E-03Q, /* bff56ffffffffffdbd0b18aac65ed000 */ + -1.90585907028351204486765167064669639E-34Q, /* bf8efaaa0c0e23e50c11b2120348054f */ + -1.37329101562499692341771212945644892E-03Q, /* bff567ffffffffff1cfd00f1b0577000 */ + -3.59631150411372589637918252836880320E-34Q, /* bf8fde08239ac74942a46298ea4fb715 */ + -1.34277343749999137467356674296739172E-03Q, /* bff55ffffffffffd839030b05d53d000 */ + -1.49571076125940368185068762485268117E-35Q, /* bf8b3e1a3d5c684b27a9f835b1d8d3c9 */ + -1.31225585937499247038404301859788734E-03Q, /* bff557fffffffffdd469936e691e3000 */ + 3.10375845385355395586146533282311300E-34Q, /* 3f8f9c8f6d63b7a4145716ffd92491fb */ + -1.28173828124999024755581675764821898E-03Q, /* bff54ffffffffffd306589b0ab21d000 */ + -1.98541096105909793397376077900810019E-34Q, /* bf8f07e808bbb1e35106c294ffbb9687 */ + -1.25122070312500340204619591143332523E-03Q, /* bff5480000000000fb06d5f16ad2c000 */ + 3.62884195935761446237911443317457521E-34Q, /* 3f8fe25b17d623178a386a6fa6c5afb2 */ + -1.22070312499999591578388993012071279E-03Q, /* bff53ffffffffffed2a356c440074000 */ + -2.96756662615653130862526710937493307E-35Q, /* bf8c3b90d8ff2a991e5bd16718fb0645 */ + -1.19018554687498821966212632349422735E-03Q, /* bff537fffffffffc9ac3b585dda89000 */ + 1.44659971891167323357060028901142644E-34Q, /* 3f8e809279ab249edf1dad9fe13fb0bf */ + -1.15966796875000160938908064907298384E-03Q, /* bff530000000000076c0800db9639000 */ + 2.50088010538742402346270685365928513E-34Q, /* 3f8f4c6c8a483b60201d30c1a83c3cb7 */ + -1.12915039062500267151512523291939657E-03Q, /* bff5280000000000c51f7e7315137000 */ + 7.56402096465615210500092443924888831E-35Q, /* 3f8d922c1e485d99aea2668ed32b55a6 */ + -1.09863281249998665006360103291051571E-03Q, /* bff51ffffffffffc26f2d4c9ce2ba000 */ + 1.43982174467233642713619821353592061E-34Q, /* 3f8e7ec530b3d92b6303bec1c81214d1 */ + -1.06811523437500522742248711752028025E-03Q, /* bff518000000000181b7380f10446000 */ + 5.41265133745862349181293024531133174E-35Q, /* 3f8d1fc9313d018b30e790e06b6be723 */ + -1.03759765624999980942114138999770552E-03Q, /* bff50ffffffffffff1f01130490e1000 */ + 1.21525139612685854366189534669623436E-34Q, /* 3f8e4311b96b6fcde412caf3f0d86fb9 */ + -1.00708007812499602697537601515759439E-03Q, /* bff507fffffffffedad7afcce7051000 */ + 1.00020246351201558505328236381833392E-34Q, /* 3f8e09e640992512b1300744a7e984ed */ + -9.76562499999992592487302113340463694E-04Q, /* bff4fffffffffffbbad8151f8adf6000 */ + -1.64984406575162932060422892046851002E-34Q, /* bf8eb69a919986e8054b86fc34300f24 */ + -9.46044921874989085824996924138179594E-04Q, /* bff4effffffffff9b55a204fd9792000 */ + -9.29539174108308550334255350011347171E-35Q, /* bf8dee3a50ed896b4656fa577a1df3d7 */ + -9.15527343750013735214860599791540029E-04Q, /* bff4e00000000007eaf5bf103f82d000 */ + 3.07557018309280519949818825519490586E-35Q, /* 3f8c470cfbef77d32c74cb8042f6ee81 */ + -8.85009765625012292294986105781516428E-04Q, /* bff4d000000000071605c65403b97000 */ + 4.77499983783821950338363358545463558E-35Q, /* 3f8cfbc3dc18884c4c4f9e07d90d7bd3 */ + -8.54492187499986941239470706817188192E-04Q, /* bff4bffffffffff878ddf9cab264a000 */ + -1.60128240346239526958630011447901568E-34Q, /* bf8ea9b1a21e19e2d5bd84b0fbffcf95 */ + -8.23974609374996290174598690241743810E-04Q, /* bff4affffffffffddc86c249ebe06000 */ + 1.61677540391961912631535763471935882E-34Q, /* 3f8eadd00841366b0dc2bc262c2c8c36 */ + -7.93457031249988696952538334288757473E-04Q, /* bff49ffffffffff97bf6f0aa85a5f000 */ + 1.22318577008381887076634753347515709E-34Q, /* 3f8e452db5b5d250878f71040da06d14 */ + -7.62939453124996723316499040007097041E-04Q, /* bff48ffffffffffe1c7265b431108000 */ + -1.03845161748762410745671891558398468E-34Q, /* bf8e14115ad884c96d1a820c73647220 */ + -7.32421874999998242520117923997325794E-04Q, /* bff47ffffffffffefca4498b7aa8a000 */ + 5.64005211953031009549514026639438083E-35Q, /* 3f8d2be06950f68f1a6d8ff829a6928e */ + -7.01904296874999772890934814265622012E-04Q, /* bff46fffffffffffde7c0fe5d8041000 */ + 5.90245467325173644235991233229525762E-35Q, /* 3f8d39d40cc49002189243c194b1db0e */ + -6.71386718750008699269643939210658742E-04Q, /* bff460000000000503c91d798b60c000 */ + -5.20515801723324452151498579012322191E-35Q, /* bf8d14c0f08a6a9285b32b8bda003eb5 */ + -6.40869140625005499535275057463709988E-04Q, /* bff45000000000032b969184e9751000 */ + -6.69469163285461870099846471658294534E-35Q, /* bf8d63f36bab7b24d936c9380e3d3fa6 */ + -6.10351562499999293780097329596079841E-04Q, /* bff43fffffffffff97c7c433e35ed000 */ + -1.16941808547394177991845382085515086E-34Q, /* bf8e36e27886f10b234a7dd8fc588bf0 */ + -5.79833984375000068291972326409994795E-04Q, /* bff43000000000000a13ff6dcf2bf000 */ + 1.17885044988246219185041488459766001E-34Q, /* 3f8e3964677e001a00412aab52790842 */ + -5.49316406249990904622170867910987793E-04Q, /* bff41ffffffffffac1c25739c716b000 */ + -3.31875702128137033065075734368960972E-35Q, /* bf8c60e928d8982c3c99aef4f885a121 */ + -5.18798828125011293653756992177727236E-04Q, /* bff410000000000682a62cff36775000 */ + -5.69971237642088463334239430962628187E-35Q, /* bf8d2f0c76f8757d61cd1abc7ea7d066 */ + -4.88281249999990512232251384917893121E-04Q, /* bff3fffffffffff50fb48992320df000 */ + 1.02144616714408655325510171265051108E-35Q, /* 3f8ab279a3626612710b9b3ac71734ac */ + -4.57763671874997554564967307956493434E-04Q, /* bff3dffffffffffd2e3c272e3cca9000 */ + -8.25484058867957231164162481843653503E-35Q, /* bf8db6e71158e7bf93e2e683f07aa841 */ + -4.27246093749991203999790346349633286E-04Q, /* bff3bffffffffff5dbe103cba0eb2000 */ + -3.51191203319375193921924105905691755E-35Q, /* bf8c757356d0f3dd7fbefc0dd419ab50 */ + -3.96728515624986649402960638705483281E-04Q, /* bff39ffffffffff09b996882706ec000 */ + -5.51925962073095883016589497244931171E-36Q, /* bf89d586d49f22289cfc860bebb99056 */ + -3.66210937499999945095511981300980754E-04Q, /* bff37fffffffffffefcb88bfc7df6000 */ + -2.11696465278144529364423332249588595E-35Q, /* bf8bc23a84d28e5496c874ef9833be25 */ + -3.35693359374992480958458008559640163E-04Q, /* bff35ffffffffff754c548a8798f2000 */ + -8.58941791799705081104736787493668352E-35Q, /* bf8dc8b1192fb7c3662826d43acb7c68 */ + -3.05175781250009811036303273640122156E-04Q, /* bff340000000000b4fb4f1aad1c76000 */ + -8.61173897858769926480551302277426632E-35Q, /* bf8dc9e0eabb1c0b33051011b64769fa */ + -2.74658203124987298321920308390303850E-04Q, /* bff31ffffffffff15b2056ac252fd000 */ + 3.35152809454778381053519808988046631E-37Q, /* 3f85c82fb59ff8d7c80d44e635420ab1 */ + -2.44140624999999992770514819575735516E-04Q, /* bff2fffffffffffffbbb82d6a7636000 */ + 3.54445837111124472730013879165516908E-35Q, /* 3f8c78e955b01378be647b1c92aa9a77 */ + -2.13623046875012756463165168672749438E-04Q, /* bff2c0000000001d6a1635fea6bbf000 */ + 1.50050816288650121729916777279129473E-35Q, /* 3f8b3f1f6f616a61129a58e131cbd31d */ + -1.83105468749991323078784464300306893E-04Q, /* bff27fffffffffebfe0cbd0c82399000 */ + -9.14919506501448661140572099029756008E-37Q, /* bf873754bacaa9d9513b6127e791eb47 */ + -1.52587890625013337032336300236461546E-04Q, /* bff240000000001ec0cb57f2cc995000 */ + 2.84906084373176180870418394956384516E-35Q, /* 3f8c2ef6d03a7e6ab087c4f099e4de89 */ + -1.22070312499990746786116828458007518E-04Q, /* bff1ffffffffffd553bbb49f35a34000 */ + 6.71618008964968339584520728412444537E-36Q, /* 3f8a1dacb99c60071fc9cd2349495bf0 */ + -9.15527343750029275602791047595142231E-05Q, /* bff180000000000d8040cd6ecde28000 */ + -1.95753652091078750312541716951402172E-35Q, /* bf8ba0526cfb24d8d59122f1c7a09a14 */ + -6.10351562499913258461494008080572701E-05Q, /* bff0ffffffffffaffebbb92d7f6a9000 */ + 5.69868489273961111703398456218119973E-36Q, /* 3f89e4ca5df09ef4a4386dd5b3bf0331 */ + -3.05175781250092882818419203884960853E-05Q, /* bff0000000000055ab55de88fac1d000 */ + 9.03341100018476837609128961872915953E-36Q, /* 3f8a803d229fa3a0e834a63abb06662b */ +#define T_EXPL_ARG2 (2*T_EXPL_ARG1 + 2 + 2*65) + 0.00000000000000000000000000000000000E+00Q, /* 00000000000000000000000000000000 */ + 0.00000000000000000000000000000000000E+00Q, /* 00000000000000000000000000000000 */ + 3.05175781249814607084128277672749162E-05Q, /* 3feffffffffffeaa02abb9102f499000 */ + 1.00271855391179733380665816525889949E-36Q, /* 3f8755351afa042ac3f58114824d4c10 */ + 6.10351562500179243748093427073421439E-05Q, /* 3ff1000000000052a95de07a4c26d000 */ + 1.67231624299180373502350811501181670E-36Q, /* 3f881c87a53691cae9d77f4e40d66616 */ + 9.15527343749970728685313252158399200E-05Q, /* 3ff17ffffffffff28040cc2acde28000 */ + 2.43665747834893104318707597514407880E-36Q, /* 3f889e9366c7c6c6a2ecb78dc9b0509e */ + 1.22070312500027751961838150070880064E-04Q, /* 3ff200000000003ffddde6c153b53000 */ + -1.73322146370624186623546452226755405E-35Q, /* bf8b709d8d658ed5dbbe943de56ee84e */ + 1.52587890624995916105682628143179430E-04Q, /* 3ff23ffffffffff6954b56e285d23000 */ + 1.23580432650945898349135528000443828E-35Q, /* 3f8b06d396601dde16de7d7bc27346e6 */ + 1.83105468750008670314358488289621794E-04Q, /* 3ff2800000000013fe0cdc8c823b7000 */ + 4.30446229148833293310207915930740796E-35Q, /* 3f8cc9ba9bfe554a4f7f2fece291eb23 */ + 2.13623046875005741337455947623248132E-04Q, /* 3ff2c0000000000d3d1662de21a3f000 */ + -3.96110759869520786681660669615255057E-35Q, /* bf8ca5379b04ff4a31aab0ceacc917e6 */ + 2.44140624999981493573336463433440506E-04Q, /* 3ff2ffffffffffd553bbdf48e0534000 */ + -1.39617373942387888957350179316792928E-35Q, /* bf8b28eeedc286015802b63f96b8c5cd */ + 2.74658203124984920706309918754626834E-04Q, /* 3ff31fffffffffee9d60c8439ec1d000 */ + -3.16168080483901830349738314447356223E-36Q, /* bf890cf74f81c77a611abc1243812444 */ + 3.05175781250008648918265055410966055E-04Q, /* 3ff3400000000009f8b5c9a346636000 */ + 8.54421306185008998867856704677221443E-35Q, /* 3f8dc649cd40922fc08adc6b6b20ead0 */ + 3.35693359374988945462612499316774515E-04Q, /* 3ff35ffffffffff34146c540f15b2000 */ + 7.96443137431639500475160850431097078E-35Q, /* 3f8da77638ed3148fc4d99d1c9e13446 */ + 3.66210937500027690542093987739604535E-04Q, /* 3ff380000000001fecce34bea89c4000 */ + 2.14507323877752361258862577769090367E-35Q, /* 3f8bc834e554d38894cf91957b0253d3 */ + 3.96728515625003928083564943615052121E-04Q, /* 3ff3a00000000004875d9a4acf6ab000 */ + 4.88358523466632050664019922448605508E-35Q, /* 3f8d03a7eaeef1a9f78c71a12c44dd28 */ + 4.27246093750017799227172345607351585E-04Q, /* 3ff3c00000000014856794c3ee850000 */ + 6.66520494592631402182216588784828935E-35Q, /* 3f8d6262118fcdb59b8f16108f5f1a6c */ + 4.57763671875002108342364320152138181E-04Q, /* 3ff3e000000000026e45d855410b9000 */ + 7.21799615960261390920033272189522298E-35Q, /* 3f8d7fc645cff8879462296af975c9fd */ + 4.88281249999999768797631616370963356E-04Q, /* 3ff3ffffffffffffbbc2d7cc004df000 */ + -5.30564629906905979452258114088325361E-35Q, /* bf8d1a18b71929a30d67a217a27ae851 */ + 5.18798828124997339054881383202487041E-04Q, /* 3ff40ffffffffffe775055eea5851000 */ + -4.03682911253647925867848180522846377E-35Q, /* bf8cad44f0f3e5199d8a589d9332acad */ + 5.49316406249980511907933706754958501E-04Q, /* 3ff41ffffffffff4c410b29bb62fb000 */ + -2.08166843948323917121806956728438051E-35Q, /* bf8bbab8cf691403249fe5b699e25143 */ + 5.79833984374989593561576568548497165E-04Q, /* 3ff42ffffffffffa0047df328d817000 */ + -1.72745033420153042445343706432627539E-34Q, /* bf8ecb3c2d7d3a9e6e960576be901fdf */ + 6.10351562500008540711511259540838154E-04Q, /* 3ff4400000000004ec62f54f8c271000 */ + 7.41889382604319545724663095428976499E-35Q, /* 3f8d8a74c002c81a47c93b8e05d15f8e */ + 6.40869140625020444702875407535884986E-04Q, /* 3ff450000000000bc91b09718515d000 */ + -4.47321009727305792048065440180490107E-35Q, /* bf8cdbac5c8fe70822081d8993eb5cb6 */ + 6.71386718750007531635964622352684074E-04Q, /* 3ff460000000000457792973db05c000 */ + 5.13698959677949336513874456684462092E-35Q, /* 3f8d112114436949c5ef38d8049004ab */ + 7.01904296875006634673332887754430334E-04Q, /* 3ff4700000000003d31adf2cb8b1d000 */ + -8.25665755717729437292989870760751482E-35Q, /* bf8db6ffcc8ef71f8e648e3a8b160f5a */ + 7.32421874999998244664170215504673504E-04Q, /* 3ff47ffffffffffefcf5498bd5c8a000 */ + -5.64005234937832153139057628112753364E-35Q, /* bf8d2be06a1dfe90e7bf90fba7c12a98 */ + 7.62939453125017456345986752604096408E-04Q, /* 3ff490000000000a101a1b093d4a8000 */ + -1.11084094120417622468550608896588329E-34Q, /* bf8e274feabd2d94f6694507a46accb1 */ + 7.93457031249987558617598988993908016E-04Q, /* 3ff49ffffffffff8d3f9dcab74bbf000 */ + -1.22966480225449015129079129940978828E-34Q, /* bf8e46e6a65eef8fa9e42eddf3da305e */ + 8.23974609374997378723747633335135819E-04Q, /* 3ff4affffffffffe7d2afbaa55b26000 */ + -1.62270010016794279091906973366704963E-34Q, /* bf8eaf633f057ebdb664a34566401c4e */ + 8.54492187500023938282350821569920958E-04Q, /* 3ff4c0000000000dccaabce399e59000 */ + -1.39076361712838158775374263169606160E-34Q, /* bf8e71ba779364b3bbdba7841f2c4ca1 */ + 8.85009765624987932362186815286691297E-04Q, /* 3ff4cffffffffff90b218886edc2a000 */ + 4.07328275060905585228261577392403980E-35Q, /* 3f8cb1254dbb6ea4b8cfa5ed4cf28d24 */ + 9.15527343749975579461305518559161974E-04Q, /* 3ff4dffffffffff1ec2a21f25df33000 */ + 1.16855112459192484947855553716334015E-35Q, /* 3f8af10bf319e9f5270cf249eeffbe5c */ + 9.46044921875016761584725882821122521E-04Q, /* 3ff4f00000000009a992c46c16d71000 */ + 9.51660680007524262741115611071680436E-35Q, /* 3f8df9fd56e81f8edf133843910ee831 */ + 9.76562499999974118878133088548272636E-04Q, /* 3ff4fffffffffff1149edc46a6df6000 */ + -5.65271128977550656964071208289181661E-36Q, /* bf89e0e12689dd721aa2314c81eb6429 */ + 1.00708007812498671732140389760347830E-03Q, /* 3ff507fffffffffc2be94b90ed091000 */ + -1.43355074891483635310132767255371379E-34Q, /* bf8e7d1a688c247b16022daab1316d55 */ + 1.03759765625002637786192745235343007E-03Q, /* 3ff51000000000079a57b966bc158000 */ + 2.95905815240957629366749917020106928E-34Q, /* 3f8f895387fc73bb38f8a1b254c01a60 */ + 1.06811523437500860568717813047520763E-03Q, /* 3ff51800000000027afcd5b35f5e6000 */ + -5.98328495358586628195372356742878314E-35Q, /* bf8d3e204130013bf6328f1b70ff8c76 */ + 1.09863281250001439958487251556220070E-03Q, /* 3ff5200000000004268077c6c66bd000 */ + 2.41371837889426603334113000868144760E-34Q, /* 3f8f40d6948edf864054ccf151f9815e */ + 1.12915039062501298413451613770002366E-03Q, /* 3ff5280000000003be0f5dd8fe81b000 */ + -1.28815268997394164973472617519705703E-34Q, /* bf8e567321172ea089dce4bc8354ecb7 */ + 1.15966796874997272036339054191407232E-03Q, /* 3ff52ffffffffff8231e3bcfff1e8000 */ + 1.02996064554316248496839462594377804E-34Q, /* 3f8e11cf7d402789244f68e2d4f985b1 */ + 1.19018554687502744121802585360546796E-03Q, /* 3ff5380000000007e8cdf3f8f6c20000 */ + -1.43453217726255628994625761307322163E-34Q, /* bf8e7d5d3370d85a374f5f4802fc517a */ + 1.22070312499997743541996266398850614E-03Q, /* 3ff53ffffffffff97f0722561f454000 */ + -1.41086259180534339713692694428211646E-34Q, /* bf8e77125519ff76244dfec5fbd58402 */ + 1.25122070312501024092560690174507039E-03Q, /* 3ff5480000000002f3a59d8820691000 */ + 3.84102646020099293168698506729765213E-34Q, /* 3f8ffe8f5b86f9c3569c8f26e19b1f50 */ + 1.28173828124997986521442660131425390E-03Q, /* 3ff54ffffffffffa3250a764439d9000 */ + 1.44644589735033114377952806106652650E-34Q, /* 3f8e808801b80dcf38323cdbfdca2549 */ + 1.31225585937501665804856968749058137E-03Q, /* 3ff5580000000004cd25a414c6d62000 */ + 1.67474574742200577294563576414361377E-34Q, /* 3f8ebd394a151dbda4f81d5d83c0f1e9 */ + 1.34277343749997290265837386401818888E-03Q, /* 3ff55ffffffffff83091b042cfd59000 */ + -1.55650565030381326742591837551559103E-34Q, /* bf8e9dca490d7fecfadba9625ffb91c5 */ + 1.37329101562497720784949380297774268E-03Q, /* 3ff567fffffffff96e3c7312f5ccf000 */ + 1.65279335325630026116581677369221748E-34Q, /* 3f8eb763496f5bd7404f2298b402074f */ + 1.40380859374999099958354100336136647E-03Q, /* 3ff56ffffffffffd67e2f09f2a381000 */ + 1.89919944388961890195706641264717076E-34Q, /* 3f8ef8e4d0ffdfeba982aa8829501389 */ + 1.43432617187497484122173130998160625E-03Q, /* 3ff577fffffffff8bf9c1d71af8a8000 */ + 2.57638517142061429772064578590009568E-34Q, /* 3f8f5675d82c1cc4ada70fd3a957b89a */ + 1.46484374999999929342158925502052945E-03Q, /* 3ff57fffffffffffcbdd1c7671b46000 */ + 1.76487201934184070490166772482073801E-34Q, /* 3f8ed52ef732458f6e4c5c07504f33cc */ + 1.49536132812502318451070466256902933E-03Q, /* 3ff5880000000006aeb7066c8ad43000 */ + 2.38068367275295804321313550609246656E-34Q, /* 3f8f3c7277ae6fc390ace5e06c0b025b */ + 1.52587890625000448053340248672949543E-03Q, /* 3ff59000000000014a9ae2104b3bc000 */ + 1.01174455568392813258454590274740959E-34Q, /* 3f8e0cf7c434762991bb38e12acee215 */ + 1.55639648437501113499837053523090913E-03Q, /* 3ff5980000000003359e2c204355e000 */ + -2.82398418808099749023517211651363693E-35Q, /* bf8c2c4c2971d88caa95e15fb1ccb1a1 */ + 1.58691406249999937955142588308171026E-03Q, /* 3ff59fffffffffffd2380ecbc87c2000 */ + -1.27361695572422741562701199136538047E-34Q, /* bf8e5295e0e206dfb0f0266c07225448 */ + 1.61743164062498000531048954475329309E-03Q, /* 3ff5a7fffffffffa3ca6fe61ed94c000 */ + -1.22606548862580061633942923016222044E-34Q, /* bf8e45f1b17bb61039d21a351bb207b8 */ + 1.64794921875001835451453858682255576E-03Q, /* 3ff5b000000000054a52fa20f6565000 */ + 1.39132339594152335892305491425264583E-34Q, /* 3f8e71e0904c5449b414ee49b191cef2 */ + 1.67846679687501263995029340691547953E-03Q, /* 3ff5b80000000003a4a9e912c910b000 */ + 6.67245854693585315412242764786197029E-35Q, /* 3f8d62c4ccac1e7511a617d469468ccd */ + 1.70898437500002646861403514115369655E-03Q, /* 3ff5c00000000007a109fbaa7e015000 */ + 6.87367172354719289559624829652240928E-36Q, /* 3f8a245fa835eceb42bae8128d9336db */ + 1.73950195312501174308226096992992128E-03Q, /* 3ff5c80000000003627c8d637a005000 */ + -2.20824271875474985927385878948759352E-34Q, /* bf8f25869b1cbefb25e735992f232f57 */ + 1.77001953124997491747605207736194513E-03Q, /* 3ff5cffffffffff8c53c84b6883b8000 */ + 3.43123048533596296514343180408963705E-34Q, /* 3f8fc816b91d173ddadbbf09b1287906 */ + 1.80053710937497698911127570705069398E-03Q, /* 3ff5d7fffffffff95e1899f4a8430000 */ + 3.99231237340890073475077494556136100E-35Q, /* 3f8ca889148f62fa854da5674df41279 */ + 1.83105468750002267094899598630423914E-03Q, /* 3ff5e0000000000688d21e62ba674000 */ + -3.22274595655810623999007524769365273E-34Q, /* bf8fac605cb9ae01eb719675ced25560 */ + 1.86157226562500499224728040579690330E-03Q, /* 3ff5e80000000001705ce28a6d89e000 */ + 3.07094985075881613489605622068441083E-34Q, /* 3f8f98330225ec7e2c8f3c0d1c432b91 */ + 1.89208984374998234666824993196980949E-03Q, /* 3ff5effffffffffae969fdc7cd8cf000 */ + -3.06287628722973914692165056776495733E-34Q, /* bf8f9720477d9cfa10e464df7f91020c */ + 1.92260742187501225343755557292811682E-03Q, /* 3ff5f800000000038824e428ed49a000 */ + 6.30049124729794620592961282769623368E-35Q, /* 3f8d4efdd7cd4336d88a6aa49e1e96bc */ + 1.95312499999998514894032051116231258E-03Q, /* 3ff5fffffffffffbb82f6a04f1ae0000 */ + -6.14610057507500948543216998736262902E-35Q, /* bf8d46c862d39255370e7974d48daa7e */ + 1.98364257812501222021119324146882732E-03Q, /* 3ff6040000000001c2d8a1aa5188d000 */ + 3.71942298418113774118754986159801984E-34Q, /* 3f8fee6567d9940495519ffe62cbc9a4 */ + + 7.06341639425619532977052017486130353E-01Q, /* 3ffe69a59c8245a9ac00000000000000 */ + 7.09106182437398424589503065362805501E-01Q, /* 3ffe6b0ff72deb89d000000000000000 */ + 7.11881545564596485142772053222870454E-01Q, /* 3ffe6c7bbce9a6d93000000000000000 */ + 7.14667771155948150507697391731198877E-01Q, /* 3ffe6de8ef213d71e000000000000000 */ + 7.17464901725936049503573599395167548E-01Q, /* 3ffe6f578f41e1a9e400000000000000 */ + 7.20272979955439790478166628417966422E-01Q, /* 3ffe70c79eba33c06c00000000000000 */ + 7.23092048692387218133958981525211129E-01Q, /* 3ffe72391efa434c7400000000000000 */ + 7.25922150952408251622927082280511968E-01Q, /* 3ffe73ac117390acd800000000000000 */ + 7.28763329919491220643124052003258839E-01Q, /* 3ffe752077990e79d000000000000000 */ + 7.31615628946641782803794740175362676E-01Q, /* 3ffe769652df22f7e000000000000000 */ + 7.34479091556544505525749855223693885E-01Q, /* 3ffe780da4bba98c4800000000000000 */ + 7.37353761442226890432394270646909717E-01Q, /* 3ffe79866ea5f432d400000000000000 */ + 7.40239682467726090031590047146892175E-01Q, /* 3ffe7b00b216ccf53000000000000000 */ + 7.43136898668758316688354170764796436E-01Q, /* 3ffe7c7c70887763c000000000000000 */ + 7.46045454253390638577059235103661194E-01Q, /* 3ffe7df9ab76b20fd000000000000000 */ + 7.48965393602715662213498148958024103E-01Q, /* 3ffe7f78645eb8076400000000000000 */ + 7.51896761271528629722027403659012634E-01Q, /* 3ffe80f89cbf42526400000000000000 */ + 7.54839601989007347171423134568613023E-01Q, /* 3ffe827a561889716000000000000000 */ + 7.57793960659394638668118204805068672E-01Q, /* 3ffe83fd91ec46ddc000000000000000 */ + 7.60759882362683631518152083117456641E-01Q, /* 3ffe858251bdb68b8c00000000000000 */ + 7.63737412355305483879774897104653064E-01Q, /* 3ffe87089711986c9400000000000000 */ + 7.66726596070820082262642358728044201E-01Q, /* 3ffe8890636e31f54400000000000000 */ + 7.69727479120609181517664865168626420E-01Q, /* 3ffe8a19b85b4fa2d800000000000000 */ + 7.72740107294572486917871856348938309E-01Q, /* 3ffe8ba4976246833800000000000000 */ + 7.75764526561826289752232810315035749E-01Q, /* 3ffe8d31020df5be4400000000000000 */ + 7.78800783071404878477039801509818062E-01Q, /* 3ffe8ebef9eac820b000000000000000 */ + 7.81848923152964780936002853195532225E-01Q, /* 3ffe904e8086b5a87800000000000000 */ + 7.84908993317491698871180005880887620E-01Q, /* 3ffe91df97714512d800000000000000 */ + 7.87981040258010162480317717381694820E-01Q, /* 3ffe9372403b8d6bcc00000000000000 */ + 7.91065110850296016042904057030682452E-01Q, /* 3ffe95067c78379f2800000000000000 */ + 7.94161252153591734614934694036492147E-01Q, /* 3ffe969c4dbb800b4800000000000000 */ + 7.97269511411324433014513601847284008E-01Q, /* 3ffe9833b59b38154400000000000000 */ + 8.00389936051826789142893403550260700E-01Q, /* 3ffe99ccb5aec7bec800000000000000 */ + 8.03522573689060742863077280162542593E-01Q, /* 3ffe9b674f8f2f3d7c00000000000000 */ + 8.06667472123343942680406826184480451E-01Q, /* 3ffe9d0384d70893f800000000000000 */ + 8.09824679342079301047618855591281317E-01Q, /* 3ffe9ea15722892c7800000000000000 */ + 8.12994243520486992160556383169023320E-01Q, /* 3ffea040c80f8374f000000000000000 */ + 8.16176213022339780422953481320291758E-01Q, /* 3ffea1e1d93d687d0000000000000000 */ + 8.19370636400700819157449927843117621E-01Q, /* 3ffea3848c4d49954c00000000000000 */ + 8.22577562398664585696650419777142815E-01Q, /* 3ffea528e2e1d9f09800000000000000 */ + 8.25797039950100647542896581398963463E-01Q, /* 3ffea6cede9f70467c00000000000000 */ + 8.29029118180400342863478613253391813E-01Q, /* 3ffea876812c0877bc00000000000000 */ + 8.32273846407226292054559735333896242E-01Q, /* 3ffeaa1fcc2f45343800000000000000 */ + 8.35531274141265073440720811959181447E-01Q, /* 3ffeabcac15271a2a400000000000000 */ + 8.38801451086982535754188461396552157E-01Q, /* 3ffead7762408309bc00000000000000 */ + 8.42084427143382358016410194068157580E-01Q, /* 3ffeaf25b0a61a7b4c00000000000000 */ + 8.45380252404767357221615498019673396E-01Q, /* 3ffeb0d5ae318680c400000000000000 */ + 8.48688977161503960155997106085123960E-01Q, /* 3ffeb2875c92c4c99400000000000000 */ + 8.52010651900789478530029441571969073E-01Q, /* 3ffeb43abd7b83db1c00000000000000 */ + 8.55345327307422548246407245642330963E-01Q, /* 3ffeb5efd29f24c26400000000000000 */ + 8.58693054264576483003423845730139874E-01Q, /* 3ffeb7a69db2bcc77800000000000000 */ + 8.62053883854575708767242758767679334E-01Q, /* 3ffeb95f206d17228000000000000000 */ + 8.65427867359675251357487013592617586E-01Q, /* 3ffebb195c86b6b29000000000000000 */ + 8.68815056262843166123843730019871145E-01Q, /* 3ffebcd553b9d7b62000000000000000 */ + 8.72215502248546159513864495238522068E-01Q, /* 3ffebe9307c271855000000000000000 */ + 8.75629257203538208242932228131394368E-01Q, /* 3ffec0527a5e384ddc00000000000000 */ + 8.79056373217652342599848225290770642E-01Q, /* 3ffec213ad4c9ed0d800000000000000 */ + 8.82496902584595399599010079327854328E-01Q, /* 3ffec3d6a24ed8221800000000000000 */ + 8.85950897802745995779361010136199184E-01Q, /* 3ffec59b5b27d9696800000000000000 */ + 8.89418411575955636383383762222365476E-01Q, /* 3ffec761d99c5ba58800000000000000 */ + 8.92899496814352794382685374330321793E-01Q, /* 3ffec92a1f72dd70d400000000000000 */ + 8.96394206635150403439382671422208659E-01Q, /* 3ffecaf42e73a4c7d800000000000000 */ + 8.99902594363456265202927397695020773E-01Q, /* 3ffeccc00868c0d18800000000000000 */ + 9.03424713533086704009278378180169966E-01Q, /* 3ffece8daf1e0ba94c00000000000000 */ + 9.06960617887383580004723171441582963E-01Q, /* 3ffed05d24612c2af000000000000000 */ + 9.10510361380034133338412516422977205E-01Q, /* 3ffed22e6a0197c02c00000000000000 */ + 9.14073998175894436579724811053893063E-01Q, /* 3ffed40181d094303400000000000000 */ + 9.17651582651815816982221463149471674E-01Q, /* 3ffed5d66da13970f400000000000000 */ + 9.21243169397474526149949269893113524E-01Q, /* 3ffed7ad2f48737a2000000000000000 */ + 9.24848813216204823639543519675498828E-01Q, /* 3ffed985c89d041a3000000000000000 */ + 9.28468569125835141431224428743007593E-01Q, /* 3ffedb603b7784cd1800000000000000 */ + 9.32102492359527579068867453315760940E-01Q, /* 3ffedd3c89b26894e000000000000000 */ + 9.35750638366620729469147477175283711E-01Q, /* 3ffedf1ab529fdd41c00000000000000 */ + 9.39413062813475779888605643463961314E-01Q, /* 3ffee0fabfbc702a3c00000000000000 */ + 9.43089821584325888048638830696290825E-01Q, /* 3ffee2dcab49ca51b400000000000000 */ + 9.46780970782128888929563004239753354E-01Q, /* 3ffee4c079b3f8000400000000000000 */ + 9.50486566729423443256052905780961737E-01Q, /* 3ffee6a62cdec7c7b000000000000000 */ + 9.54206665969188322362626308859034907E-01Q, /* 3ffee88dc6afecfbfc00000000000000 */ + 9.57941325265705301283958306157728657E-01Q, /* 3ffeea77490f0196b000000000000000 */ + 9.61690601605425299247542625380447134E-01Q, /* 3ffeec62b5e5881fb000000000000000 */ + 9.65454552197837823079851204965962097E-01Q, /* 3ffeee500f1eed967000000000000000 */ + 9.69233234476344074348475032820715569E-01Q, /* 3ffef03f56a88b5d7800000000000000 */ + 9.73026706099133165128733935489435680E-01Q, /* 3ffef2308e71a927a800000000000000 */ + 9.76835024950062025261843245971249416E-01Q, /* 3ffef423b86b7ee79000000000000000 */ + 9.80658249139538557015427500118676107E-01Q, /* 3ffef618d68936c09c00000000000000 */ + 9.84496437005408397968864164795377292E-01Q, /* 3ffef80feabfeefa4800000000000000 */ + 9.88349647113845042323276857132441364E-01Q, /* 3ffefa08f706bbf53800000000000000 */ + 9.92217938260243514925207364285597578E-01Q, /* 3ffefc03fd56aa225000000000000000 */ + 9.96101369470117486981664001177705359E-01Q, /* 3ffefe00ffaabffbbc00000000000000 */ +#define T_EXPL_RES1 (T_EXPL_ARG2 + 2 + 2*65 + 89) + 1.00000000000000000000000000000000000E+00Q, /* 3fff0000000000000000000000000000 */ + 1.00391388933834757590801700644078664E+00Q, /* 3fff0100802ab5577800000000000000 */ + 1.00784309720644799091004983893071767E+00Q, /* 3fff0202015600445c00000000000000 */ + 1.01178768355933151879000320150225889E+00Q, /* 3fff0304848362076c00000000000000 */ + 1.01574770858668572692806719715008512E+00Q, /* 3fff04080ab55de39000000000000000 */ + 1.01972323271377413034244341361045372E+00Q, /* 3fff050c94ef7a206c00000000000000 */ + 1.02371431660235789884438872832106426E+00Q, /* 3fff06122436410dd000000000000000 */ + 1.02772102115162167201845022646011785E+00Q, /* 3fff0718b98f42085000000000000000 */ + 1.03174340749910264936062276319717057E+00Q, /* 3fff08205601127ec800000000000000 */ + 1.03578153702162378824169763902318664E+00Q, /* 3fff0928fa934ef90800000000000000 */ + 1.03983547133622999947277776300325058E+00Q, /* 3fff0a32a84e9c1f5800000000000000 */ + 1.04390527230112850620713516036630608E+00Q, /* 3fff0b3d603ca7c32800000000000000 */ + 1.04799100201663270004459604933799710E+00Q, /* 3fff0c49236829e8bc00000000000000 */ + 1.05209272282610977189420964350574650E+00Q, /* 3fff0d55f2dce5d1e800000000000000 */ + 1.05621049731693195106174698594259098E+00Q, /* 3fff0e63cfa7ab09d000000000000000 */ + 1.06034438832143151909548350886325352E+00Q, /* 3fff0f72bad65671b800000000000000 */ + 1.06449445891785943185681162503897212E+00Q, /* 3fff1082b577d34ed800000000000000 */ + 1.06866077243134810492719566354935523E+00Q, /* 3fff1193c09c1c595c00000000000000 */ + 1.07284339243487741866189821848820429E+00Q, /* 3fff12a5dd543ccc4c00000000000000 */ + 1.07704238275024494209120007326419000E+00Q, /* 3fff13b90cb25176a400000000000000 */ + 1.08125780744903959851299646288680378E+00Q, /* 3fff14cd4fc989cd6400000000000000 */ + 1.08548973085361949442173568058933597E+00Q, /* 3fff15e2a7ae28fecc00000000000000 */ + 1.08973821753809324563988525369495619E+00Q, /* 3fff16f9157587069400000000000000 */ + 1.09400333232930546678574046381982043E+00Q, /* 3fff18109a3611c35000000000000000 */ + 1.09828514030782586896606289883493446E+00Q, /* 3fff192937074e0cd800000000000000 */ + 1.10258370680894224324930519287590869E+00Q, /* 3fff1a42ed01d8cbc800000000000000 */ + 1.10689909742365749645287564817408565E+00Q, /* 3fff1b5dbd3f68122400000000000000 */ + 1.11123137799969046168868658241990488E+00Q, /* 3fff1c79a8dacc350c00000000000000 */ + 1.11558061464248076122274255794764031E+00Q, /* 3fff1d96b0eff0e79400000000000000 */ + 1.11994687371619722204840741142106708E+00Q, /* 3fff1eb4d69bde569c00000000000000 */ + 1.12433022184475073235176978414529003E+00Q, /* 3fff1fd41afcba45e800000000000000 */ + 1.12873072591281087273529237791080959E+00Q, /* 3fff20f47f31c92e4800000000000000 */ + 1.13314845306682632219974493636982515E+00Q, /* 3fff2216045b6f5cd000000000000000 */ + 1.13758347071604959399593326452304609E+00Q, /* 3fff2338ab9b32134800000000000000 */ + 1.14203584653356560174586320499656722E+00Q, /* 3fff245c7613b8a9b000000000000000 */ + 1.14650564845732405583333957110880874E+00Q, /* 3fff258164e8cdb0d800000000000000 */ + 1.15099294469117646722011727433709893E+00Q, /* 3fff26a7793f60164400000000000000 */ + 1.15549780370591653744227755851170514E+00Q, /* 3fff27ceb43d84490400000000000000 */ + 1.16002029424032515603215642840950750E+00Q, /* 3fff28f7170a755fd800000000000000 */ + 1.16456048530221917269855680387991015E+00Q, /* 3fff2a20a2ce96406400000000000000 */ + 1.16911844616950438835445424956560601E+00Q, /* 3fff2b4b58b372c79400000000000000 */ + 1.17369424639123270948104504896036815E+00Q, /* 3fff2c7739e3c0f32c00000000000000 */ + 1.17828795578866324378353169777255971E+00Q, /* 3fff2da4478b620c7400000000000000 */ + 1.18289964445632783673900689791480545E+00Q, /* 3fff2ed282d763d42400000000000000 */ + 1.18752938276310060494722620205720887E+00Q, /* 3fff3001ecf601af7000000000000000 */ + 1.19217724135327157730657177125976887E+00Q, /* 3fff31328716a5d63c00000000000000 */ + 1.19684329114762477708211463323095813E+00Q, /* 3fff32645269ea829000000000000000 */ + 1.20152760334452030077656559114984702E+00Q, /* 3fff339750219b212c00000000000000 */ + 1.20623024942098072687102217059873510E+00Q, /* 3fff34cb8170b5835400000000000000 */ + 1.21095130113378179892436037334846333E+00Q, /* 3fff3600e78b6b11d000000000000000 */ + 1.21569083052054743854242246925423387E+00Q, /* 3fff373783a722012400000000000000 */ + 1.22044890990084875515009343871497549E+00Q, /* 3fff386f56fa7686e800000000000000 */ + 1.22522561187730755216662714701669756E+00Q, /* 3fff39a862bd3c106400000000000000 */ + 1.23002100933670455162882717559114099E+00Q, /* 3fff3ae2a8287e7a8000000000000000 */ + 1.23483517545109100499445276000187732E+00Q, /* 3fff3c1e2876834aa800000000000000 */ + 1.23966818367890557750499169742397498E+00Q, /* 3fff3d5ae4e2cae92c00000000000000 */ + 1.24452010776609517384017067342938390E+00Q, /* 3fff3e98deaa11dcbc00000000000000 */ + 1.24939102174724003813111039562500082E+00Q, /* 3fff3fd8170a52071800000000000000 */ + 1.25428099994668373895478907797951251E+00Q, /* 3fff41188f42c3e32000000000000000 */ + 1.25919011697966698459794088194030337E+00Q, /* 3fff425a4893dfc3f800000000000000 */ + 1.26411844775346637881341393949696794E+00Q, /* 3fff439d443f5f159000000000000000 */ + 1.26906606746853711786826579555054195E+00Q, /* 3fff44e183883d9e4800000000000000 */ + 1.27403305161966090564007458851847332E+00Q, /* 3fff462707b2bac20c00000000000000 */ + 1.27901947599709753244923149395617656E+00Q, /* 3fff476dd2045ac67800000000000000 */ + 1.28402541668774150540599521264084615E+00Q, /* 3fff48b5e3c3e8186800000000000000 */ + 1.28905095007628295311619126550795045E+00Q, /* 3fff49ff3e397492bc00000000000000 */ + 1.29409615284637330434591717676084954E+00Q, /* 3fff4b49e2ae5ac67400000000000000 */ + 1.29916110198179535206719492634874769E+00Q, /* 3fff4c95d26d3f440800000000000000 */ + 1.30424587476763775839572190307080746E+00Q, /* 3fff4de30ec211e60000000000000000 */ + 1.30935054879147461104338390214252286E+00Q, /* 3fff4f3198fa0f1cf800000000000000 */ + 1.31447520194454914310711046709911898E+00Q, /* 3fff50817263c13cd000000000000000 */ + 1.31961991242296217130558488861424848E+00Q, /* 3fff51d29c4f01cb3000000000000000 */ + 1.32478475872886558573071624778094701E+00Q, /* 3fff5325180cfacf7800000000000000 */ + 1.32996981967165983640200010995613411E+00Q, /* 3fff5478e6f02823d000000000000000 */ + 1.33517517436919680440254865061433520E+00Q, /* 3fff55ce0a4c58c7bc00000000000000 */ + 1.34040090224898678084031189428060316E+00Q, /* 3fff57248376b033d800000000000000 */ + 1.34564708304941055283521222918352578E+00Q, /* 3fff587c53c5a7af0400000000000000 */ + 1.35091379682093615244298234756570309E+00Q, /* 3fff59d57c910fa4e000000000000000 */ + 1.35620112392734021300455538039386738E+00Q, /* 3fff5b2fff3210fd9400000000000000 */ + 1.36150914504693443252136830778908916E+00Q, /* 3fff5c8bdd032e770800000000000000 */ + 1.36683794117379636690046140756749082E+00Q, /* 3fff5de9176045ff5400000000000000 */ + 1.37218759361900544124779344201670028E+00Q, /* 3fff5f47afa69210a800000000000000 */ + 1.37755818401188367960941150158760138E+00Q, /* 3fff60a7a734ab0e8800000000000000 */ + 1.38294979430124120867162673675920814E+00Q, /* 3fff6208ff6a88a46000000000000000 */ + 1.38836250675662681297595213436579797E+00Q, /* 3fff636bb9a983258400000000000000 */ + 1.39379640396958309755959248832368758E+00Q, /* 3fff64cfd75454ee7c00000000000000 */ + 1.39925156885490681313299887733592186E+00Q, /* 3fff663559cf1bc7c400000000000000 */ + 1.40472808465191417726103395580139477E+00Q, /* 3fff679c427f5a49f400000000000000 */ + 1.41022603492571069194738697660795879E+00Q, /* 3fff690492cbf9432c00000000000000 */ + 1.41574550356846662335641440222389065E+00Q, /* 3fff6a6e4c1d491e1800000000000000 */ + + 9.98018323540573404351050612604012713E-01Q, /* 3ffefefc41f8d4bdb000000000000000 */ + 9.98048781107475468932221929208026268E-01Q, /* 3ffeff003ff556aa8800000000000000 */ + 9.98079239603882895082165305211674422E-01Q, /* 3ffeff043df9d4986000000000000000 */ + 9.98109699029824021243584297735651489E-01Q, /* 3ffeff083c064e972c00000000000000 */ + 9.98140159385327269125909310787392315E-01Q, /* 3ffeff0c3a1ac4b6ec00000000000000 */ + 9.98170620670420977171843901487591211E-01Q, /* 3ffeff10383737079400000000000000 */ + 9.98201082885133511579667242585856002E-01Q, /* 3ffeff14365ba5991c00000000000000 */ + 9.98231546029493238547658506831794512E-01Q, /* 3ffeff183488107b7c00000000000000 */ + 9.98262010103528552029672482603928074E-01Q, /* 3ffeff1c32bc77beb000000000000000 */ + 9.98292475107267818223988342651864514E-01Q, /* 3ffeff2030f8db72b000000000000000 */ + 9.98322941040739375573309644096298143E-01Q, /* 3ffeff242f3d3ba77000000000000000 */ + 9.98353407903971645787066790944663808E-01Q, /* 3ffeff282d89986cf000000000000000 */ + 9.98383875696992967307963340317655820E-01Q, /* 3ffeff2c2bddf1d32400000000000000 */ + 9.98414344419831761845429696222709026E-01Q, /* 3ffeff302a3a47ea0c00000000000000 */ + 9.98444814072516340086593800151604228E-01Q, /* 3ffeff34289e9ac19800000000000000 */ + 9.98475284655075123740886056111776270E-01Q, /* 3ffeff38270aea69c800000000000000 */ + 9.98505756167536479006585636852832977E-01Q, /* 3ffeff3c257f36f29400000000000000 */ + 9.98536228609928799837547330753295682E-01Q, /* 3ffeff4023fb806bf800000000000000 */ + 9.98566701982280452432050310562772211E-01Q, /* 3ffeff44227fc6e5ec00000000000000 */ + 9.98597176284619802988373749030870385E-01Q, /* 3ffeff48210c0a706800000000000000 */ + 9.98627651516975245460372434536111541E-01Q, /* 3ffeff4c1fa04b1b6800000000000000 */ + 9.98658127679375173801901155457017012E-01Q, /* 3ffeff501e3c88f6e800000000000000 */ + 9.98688604771847954211239084543194622E-01Q, /* 3ffeff541ce0c412e000000000000000 */ + 9.98719082794421980642241010173165705E-01Q, /* 3ffeff581b8cfc7f4c00000000000000 */ + 9.98749561747125619293186105096538085E-01Q, /* 3ffeff5c1a41324c2400000000000000 */ + 9.98780041629987291873504773320746608E-01Q, /* 3ffeff6018fd65896800000000000000 */ + 9.98810522443035364581476187595399097E-01Q, /* 3ffeff6417c196471000000000000000 */ + 9.98841004186298203615379520670103375E-01Q, /* 3ffeff68168dc4951400000000000000 */ + 9.98871486859804230684645176552294288E-01Q, /* 3ffeff6c1561f0837400000000000000 */ + 9.98901970463581839743127943620493170E-01Q, /* 3ffeff70143e1a222c00000000000000 */ + 9.98932454997659369233531378995394334E-01Q, /* 3ffeff74132241813000000000000000 */ + 9.98962940462065268620861502313346136E-01Q, /* 3ffeff78120e66b08400000000000000 */ + 9.98993426856827904103397486323956400E-01Q, /* 3ffeff7c110289c02000000000000000 */ + 9.99023914181975669634994119405746460E-01Q, /* 3ffeff800ffeaac00000000000000000 */ + 9.99054402437536959169506189937237650E-01Q, /* 3ffeff840f02c9c02000000000000000 */ + 9.99084891623540138905212870668037795E-01Q, /* 3ffeff880e0ee6d07800000000000000 */ + 9.99115381740013658307120181234495249E-01Q, /* 3ffeff8c0d2302010c00000000000000 */ + 9.99145872786985911329082910015131347E-01Q, /* 3ffeff900c3f1b61d800000000000000 */ + 9.99176364764485236413804614130640402E-01Q, /* 3ffeff940b633302d000000000000000 */ + 9.99206857672540083026291313217370771E-01Q, /* 3ffeff980a8f48f3f800000000000000 */ + 9.99237351511178817364822180024930276E-01Q, /* 3ffeff9c09c35d454800000000000000 */ + 9.99267846280429861138827618560753763E-01Q, /* 3ffeffa008ff7006c000000000000000 */ + 9.99298341980321608302162417203362565E-01Q, /* 3ffeffa4084381485c00000000000000 */ + 9.99328838610882452808681364331278019E-01Q, /* 3ffeffa8078f911a1800000000000000 */ + 9.99359336172140816367814863951934967E-01Q, /* 3ffeffac06e39f8bf400000000000000 */ + 9.99389834664125092933417704443854745E-01Q, /* 3ffeffb0063facadec00000000000000 */ + 9.99420334086863676459344674185558688E-01Q, /* 3ffeffb405a3b88ffc00000000000000 */ + 9.99450834440384988655026177184481639E-01Q, /* 3ffeffb8050fc3422400000000000000 */ + 9.99481335724717395718741386190231424E-01Q, /* 3ffeffbc0483ccd45c00000000000000 */ + 9.99511837939889374871071936468069907E-01Q, /* 3ffeffc003ffd556ac00000000000000 */ + 9.99542341085929264554721385138691403E-01Q, /* 3ffeffc40383dcd90800000000000000 */ + 9.99572845162865514234695751838444266E-01Q, /* 3ffeffc8030fe36b7400000000000000 */ + 9.99603350170726517864849824945849832E-01Q, /* 3ffeffcc02a3e91dec00000000000000 */ + 9.99633856109540669399038392839429434E-01Q, /* 3ffeffd0023fee006c00000000000000 */ + 9.99664362979336418302267475155531429E-01Q, /* 3ffeffd401e3f222f800000000000000 */ + 9.99694870780142130772816244643763639E-01Q, /* 3ffeffd8018ff5958800000000000000 */ + 9.99725379511986284031266336569387931E-01Q, /* 3ffeffdc0143f8682400000000000000 */ + 9.99755889174897216520321308053098619E-01Q, /* 3ffeffe000fffaaac000000000000000 */ + 9.99786399768903377704987178731244057E-01Q, /* 3ffeffe400c3fc6d6000000000000000 */ + 9.99816911294033217050269968240172602E-01Q, /* 3ffeffe8008ffdc00800000000000000 */ + 9.99847423750315072998873233700578567E-01Q, /* 3ffeffec0063feb2ac00000000000000 */ + 9.99877937137777450526954226006637327E-01Q, /* 3ffefff0003fff555800000000000000 */ + 9.99908451456448688077216502279043198E-01Q, /* 3ffefff40023ffb80000000000000000 */ + 9.99938966706357262870241697783058044E-01Q, /* 3ffefff8000fffeaac00000000000000 */ + 9.99969482887531541104308985268289689E-01Q, /* 3ffefffc0003fffd5400000000000000 */ +#define T_EXPL_RES2 (T_EXPL_RES1 + 1 + 89 + 65) + 1.00000000000000000000000000000000000E+00Q, /* 3fff0000000000000000000000000000 */ + 1.00003051804379100575559391472779680E+00Q, /* 3fff0002000200015400000000000000 */ + 1.00006103701893306334724798034585547E+00Q, /* 3fff00040008000aac00000000000000 */ + 1.00009155692545448346209013834595680E+00Q, /* 3fff0006001200240000000000000000 */ + 1.00012207776338379883185325525118969E+00Q, /* 3fff0008002000555800000000000000 */ + 1.00015259953274932014366527255333494E+00Q, /* 3fff000a003200a6ac00000000000000 */ + 1.00018312223357958012925905677548144E+00Q, /* 3fff000c004801200400000000000000 */ + 1.00021364586590294498691378066723701E+00Q, /* 3fff000e006201c95c00000000000000 */ + 1.00024417042974783642605984823603649E+00Q, /* 3fff0010008002aab400000000000000 */ + 1.00027469592514273166727889474714175E+00Q, /* 3fff001200a203cc1000000000000000 */ + 1.00030522235211605242000132420798764E+00Q, /* 3fff001400c805357000000000000000 */ + 1.00033574971069616488250630936818197E+00Q, /* 3fff001600f206eed000000000000000 */ + 1.00036627800091160178652671675081365E+00Q, /* 3fff0018012009003800000000000000 */ + 1.00039680722279067381919048784766346E+00Q, /* 3fff001a01520b71a000000000000000 */ + 1.00042733737636191371223048918182030E+00Q, /* 3fff001c01880e4b1000000000000000 */ + 1.00045786846165368766392589350289200E+00Q, /* 3fff001e01c211948400000000000000 */ + 1.00048840047869447289485833607614040E+00Q, /* 3fff0020020015560000000000000000 */ + 1.00051893342751269111445822090900037E+00Q, /* 3fff0022024219978400000000000000 */ + 1.00054946730813676403215595200890675E+00Q, /* 3fff002402881e611000000000000000 */ + 1.00058000212059516886853316464112140E+00Q, /* 3fff002602d223baa800000000000000 */ + 1.00061053786491632733302026281307917E+00Q, /* 3fff0028032029ac4c00000000000000 */ + 1.00064107454112866113504765053221490E+00Q, /* 3fff002a0372303dfc00000000000000 */ + 1.00067161214926059198404573180596344E+00Q, /* 3fff002c03c83777b800000000000000 */ + 1.00070215068934059710059614189958666E+00Q, /* 3fff002e04223f618400000000000000 */ + 1.00073269016139709819412928482051939E+00Q, /* 3fff0030048048036000000000000000 */ + 1.00076323056545857248522679583402351E+00Q, /* 3fff003204e251655000000000000000 */ + 1.00079377190155338617216784768970683E+00Q, /* 3fff003405485b8f5000000000000000 */ + 1.00082431416971007198668530691065826E+00Q, /* 3fff003605b266896800000000000000 */ + 1.00085485736995705163820957750431262E+00Q, /* 3fff00380620725b9800000000000000 */ + 1.00088540150232269132501983222027775E+00Q, /* 3fff003a06927f0ddc00000000000000 */ + 1.00091594656683552377884893758164253E+00Q, /* 3fff003c07088ca83c00000000000000 */ + 1.00094649256352402622027852885366883E+00Q, /* 3fff003e07829b32bc00000000000000 */ + 1.00097703949241650933643654752813745E+00Q, /* 3fff00400800aab55400000000000000 */ + 1.00100758735354156137020709138596430E+00Q, /* 3fff00420882bb381000000000000000 */ + 1.00103813614692760403102056443458423E+00Q, /* 3fff00440908ccc2f000000000000000 */ + 1.00106868587260300351715613942360505E+00Q, /* 3fff00460992df5df000000000000000 */ + 1.00109923653059629256034668287611566E+00Q, /* 3fff00480a20f3111800000000000000 */ + 1.00112978812093589287002259879955091E+00Q, /* 3fff004a0ab307e46800000000000000 */ + 1.00116034064365022615561429120134562E+00Q, /* 3fff004c0b491ddfe000000000000000 */ + 1.00119089409876788066000585786241572E+00Q, /* 3fff004e0be3350b8c00000000000000 */ + 1.00122144848631711155917400901671499E+00Q, /* 3fff00500c814d6f6000000000000000 */ + 1.00125200380632656260715407370298635E+00Q, /* 3fff00520d2367136c00000000000000 */ + 1.00128256005882454449107399341301061E+00Q, /* 3fff00540dc981ffa800000000000000 */ + 1.00131311724383964545381786592770368E+00Q, /* 3fff00560e739e3c2000000000000000 */ + 1.00134367536140017618251363273884635E+00Q, /* 3fff00580f21bbd0cc00000000000000 */ + 1.00137423441153472492004539162735455E+00Q, /* 3fff005a0fd3dac5b800000000000000 */ + 1.00140479439427171337584354660066310E+00Q, /* 3fff005c1089fb22e400000000000000 */ + 1.00143535530963956325933850166620687E+00Q, /* 3fff005e11441cf05000000000000000 */ + 1.00146591715766680730226312334707472E+00Q, /* 3fff0060120240360400000000000000 */ + 1.00149647993838186721404781565070152E+00Q, /* 3fff006212c464fc0000000000000000 */ + 1.00152704365181316470412298258452211E+00Q, /* 3fff0064138a8b4a4400000000000000 */ + 1.00155760829798923250422149067162536E+00Q, /* 3fff00661454b328d800000000000000 */ + 1.00158817387693849232377374391944613E+00Q, /* 3fff00681522dc9fbc00000000000000 */ + 1.00161874038868942138336137759324629E+00Q, /* 3fff006a15f507b6f400000000000000 */ + 1.00164930783327055241471725821611471E+00Q, /* 3fff006c16cb34768800000000000000 */ + 1.00167987621071025161612055853765924E+00Q, /* 3fff006e17a562e67400000000000000 */ + 1.00171044552103705171930414508096874E+00Q, /* 3fff00701883930ec000000000000000 */ + 1.00174101576427937443369842185347807E+00Q, /* 3fff00721965c4f76c00000000000000 */ + 1.00177158694046569697988502412044909E+00Q, /* 3fff00741a4bf8a87c00000000000000 */ + 1.00180215904962455208959681840497069E+00Q, /* 3fff00761b362e29f800000000000000 */ + 1.00183273209178441698341543997230474E+00Q, /* 3fff00781c246583e400000000000000 */ + 1.00186330606697365785962006157205906E+00Q, /* 3fff007a1d169ebe3c00000000000000 */ + 1.00189388097522080744994354972732253E+00Q, /* 3fff007c1e0cd9e10800000000000000 */ + 1.00192445681655439848611877096118405E+00Q, /* 3fff007e1f0716f45000000000000000 */ + 1.00195503359100279716642489802325144E+00Q, /* 3fff0080200556001000000000000000 */ + 1.00198561129859459173374602869444061E+00Q, /* 3fff00822107970c5400000000000000 */ +}; + + + + + + + +/* The basic design here is from + Abraham Ziv, "Fast Evaluation of Elementary Mathematical Functions with + Correctly Rounded Last Bit", ACM Trans. Math. Soft., 17 (3), September 1991, + pp. 410-423. + + We work with number pairs where the first number is the high part and + the second one is the low part. Arithmetic with the high part numbers must + be exact, without any roundoff errors. + + The input value, X, is written as + X = n * ln(2)_0 + arg1[t1]_0 + arg2[t2]_0 + x + - n * ln(2)_1 + arg1[t1]_1 + arg2[t2]_1 + xl + + where: + - n is an integer, 16384 >= n >= -16495; + - ln(2)_0 is the first 93 bits of ln(2), and |ln(2)_0-ln(2)-ln(2)_1| < 2^-205 + - t1 is an integer, 89 >= t1 >= -89 + - t2 is an integer, 65 >= t2 >= -65 + - |arg1[t1]-t1/256.0| < 2^-53 + - |arg2[t2]-t2/32768.0| < 2^-53 + - x + xl is whatever is left, |x + xl| < 2^-16 + 2^-53 + + Then e^x is approximated as + + e^x = 2^n_1 ( 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1) + + 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1) + * p (x + xl + n * ln(2)_1)) + where: + - p(x) is a polynomial approximating e(x)-1 + - e^(arg1[t1]_0 + arg1[t1]_1) is obtained from a table + - e^(arg2[t2]_0 + arg2[t2]_1) likewise + - n_1 + n_0 = n, so that |n_0| < -LDBL_MIN_EXP-1. + + If it happens that n_1 == 0 (this is the usual case), that multiplication + is omitted. + */ + + +static const __float128 C[] = { +/* Smallest integer x for which e^x overflows. */ +#define himark C[0] + 11356.523406294143949491931077970765Q, + +/* Largest integer x for which e^x underflows. */ +#define lomark C[1] +-11433.4627433362978788372438434526231Q, + +/* 3x2^96 */ +#define THREEp96 C[2] + 59421121885698253195157962752.0Q, + +/* 3x2^103 */ +#define THREEp103 C[3] + 30423614405477505635920876929024.0Q, + +/* 3x2^111 */ +#define THREEp111 C[4] + 7788445287802241442795744493830144.0Q, + +/* 1/ln(2) */ +#define M_1_LN2 C[5] + 1.44269504088896340735992468100189204Q, + +/* first 93 bits of ln(2) */ +#define M_LN2_0 C[6] + 0.693147180559945309417232121457981864Q, + +/* ln2_0 - ln(2) */ +#define M_LN2_1 C[7] +-1.94704509238074995158795957333327386E-31Q, + +/* very small number */ +#define TINY C[8] + 1.0e-4900Q, + +/* 2^16383 */ +#define TWO16383 C[9] + 5.94865747678615882542879663314003565E+4931Q, + +/* 256 */ +#define TWO8 C[10] + 256.0Q, + +/* 32768 */ +#define TWO15 C[11] + 32768.0Q, + +/* Chebyshev polynom coeficients for (exp(x)-1)/x */ +#define P1 C[12] +#define P2 C[13] +#define P3 C[14] +#define P4 C[15] +#define P5 C[16] +#define P6 C[17] + 0.5Q, + 1.66666666666666666666666666666666683E-01Q, + 4.16666666666666666666654902320001674E-02Q, + 8.33333333333333333333314659767198461E-03Q, + 1.38888888889899438565058018857254025E-03Q, + 1.98412698413981650382436541785404286E-04Q, +}; + + +__float128 +expq (__float128 x) +{ + /* Check for usual case. */ + if (__builtin_isless (x, himark) && __builtin_isgreater (x, lomark)) + { + int tval1, tval2, unsafe, n_i; + __float128 x22, n, t, result, xl; + ieee854_float128 ex2_u, scale_u; +#ifdef USE_FENV_H + fenv_t oldenv; + + feholdexcept (&oldenv); +# ifdef FE_TONEAREST + fesetround (FE_TONEAREST); +# endif +#endif + + /* Calculate n. */ + n = x * M_1_LN2 + THREEp111; + n -= THREEp111; + x = x - n * M_LN2_0; + xl = n * M_LN2_1; + + /* Calculate t/256. */ + t = x + THREEp103; + t -= THREEp103; + + /* Compute tval1 = t. */ + tval1 = (int) (t * TWO8); + + x -= __expl_table[T_EXPL_ARG1+2*tval1]; + xl -= __expl_table[T_EXPL_ARG1+2*tval1+1]; + + /* Calculate t/32768. */ + t = x + THREEp96; + t -= THREEp96; + + /* Compute tval2 = t. */ + tval2 = (int) (t * TWO15); + + x -= __expl_table[T_EXPL_ARG2+2*tval2]; + xl -= __expl_table[T_EXPL_ARG2+2*tval2+1]; + + x = x + xl; + + /* Compute ex2 = 2^n_0 e^(argtable[tval1]) e^(argtable[tval2]). */ + ex2_u.value = __expl_table[T_EXPL_RES1 + tval1] + * __expl_table[T_EXPL_RES2 + tval2]; + n_i = (int)n; + /* 'unsafe' is 1 iff n_1 != 0. */ + unsafe = abs(n_i) >= -FLT128_MIN_EXP - 1; + ex2_u.ieee.exponent += n_i >> unsafe; + + /* Compute scale = 2^n_1. */ + scale_u.value = 1.0Q; + scale_u.ieee.exponent += n_i - (n_i >> unsafe); + + /* Approximate e^x2 - 1, using a seventh-degree polynomial, + with maximum error in [-2^-16-2^-53,2^-16+2^-53] + less than 4.8e-39. */ + x22 = x + x*x*(P1+x*(P2+x*(P3+x*(P4+x*(P5+x*P6))))); + + /* Return result. */ +#ifdef USE_FENV_H + fesetenv (&oldenv); +#endif + result = x22 * ex2_u.value + ex2_u.value; + + /* Now we can test whether the result is ultimate or if we are unsure. + In the later case we should probably call a mpn based routine to give + the ultimate result. + Empirically, this routine is already ultimate in about 99.9986% of + cases, the test below for the round to nearest case will be false + in ~ 99.9963% of cases. + Without proc2 routine maximum error which has been seen is + 0.5000262 ulp. + + union ieee854_long_double ex3_u; + +#ifdef USE_FENV_H + #ifdef FE_TONEAREST + fesetround (FE_TONEAREST); + #endif +#endif + ex3_u.d = (result - ex2_u.d) - x22 * ex2_u.d; + ex2_u.d = result; + ex3_u.ieee.exponent += LDBL_MANT_DIG + 15 + IEEE854_LONG_DOUBLE_BIAS + - ex2_u.ieee.exponent; + n_i = abs (ex3_u.d); + n_i = (n_i + 1) / 2; +#ifdef USE_FENV_H + fesetenv (&oldenv); + #ifdef FE_TONEAREST + if (fegetround () == FE_TONEAREST) + n_i -= 0x4000; + #endif +#endif + if (!n_i) { + return __ieee754_expl_proc2 (origx); + } + */ + if (!unsafe) + return result; + else + return result * scale_u.value; + } + /* Exceptional cases: */ + else if (__builtin_isless (x, himark)) + { + if (isinfq (x)) + /* e^-inf == 0, with no error. */ + return 0; + else + /* Underflow */ + return TINY * TINY; + } + + /* Return x, if x is a NaN or Inf; or overflow, otherwise. */ + return TWO16383*x; +} diff --git a/libquadmath/math/fabsq.c b/libquadmath/math/fabsq.c new file mode 100644 index 000000000..7ec2850fe --- /dev/null +++ b/libquadmath/math/fabsq.c @@ -0,0 +1,25 @@ +/* s_fabsl.c -- __float128 version of s_fabs.c. + * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include "quadmath-imp.h" + +__float128 +fabsq (__float128 x) +{ + uint64_t hx; + GET_FLT128_MSW64(hx,x); + SET_FLT128_MSW64(x,hx&0x7fffffffffffffffLL); + return x; +} diff --git a/libquadmath/math/fdimq.c b/libquadmath/math/fdimq.c new file mode 100644 index 000000000..539fb08c6 --- /dev/null +++ b/libquadmath/math/fdimq.c @@ -0,0 +1,43 @@ +/* Return positive difference between arguments. + Copyright (C) 1997, 2004, 2009 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include <errno.h> +#include "quadmath-imp.h" + +__float128 +fdimq (__float128 x, __float128 y) +{ + int clsx = fpclassifyq (x); + int clsy = fpclassifyq (y); + + if (clsx == QUADFP_NAN || clsy == QUADFP_NAN + || (y < 0 && clsx == QUADFP_INFINITE && clsy == QUADFP_INFINITE)) + /* Raise invalid flag. */ + return x - y; + + if (x <= y) + return 0.0Q; + + __float128 r = x - y; + if (isinfq (r)) + errno = ERANGE; + + return r; +} diff --git a/libquadmath/math/finiteq.c b/libquadmath/math/finiteq.c new file mode 100644 index 000000000..f22e9d7f2 --- /dev/null +++ b/libquadmath/math/finiteq.c @@ -0,0 +1,25 @@ +/* s_finitel.c -- long double version of s_finite.c. + * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include "quadmath-imp.h" + +int +finiteq (const __float128 x) +{ + int64_t hx; + GET_FLT128_MSW64(hx,x); + return (int)((uint64_t)((hx&0x7fffffffffffffffLL) + -0x7fff000000000000LL)>>63); +} diff --git a/libquadmath/math/floorq.c b/libquadmath/math/floorq.c new file mode 100644 index 000000000..781cf40e3 --- /dev/null +++ b/libquadmath/math/floorq.c @@ -0,0 +1,62 @@ +/* s_floorl.c -- long double version of s_floor.c. + * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include "quadmath-imp.h" + +static const __float128 huge = 1.0e4930Q; + +__float128 +floorq (__float128 x) +{ + int64_t i0,i1,j0; + uint64_t i,j; + GET_FLT128_WORDS64(i0,i1,x); + j0 = ((i0>>48)&0x7fff)-0x3fff; + if(j0<48) { + if(j0<0) { /* raise inexact if x != 0 */ + if(huge+x>0.0) {/* return 0*sign(x) if |x|<1 */ + if(i0>=0) {i0=i1=0;} + else if(((i0&0x7fffffffffffffffLL)|i1)!=0) + { i0=0xbfff000000000000ULL;i1=0;} + } + } else { + i = (0x0000ffffffffffffULL)>>j0; + if(((i0&i)|i1)==0) return x; /* x is integral */ + if(huge+x>0.0) { /* raise inexact flag */ + if(i0<0) i0 += (0x0001000000000000LL)>>j0; + i0 &= (~i); i1=0; + } + } + } else if (j0>111) { + if(j0==0x4000) return x+x; /* inf or NaN */ + else return x; /* x is integral */ + } else { + i = -1ULL>>(j0-48); + if((i1&i)==0) return x; /* x is integral */ + if(huge+x>0.0) { /* raise inexact flag */ + if(i0<0) { + if(j0==48) i0+=1; + else { + j = i1+(1LL<<(112-j0)); + if(j<i1) i0 +=1 ; /* got a carry */ + i1=j; + } + } + i1 &= (~i); + } + } + SET_FLT128_WORDS64(x,i0,i1); + return x; +} diff --git a/libquadmath/math/fmaq.c b/libquadmath/math/fmaq.c new file mode 100644 index 000000000..126b0a2d2 --- /dev/null +++ b/libquadmath/math/fmaq.c @@ -0,0 +1,241 @@ +/* Compute x * y + z as ternary operation. + Copyright (C) 2010 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Jakub Jelinek <jakub@redhat.com>, 2010. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" +#include <math.h> +#include <float.h> +#ifdef HAVE_FENV_H +# include <fenv.h> +# if defined HAVE_FEHOLDEXCEPT && defined HAVE_FESETROUND \ + && defined HAVE_FEUPDATEENV && defined HAVE_FETESTEXCEPT \ + && defined FE_TOWARDZERO && defined FE_INEXACT +# define USE_FENV_H +# endif +#endif + +/* This implementation uses rounding to odd to avoid problems with + double rounding. See a paper by Boldo and Melquiond: + http://www.lri.fr/~melquion/doc/08-tc.pdf */ + +__float128 +fmaq (__float128 x, __float128 y, __float128 z) +{ + ieee854_float128 u, v, w; + int adjust = 0; + u.value = x; + v.value = y; + w.value = z; + if (__builtin_expect (u.ieee.exponent + v.ieee.exponent + >= 0x7fff + IEEE854_FLOAT128_BIAS + - FLT128_MANT_DIG, 0) + || __builtin_expect (u.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0) + || __builtin_expect (v.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0) + || __builtin_expect (w.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0) + || __builtin_expect (u.ieee.exponent + v.ieee.exponent + <= IEEE854_FLOAT128_BIAS + FLT128_MANT_DIG, 0)) + { + /* If z is Inf, but x and y are finite, the result should be + z rather than NaN. */ + if (w.ieee.exponent == 0x7fff + && u.ieee.exponent != 0x7fff + && v.ieee.exponent != 0x7fff) + return (z + x) + y; + /* If x or y or z is Inf/NaN, or if fma will certainly overflow, + or if x * y is less than half of FLT128_DENORM_MIN, + compute as x * y + z. */ + if (u.ieee.exponent == 0x7fff + || v.ieee.exponent == 0x7fff + || w.ieee.exponent == 0x7fff + || u.ieee.exponent + v.ieee.exponent + > 0x7fff + IEEE854_FLOAT128_BIAS + || u.ieee.exponent + v.ieee.exponent + < IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG - 2) + return x * y + z; + if (u.ieee.exponent + v.ieee.exponent + >= 0x7fff + IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG) + { + /* Compute 1p-113 times smaller result and multiply + at the end. */ + if (u.ieee.exponent > v.ieee.exponent) + u.ieee.exponent -= FLT128_MANT_DIG; + else + v.ieee.exponent -= FLT128_MANT_DIG; + /* If x + y exponent is very large and z exponent is very small, + it doesn't matter if we don't adjust it. */ + if (w.ieee.exponent > FLT128_MANT_DIG) + w.ieee.exponent -= FLT128_MANT_DIG; + adjust = 1; + } + else if (w.ieee.exponent >= 0x7fff - FLT128_MANT_DIG) + { + /* Similarly. + If z exponent is very large and x and y exponents are + very small, it doesn't matter if we don't adjust it. */ + if (u.ieee.exponent > v.ieee.exponent) + { + if (u.ieee.exponent > FLT128_MANT_DIG) + u.ieee.exponent -= FLT128_MANT_DIG; + } + else if (v.ieee.exponent > FLT128_MANT_DIG) + v.ieee.exponent -= FLT128_MANT_DIG; + w.ieee.exponent -= FLT128_MANT_DIG; + adjust = 1; + } + else if (u.ieee.exponent >= 0x7fff - FLT128_MANT_DIG) + { + u.ieee.exponent -= FLT128_MANT_DIG; + if (v.ieee.exponent) + v.ieee.exponent += FLT128_MANT_DIG; + else + v.value *= 0x1p113Q; + } + else if (v.ieee.exponent >= 0x7fff - FLT128_MANT_DIG) + { + v.ieee.exponent -= FLT128_MANT_DIG; + if (u.ieee.exponent) + u.ieee.exponent += FLT128_MANT_DIG; + else + u.value *= 0x1p113Q; + } + else /* if (u.ieee.exponent + v.ieee.exponent + <= IEEE854_FLOAT128_BIAS + FLT128_MANT_DIG) */ + { + if (u.ieee.exponent > v.ieee.exponent) + u.ieee.exponent += 2 * FLT128_MANT_DIG; + else + v.ieee.exponent += 2 * FLT128_MANT_DIG; + if (w.ieee.exponent <= 4 * FLT128_MANT_DIG + 4) + { + if (w.ieee.exponent) + w.ieee.exponent += 2 * FLT128_MANT_DIG; + else + w.value *= 0x1p226Q; + adjust = -1; + } + /* Otherwise x * y should just affect inexact + and nothing else. */ + } + x = u.value; + y = v.value; + z = w.value; + } + /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */ +#define C ((1LL << (FLT128_MANT_DIG + 1) / 2) + 1) + __float128 x1 = x * C; + __float128 y1 = y * C; + __float128 m1 = x * y; + x1 = (x - x1) + x1; + y1 = (y - y1) + y1; + __float128 x2 = x - x1; + __float128 y2 = y - y1; + __float128 m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2; + + /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */ + __float128 a1 = z + m1; + __float128 t1 = a1 - z; + __float128 t2 = a1 - t1; + t1 = m1 - t1; + t2 = z - t2; + __float128 a2 = t1 + t2; + +#ifdef USE_FENV_H + fenv_t env; + feholdexcept (&env); + fesetround (FE_TOWARDZERO); +#endif + /* Perform m2 + a2 addition with round to odd. */ + u.value = a2 + m2; + + if (__builtin_expect (adjust == 0, 1)) + { +#ifdef USE_FENV_H + if ((u.ieee.mant_low & 1) == 0 && u.ieee.exponent != 0x7fff) + u.ieee.mant_low |= fetestexcept (FE_INEXACT) != 0; + feupdateenv (&env); +#endif + /* Result is a1 + u.value. */ + return a1 + u.value; + } + else if (__builtin_expect (adjust > 0, 1)) + { +#ifdef USE_FENV_H + if ((u.ieee.mant_low & 1) == 0 && u.ieee.exponent != 0x7fff) + u.ieee.mant_low |= fetestexcept (FE_INEXACT) != 0; + feupdateenv (&env); +#endif + /* Result is a1 + u.value, scaled up. */ + return (a1 + u.value) * 0x1p113Q; + } + else + { +#ifdef USE_FENV_H + if ((u.ieee.mant_low & 1) == 0) + u.ieee.mant_low |= fetestexcept (FE_INEXACT) != 0; +#endif + v.value = a1 + u.value; + /* Ensure the addition is not scheduled after fetestexcept call. */ + asm volatile ("" : : "m" (v)); +#ifdef USE_FENV_H + int j = fetestexcept (FE_INEXACT) != 0; + feupdateenv (&env); +#else + int j = 0; +#endif + /* Ensure the following computations are performed in default rounding + mode instead of just reusing the round to zero computation. */ + asm volatile ("" : "=m" (u) : "m" (u)); + /* If a1 + u.value is exact, the only rounding happens during + scaling down. */ + if (j == 0) + return v.value * 0x1p-226Q; + /* If result rounded to zero is not subnormal, no double + rounding will occur. */ + if (v.ieee.exponent > 226) + return (a1 + u.value) * 0x1p-226Q; + /* If v.value * 0x1p-226Q with round to zero is a subnormal above + or equal to FLT128_MIN / 2, then v.value * 0x1p-226Q shifts mantissa + down just by 1 bit, which means v.ieee.mant_low |= j would + change the round bit, not sticky or guard bit. + v.value * 0x1p-226Q never normalizes by shifting up, + so round bit plus sticky bit should be already enough + for proper rounding. */ + if (v.ieee.exponent == 226) + { + /* v.ieee.mant_low & 2 is LSB bit of the result before rounding, + v.ieee.mant_low & 1 is the round bit and j is our sticky + bit. In round-to-nearest 001 rounds down like 00, + 011 rounds up, even though 01 rounds down (thus we need + to adjust), 101 rounds down like 10 and 111 rounds up + like 11. */ + if ((v.ieee.mant_low & 3) == 1) + { + v.value *= 0x1p-226Q; + if (v.ieee.negative) + return v.value - 0x1p-16494Q /* __FLT128_DENORM_MIN__ */; + else + return v.value + 0x1p-16494Q /* __FLT128_DENORM_MIN__ */; + } + else + return v.value * 0x1p-226Q; + } + v.ieee.mant_low |= j; + return v.value * 0x1p-226Q; + } +} diff --git a/libquadmath/math/fmaxq.c b/libquadmath/math/fmaxq.c new file mode 100644 index 000000000..e8ed6f440 --- /dev/null +++ b/libquadmath/math/fmaxq.c @@ -0,0 +1,28 @@ +/* Return maximum numeric value of X and Y. + Copyright (C) 1997 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + + +__float128 +fmaxq (__float128 x, __float128 y) +{ + return (__builtin_isgreaterequal (x, y) || isnanq (y)) ? x : y; +} diff --git a/libquadmath/math/fminq.c b/libquadmath/math/fminq.c new file mode 100644 index 000000000..2fbe4116a --- /dev/null +++ b/libquadmath/math/fminq.c @@ -0,0 +1,28 @@ +/* Return minimum numeric value of X and Y. + Copyright (C) 1997 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + + +__float128 +fminq (__float128 x, __float128 y) +{ + return (__builtin_islessequal (x, y) || isnanq (y)) ? x : y; +} diff --git a/libquadmath/math/fmodq.c b/libquadmath/math/fmodq.c new file mode 100644 index 000000000..036e3dd53 --- /dev/null +++ b/libquadmath/math/fmodq.c @@ -0,0 +1,129 @@ +/* e_fmodl.c -- long double version of e_fmod.c. + * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz. + */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* + * fmodq(x,y) + * Return x mod y in exact arithmetic + * Method: shift and subtract + */ + +#include "quadmath-imp.h" + +static const __float128 one = 1.0, Zero[] = {0.0, -0.0,}; + +__float128 +fmodq (__float128 x, __float128 y) +{ + int64_t n,hx,hy,hz,ix,iy,sx,i; + uint64_t lx,ly,lz; + + GET_FLT128_WORDS64(hx,lx,x); + GET_FLT128_WORDS64(hy,ly,y); + sx = hx&0x8000000000000000ULL; /* sign of x */ + hx ^=sx; /* |x| */ + hy &= 0x7fffffffffffffffLL; /* |y| */ + + /* purge off exception values */ + if((hy|ly)==0||(hx>=0x7fff000000000000LL)|| /* y=0,or x not finite */ + ((hy|((ly|-ly)>>63))>0x7fff000000000000LL)) /* or y is NaN */ + return (x*y)/(x*y); + if(hx<=hy) { + if((hx<hy)||(lx<ly)) return x; /* |x|<|y| return x */ + if(lx==ly) + return Zero[(uint64_t)sx>>63]; /* |x|=|y| return x*0*/ + } + + /* determine ix = ilogb(x) */ + if(hx<0x0001000000000000LL) { /* subnormal x */ + if(hx==0) { + for (ix = -16431, i=lx; i>0; i<<=1) ix -=1; + } else { + for (ix = -16382, i=hx<<15; i>0; i<<=1) ix -=1; + } + } else ix = (hx>>48)-0x3fff; + + /* determine iy = ilogb(y) */ + if(hy<0x0001000000000000LL) { /* subnormal y */ + if(hy==0) { + for (iy = -16431, i=ly; i>0; i<<=1) iy -=1; + } else { + for (iy = -16382, i=hy<<15; i>0; i<<=1) iy -=1; + } + } else iy = (hy>>48)-0x3fff; + + /* set up {hx,lx}, {hy,ly} and align y to x */ + if(ix >= -16382) + hx = 0x0001000000000000LL|(0x0000ffffffffffffLL&hx); + else { /* subnormal x, shift x to normal */ + n = -16382-ix; + if(n<=63) { + hx = (hx<<n)|(lx>>(64-n)); + lx <<= n; + } else { + hx = lx<<(n-64); + lx = 0; + } + } + if(iy >= -16382) + hy = 0x0001000000000000LL|(0x0000ffffffffffffLL&hy); + else { /* subnormal y, shift y to normal */ + n = -16382-iy; + if(n<=63) { + hy = (hy<<n)|(ly>>(64-n)); + ly <<= n; + } else { + hy = ly<<(n-64); + ly = 0; + } + } + + /* fix point fmod */ + n = ix - iy; + while(n--) { + hz=hx-hy;lz=lx-ly; if(lx<ly) hz -= 1; + if(hz<0){hx = hx+hx+(lx>>63); lx = lx+lx;} + else { + if((hz|lz)==0) /* return sign(x)*0 */ + return Zero[(uint64_t)sx>>63]; + hx = hz+hz+(lz>>63); lx = lz+lz; + } + } + hz=hx-hy;lz=lx-ly; if(lx<ly) hz -= 1; + if(hz>=0) {hx=hz;lx=lz;} + + /* convert back to floating value and restore the sign */ + if((hx|lx)==0) /* return sign(x)*0 */ + return Zero[(uint64_t)sx>>63]; + while(hx<0x0001000000000000LL) { /* normalize x */ + hx = hx+hx+(lx>>63); lx = lx+lx; + iy -= 1; + } + if(iy>= -16382) { /* normalize output */ + hx = ((hx-0x0001000000000000LL)|((iy+16383)<<48)); + SET_FLT128_WORDS64(x,hx|sx,lx); + } else { /* subnormal output */ + n = -16382 - iy; + if(n<=48) { + lx = (lx>>n)|((uint64_t)hx<<(64-n)); + hx >>= n; + } else if (n<=63) { + lx = (hx<<(64-n))|(lx>>n); hx = sx; + } else { + lx = hx>>(n-64); hx = sx; + } + SET_FLT128_WORDS64(x,hx|sx,lx); + x *= one; /* create necessary signal */ + } + return x; /* exact output */ +} diff --git a/libquadmath/math/frexpq.c b/libquadmath/math/frexpq.c new file mode 100644 index 000000000..fa3d7836d --- /dev/null +++ b/libquadmath/math/frexpq.c @@ -0,0 +1,49 @@ +/* s_frexpl.c -- long double version of s_frexp.c. + * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* + * for non-zero x + * x = frexpq(arg,&exp); + * return a __float128 fp quantity x such that 0.5 <= |x| <1.0 + * and the corresponding binary exponent "exp". That is + * arg = x*2^exp. + * If arg is inf, 0.0, or NaN, then frexpq(arg,&exp) returns arg + * with *exp=0. + */ + +#include "quadmath-imp.h" + +static const __float128 +two114 = 2.0769187434139310514121985316880384E+34Q; /* 0x4071000000000000, 0 */ + +__float128 +frexpq (__float128 x, int *eptr) +{ + uint64_t hx, lx, ix; + GET_FLT128_WORDS64(hx,lx,x); + ix = 0x7fffffffffffffffULL&hx; + *eptr = 0; + if(ix>=0x7fff000000000000ULL||((ix|lx)==0)) return x; /* 0,inf,nan */ + if (ix<0x0001000000000000ULL) { /* subnormal */ + x *= two114; + GET_FLT128_MSW64(hx,x); + ix = hx&0x7fffffffffffffffULL; + *eptr = -114; + } + *eptr += (ix>>48)-16382; + hx = (hx&0x8000ffffffffffffULL) | 0x3ffe000000000000ULL; + SET_FLT128_MSW64(x,hx); + return x; +} diff --git a/libquadmath/math/hypotq.c b/libquadmath/math/hypotq.c new file mode 100644 index 000000000..2df317f36 --- /dev/null +++ b/libquadmath/math/hypotq.c @@ -0,0 +1,124 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* From e_hypotl.c -- long double version of e_hypot.c. + * Conversion to long double by Jakub Jelinek, jakub@redhat.com. + * Conversion to __float128 by FX Coudert, fxcoudert@gcc.gnu.org. + */ + +/* hypotq(x,y) + * + * Method : + * If (assume round-to-nearest) z=x*x+y*y + * has error less than sqrtl(2)/2 ulp, than + * sqrtl(z) has error less than 1 ulp (exercise). + * + * So, compute sqrtl(x*x+y*y) with some care as + * follows to get the error below 1 ulp: + * + * Assume x>y>0; + * (if possible, set rounding to round-to-nearest) + * 1. if x > 2y use + * x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y + * where x1 = x with lower 64 bits cleared, x2 = x-x1; else + * 2. if x <= 2y use + * t1*y1+((x-y)*(x-y)+(t1*y2+t2*y)) + * where t1 = 2x with lower 64 bits cleared, t2 = 2x-t1, + * y1= y with lower 64 bits chopped, y2 = y-y1. + * + * NOTE: scaling may be necessary if some argument is too + * large or too tiny + * + * Special cases: + * hypotq(x,y) is INF if x or y is +INF or -INF; else + * hypotq(x,y) is NAN if x or y is NAN. + * + * Accuracy: + * hypotq(x,y) returns sqrtl(x^2+y^2) with error less + * than 1 ulps (units in the last place) + */ + +#include "quadmath-imp.h" + +__float128 +hypotq (__float128 x, __float128 y) +{ + __float128 a, b, t1, t2, y1, y2, w; + int64_t j, k, ha, hb; + + GET_FLT128_MSW64(ha,x); + ha &= 0x7fffffffffffffffLL; + GET_FLT128_MSW64(hb,y); + hb &= 0x7fffffffffffffffLL; + if(hb > ha) {a=y;b=x;j=ha; ha=hb;hb=j;} else {a=x;b=y;} + SET_FLT128_MSW64(a,ha); /* a <- |a| */ + SET_FLT128_MSW64(b,hb); /* b <- |b| */ + if((ha-hb)>0x78000000000000LL) {return a+b;} /* x/y > 2**120 */ + k=0; + if(ha > 0x5f3f000000000000LL) { /* a>2**8000 */ + if(ha >= 0x7fff000000000000LL) { /* Inf or NaN */ + uint64_t low; + w = a+b; /* for sNaN */ + GET_FLT128_LSW64(low,a); + if(((ha&0xffffffffffffLL)|low)==0) w = a; + GET_FLT128_LSW64(low,b); + if(((hb^0x7fff000000000000LL)|low)==0) w = b; + return w; + } + /* scale a and b by 2**-9600 */ + ha -= 0x2580000000000000LL; + hb -= 0x2580000000000000LL; k += 9600; + SET_FLT128_MSW64(a,ha); + SET_FLT128_MSW64(b,hb); + } + if(hb < 0x20bf000000000000LL) { /* b < 2**-8000 */ + if(hb <= 0x0000ffffffffffffLL) { /* subnormal b or 0 */ + uint64_t low; + GET_FLT128_LSW64(low,b); + if((hb|low)==0) return a; + t1=0; + SET_FLT128_MSW64(t1,0x7ffd000000000000LL); /* t1=2^16382 */ + b *= t1; + a *= t1; + k -= 16382; + } else { /* scale a and b by 2^9600 */ + ha += 0x2580000000000000LL; /* a *= 2^9600 */ + hb += 0x2580000000000000LL; /* b *= 2^9600 */ + k -= 9600; + SET_FLT128_MSW64(a,ha); + SET_FLT128_MSW64(b,hb); + } + } + /* medium size a and b */ + w = a-b; + if (w>b) { + t1 = 0; + SET_FLT128_MSW64(t1,ha); + t2 = a-t1; + w = sqrtq(t1*t1-(b*(-b)-t2*(a+t1))); + } else { + a = a+a; + y1 = 0; + SET_FLT128_MSW64(y1,hb); + y2 = b - y1; + t1 = 0; + SET_FLT128_MSW64(t1,ha+0x0001000000000000LL); + t2 = a - t1; + w = sqrtq(t1*y1-(w*(-w)-(t1*y2+t2*b))); + } + if(k!=0) { + uint64_t high; + t1 = 1.0Q; + GET_FLT128_MSW64(high,t1); + SET_FLT128_MSW64(t1,high+(k<<48)); + return t1*w; + } else return w; +} diff --git a/libquadmath/math/ilogbq.c b/libquadmath/math/ilogbq.c new file mode 100644 index 000000000..47986f5b3 --- /dev/null +++ b/libquadmath/math/ilogbq.c @@ -0,0 +1,64 @@ +/* s_ilogbl.c -- long double version of s_ilogb.c. + * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#if defined(LIBM_SCCS) && !defined(lint) +static char rcsid[] = "$NetBSD: $"; +#endif + +/* ilogbl(long double x) + * return the binary exponent of non-zero x + * ilogbl(0) = FP_ILOGB0 + * ilogbl(NaN) = FP_ILOGBNAN (no signal is raised) + * ilogbl(+-Inf) = INT_MAX (no signal is raised) + */ + +#include <limits.h> +#include <math.h> +#include "quadmath-imp.h" + +#ifndef FP_ILOGB0 +# define FP_ILOGB0 INT_MIN +#endif +#ifndef FP_ILOGBNAN +# define FP_ILOGBNAN INT_MAX +#endif + +int +ilogbq (__float128 x) +{ + int64_t hx,lx; + int ix; + + GET_FLT128_WORDS64(hx,lx,x); + hx &= 0x7fffffffffffffffLL; + if(hx <= 0x0001000000000000LL) { + if((hx|lx)==0) + return FP_ILOGB0; /* ilogbl(0) = FP_ILOGB0 */ + else /* subnormal x */ + if(hx==0) { + for (ix = -16431; lx>0; lx<<=1) ix -=1; + } else { + for (ix = -16382, hx<<=15; hx>0; hx<<=1) ix -=1; + } + return ix; + } + else if (hx<0x7fff000000000000LL) return (hx>>48)-0x3fff; + else if (FP_ILOGBNAN != INT_MAX) { + /* ISO C99 requires ilogbl(+-Inf) == INT_MAX. */ + if (((hx^0x7fff000000000000LL)|lx) == 0) + return INT_MAX; + } + return FP_ILOGBNAN; +} diff --git a/libquadmath/math/isinfq.c b/libquadmath/math/isinfq.c new file mode 100644 index 000000000..46996b54c --- /dev/null +++ b/libquadmath/math/isinfq.c @@ -0,0 +1,17 @@ +/* + * Written by J.T. Conklin <jtc@netbsd.org>. + * Change for long double by Jakub Jelinek <jj@ultra.linux.cz> + * Public domain. + */ + +#include "quadmath-imp.h" + +int +isinfq (__float128 x) +{ + int64_t hx,lx; + GET_FLT128_WORDS64(hx,lx,x); + lx |= (hx & 0x7fffffffffffffffLL) ^ 0x7fff000000000000LL; + lx |= -lx; + return ~(lx >> 63) & (hx >> 62); +} diff --git a/libquadmath/math/isnanq.c b/libquadmath/math/isnanq.c new file mode 100644 index 000000000..ab9df1658 --- /dev/null +++ b/libquadmath/math/isnanq.c @@ -0,0 +1,27 @@ +/* s_isnanl.c -- long double version of s_isnan.c. + * Conversion to long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include "quadmath-imp.h" + +int +isnanq (const __float128 x) +{ + int64_t hx,lx; + GET_FLT128_WORDS64(hx,lx,x); + hx &= 0x7fffffffffffffffLL; + hx |= (uint64_t)(lx|(-lx))>>63; + hx = 0x7fff000000000000LL - hx; + return (int)((uint64_t)hx>>63); +} diff --git a/libquadmath/math/j0q.c b/libquadmath/math/j0q.c new file mode 100644 index 000000000..fecbe6277 --- /dev/null +++ b/libquadmath/math/j0q.c @@ -0,0 +1,919 @@ +/* j0l.c + * + * Bessel function of order zero + * + * + * + * SYNOPSIS: + * + * long double x, y, j0l(); + * + * y = j0l( x ); + * + * + * + * DESCRIPTION: + * + * Returns Bessel function of first kind, order zero of the argument. + * + * The domain is divided into two major intervals [0, 2] and + * (2, infinity). In the first interval the rational approximation + * is J0(x) = 1 - x^2 / 4 + x^4 R(x^2) + * The second interval is further partitioned into eight equal segments + * of 1/x. + * + * J0(x) = sqrt(2/(pi x)) (P0(x) cos(X) - Q0(x) sin(X)), + * X = x - pi/4, + * + * and the auxiliary functions are given by + * + * J0(x)cos(X) + Y0(x)sin(X) = sqrt( 2/(pi x)) P0(x), + * P0(x) = 1 + 1/x^2 R(1/x^2) + * + * Y0(x)cos(X) - J0(x)sin(X) = sqrt( 2/(pi x)) Q0(x), + * Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2)) + * + * + * + * ACCURACY: + * + * Absolute error: + * arithmetic domain # trials peak rms + * IEEE 0, 30 100000 1.7e-34 2.4e-35 + * + * + */ + +/* y0l.c + * + * Bessel function of the second kind, order zero + * + * + * + * SYNOPSIS: + * + * double x, y, y0l(); + * + * y = y0l( x ); + * + * + * + * DESCRIPTION: + * + * Returns Bessel function of the second kind, of order + * zero, of the argument. + * + * The approximation is the same as for J0(x), and + * Y0(x) = sqrt(2/(pi x)) (P0(x) sin(X) + Q0(x) cos(X)). + * + * ACCURACY: + * + * Absolute error, when y0(x) < 1; else relative error: + * + * arithmetic domain # trials peak rms + * IEEE 0, 30 100000 3.0e-34 2.7e-35 + * + */ + +/* Copyright 2001 by Stephen L. Moshier (moshier@na-net.ornl.gov). + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + +#include "quadmath-imp.h" + +/* 1 / sqrt(pi) */ +static const __float128 ONEOSQPI = 5.6418958354775628694807945156077258584405E-1Q; +/* 2 / pi */ +static const __float128 TWOOPI = 6.3661977236758134307553505349005744813784E-1Q; +static const __float128 zero = 0.0Q; + +/* J0(x) = 1 - x^2/4 + x^2 x^2 R(x^2) + Peak relative error 3.4e-37 + 0 <= x <= 2 */ +#define NJ0_2N 6 +static const __float128 J0_2N[NJ0_2N + 1] = { + 3.133239376997663645548490085151484674892E16Q, + -5.479944965767990821079467311839107722107E14Q, + 6.290828903904724265980249871997551894090E12Q, + -3.633750176832769659849028554429106299915E10Q, + 1.207743757532429576399485415069244807022E8Q, + -2.107485999925074577174305650549367415465E5Q, + 1.562826808020631846245296572935547005859E2Q, +}; +#define NJ0_2D 6 +static const __float128 J0_2D[NJ0_2D + 1] = { + 2.005273201278504733151033654496928968261E18Q, + 2.063038558793221244373123294054149790864E16Q, + 1.053350447931127971406896594022010524994E14Q, + 3.496556557558702583143527876385508882310E11Q, + 8.249114511878616075860654484367133976306E8Q, + 1.402965782449571800199759247964242790589E6Q, + 1.619910762853439600957801751815074787351E3Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2), + 0 <= 1/x <= .0625 + Peak relative error 3.3e-36 */ +#define NP16_IN 9 +static const __float128 P16_IN[NP16_IN + 1] = { + -1.901689868258117463979611259731176301065E-16Q, + -1.798743043824071514483008340803573980931E-13Q, + -6.481746687115262291873324132944647438959E-11Q, + -1.150651553745409037257197798528294248012E-8Q, + -1.088408467297401082271185599507222695995E-6Q, + -5.551996725183495852661022587879817546508E-5Q, + -1.477286941214245433866838787454880214736E-3Q, + -1.882877976157714592017345347609200402472E-2Q, + -9.620983176855405325086530374317855880515E-2Q, + -1.271468546258855781530458854476627766233E-1Q, +}; +#define NP16_ID 9 +static const __float128 P16_ID[NP16_ID + 1] = { + 2.704625590411544837659891569420764475007E-15Q, + 2.562526347676857624104306349421985403573E-12Q, + 9.259137589952741054108665570122085036246E-10Q, + 1.651044705794378365237454962653430805272E-7Q, + 1.573561544138733044977714063100859136660E-5Q, + 8.134482112334882274688298469629884804056E-4Q, + 2.219259239404080863919375103673593571689E-2Q, + 2.976990606226596289580242451096393862792E-1Q, + 1.713895630454693931742734911930937246254E0Q, + 3.231552290717904041465898249160757368855E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2) + 0.0625 <= 1/x <= 0.125 + Peak relative error 2.4e-35 */ +#define NP8_16N 10 +static const __float128 P8_16N[NP8_16N + 1] = { + -2.335166846111159458466553806683579003632E-15Q, + -1.382763674252402720401020004169367089975E-12Q, + -3.192160804534716696058987967592784857907E-10Q, + -3.744199606283752333686144670572632116899E-8Q, + -2.439161236879511162078619292571922772224E-6Q, + -9.068436986859420951664151060267045346549E-5Q, + -1.905407090637058116299757292660002697359E-3Q, + -2.164456143936718388053842376884252978872E-2Q, + -1.212178415116411222341491717748696499966E-1Q, + -2.782433626588541494473277445959593334494E-1Q, + -1.670703190068873186016102289227646035035E-1Q, +}; +#define NP8_16D 10 +static const __float128 P8_16D[NP8_16D + 1] = { + 3.321126181135871232648331450082662856743E-14Q, + 1.971894594837650840586859228510007703641E-11Q, + 4.571144364787008285981633719513897281690E-9Q, + 5.396419143536287457142904742849052402103E-7Q, + 3.551548222385845912370226756036899901549E-5Q, + 1.342353874566932014705609788054598013516E-3Q, + 2.899133293006771317589357444614157734385E-2Q, + 3.455374978185770197704507681491574261545E-1Q, + 2.116616964297512311314454834712634820514E0Q, + 5.850768316827915470087758636881584174432E0Q, + 5.655273858938766830855753983631132928968E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2) + 0.125 <= 1/x <= 0.1875 + Peak relative error 2.7e-35 */ +#define NP5_8N 10 +static const __float128 P5_8N[NP5_8N + 1] = { + -1.270478335089770355749591358934012019596E-12Q, + -4.007588712145412921057254992155810347245E-10Q, + -4.815187822989597568124520080486652009281E-8Q, + -2.867070063972764880024598300408284868021E-6Q, + -9.218742195161302204046454768106063638006E-5Q, + -1.635746821447052827526320629828043529997E-3Q, + -1.570376886640308408247709616497261011707E-2Q, + -7.656484795303305596941813361786219477807E-2Q, + -1.659371030767513274944805479908858628053E-1Q, + -1.185340550030955660015841796219919804915E-1Q, + -8.920026499909994671248893388013790366712E-3Q, +}; +#define NP5_8D 9 +static const __float128 P5_8D[NP5_8D + 1] = { + 1.806902521016705225778045904631543990314E-11Q, + 5.728502760243502431663549179135868966031E-9Q, + 6.938168504826004255287618819550667978450E-7Q, + 4.183769964807453250763325026573037785902E-5Q, + 1.372660678476925468014882230851637878587E-3Q, + 2.516452105242920335873286419212708961771E-2Q, + 2.550502712902647803796267951846557316182E-1Q, + 1.365861559418983216913629123778747617072E0Q, + 3.523825618308783966723472468855042541407E0Q, + 3.656365803506136165615111349150536282434E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2) + Peak relative error 3.5e-35 + 0.1875 <= 1/x <= 0.25 */ +#define NP4_5N 9 +static const __float128 P4_5N[NP4_5N + 1] = { + -9.791405771694098960254468859195175708252E-10Q, + -1.917193059944531970421626610188102836352E-7Q, + -1.393597539508855262243816152893982002084E-5Q, + -4.881863490846771259880606911667479860077E-4Q, + -8.946571245022470127331892085881699269853E-3Q, + -8.707474232568097513415336886103899434251E-2Q, + -4.362042697474650737898551272505525973766E-1Q, + -1.032712171267523975431451359962375617386E0Q, + -9.630502683169895107062182070514713702346E-1Q, + -2.251804386252969656586810309252357233320E-1Q, +}; +#define NP4_5D 9 +static const __float128 P4_5D[NP4_5D + 1] = { + 1.392555487577717669739688337895791213139E-8Q, + 2.748886559120659027172816051276451376854E-6Q, + 2.024717710644378047477189849678576659290E-4Q, + 7.244868609350416002930624752604670292469E-3Q, + 1.373631762292244371102989739300382152416E-1Q, + 1.412298581400224267910294815260613240668E0Q, + 7.742495637843445079276397723849017617210E0Q, + 2.138429269198406512028307045259503811861E1Q, + 2.651547684548423476506826951831712762610E1Q, + 1.167499382465291931571685222882909166935E1Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2) + Peak relative error 2.3e-36 + 0.25 <= 1/x <= 0.3125 */ +#define NP3r2_4N 9 +static const __float128 P3r2_4N[NP3r2_4N + 1] = { + -2.589155123706348361249809342508270121788E-8Q, + -3.746254369796115441118148490849195516593E-6Q, + -1.985595497390808544622893738135529701062E-4Q, + -5.008253705202932091290132760394976551426E-3Q, + -6.529469780539591572179155511840853077232E-2Q, + -4.468736064761814602927408833818990271514E-1Q, + -1.556391252586395038089729428444444823380E0Q, + -2.533135309840530224072920725976994981638E0Q, + -1.605509621731068453869408718565392869560E0Q, + -2.518966692256192789269859830255724429375E-1Q, +}; +#define NP3r2_4D 9 +static const __float128 P3r2_4D[NP3r2_4D + 1] = { + 3.682353957237979993646169732962573930237E-7Q, + 5.386741661883067824698973455566332102029E-5Q, + 2.906881154171822780345134853794241037053E-3Q, + 7.545832595801289519475806339863492074126E-2Q, + 1.029405357245594877344360389469584526654E0Q, + 7.565706120589873131187989560509757626725E0Q, + 2.951172890699569545357692207898667665796E1Q, + 5.785723537170311456298467310529815457536E1Q, + 5.095621464598267889126015412522773474467E1Q, + 1.602958484169953109437547474953308401442E1Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2) + Peak relative error 1.0e-35 + 0.3125 <= 1/x <= 0.375 */ +#define NP2r7_3r2N 9 +static const __float128 P2r7_3r2N[NP2r7_3r2N + 1] = { + -1.917322340814391131073820537027234322550E-7Q, + -1.966595744473227183846019639723259011906E-5Q, + -7.177081163619679403212623526632690465290E-4Q, + -1.206467373860974695661544653741899755695E-2Q, + -1.008656452188539812154551482286328107316E-1Q, + -4.216016116408810856620947307438823892707E-1Q, + -8.378631013025721741744285026537009814161E-1Q, + -6.973895635309960850033762745957946272579E-1Q, + -1.797864718878320770670740413285763554812E-1Q, + -4.098025357743657347681137871388402849581E-3Q, +}; +#define NP2r7_3r2D 8 +static const __float128 P2r7_3r2D[NP2r7_3r2D + 1] = { + 2.726858489303036441686496086962545034018E-6Q, + 2.840430827557109238386808968234848081424E-4Q, + 1.063826772041781947891481054529454088832E-2Q, + 1.864775537138364773178044431045514405468E-1Q, + 1.665660052857205170440952607701728254211E0Q, + 7.723745889544331153080842168958348568395E0Q, + 1.810726427571829798856428548102077799835E1Q, + 1.986460672157794440666187503833545388527E1Q, + 8.645503204552282306364296517220055815488E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2) + Peak relative error 1.3e-36 + 0.3125 <= 1/x <= 0.4375 */ +#define NP2r3_2r7N 9 +static const __float128 P2r3_2r7N[NP2r3_2r7N + 1] = { + -1.594642785584856746358609622003310312622E-6Q, + -1.323238196302221554194031733595194539794E-4Q, + -3.856087818696874802689922536987100372345E-3Q, + -5.113241710697777193011470733601522047399E-2Q, + -3.334229537209911914449990372942022350558E-1Q, + -1.075703518198127096179198549659283422832E0Q, + -1.634174803414062725476343124267110981807E0Q, + -1.030133247434119595616826842367268304880E0Q, + -1.989811539080358501229347481000707289391E-1Q, + -3.246859189246653459359775001466924610236E-3Q, +}; +#define NP2r3_2r7D 8 +static const __float128 P2r3_2r7D[NP2r3_2r7D + 1] = { + 2.267936634217251403663034189684284173018E-5Q, + 1.918112982168673386858072491437971732237E-3Q, + 5.771704085468423159125856786653868219522E-2Q, + 8.056124451167969333717642810661498890507E-1Q, + 5.687897967531010276788680634413789328776E0Q, + 2.072596760717695491085444438270778394421E1Q, + 3.801722099819929988585197088613160496684E1Q, + 3.254620235902912339534998592085115836829E1Q, + 1.104847772130720331801884344645060675036E1Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2) + Peak relative error 1.2e-35 + 0.4375 <= 1/x <= 0.5 */ +#define NP2_2r3N 8 +static const __float128 P2_2r3N[NP2_2r3N + 1] = { + -1.001042324337684297465071506097365389123E-4Q, + -6.289034524673365824853547252689991418981E-3Q, + -1.346527918018624234373664526930736205806E-1Q, + -1.268808313614288355444506172560463315102E0Q, + -5.654126123607146048354132115649177406163E0Q, + -1.186649511267312652171775803270911971693E1Q, + -1.094032424931998612551588246779200724257E1Q, + -3.728792136814520055025256353193674625267E0Q, + -3.000348318524471807839934764596331810608E-1Q, +}; +#define NP2_2r3D 8 +static const __float128 P2_2r3D[NP2_2r3D + 1] = { + 1.423705538269770974803901422532055612980E-3Q, + 9.171476630091439978533535167485230575894E-2Q, + 2.049776318166637248868444600215942828537E0Q, + 2.068970329743769804547326701946144899583E1Q, + 1.025103500560831035592731539565060347709E2Q, + 2.528088049697570728252145557167066708284E2Q, + 2.992160327587558573740271294804830114205E2Q, + 1.540193761146551025832707739468679973036E2Q, + 2.779516701986912132637672140709452502650E1Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x), + Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2)) + Peak relative error 2.2e-35 + 0 <= 1/x <= .0625 */ +#define NQ16_IN 10 +static const __float128 Q16_IN[NQ16_IN + 1] = { + 2.343640834407975740545326632205999437469E-18Q, + 2.667978112927811452221176781536278257448E-15Q, + 1.178415018484555397390098879501969116536E-12Q, + 2.622049767502719728905924701288614016597E-10Q, + 3.196908059607618864801313380896308968673E-8Q, + 2.179466154171673958770030655199434798494E-6Q, + 8.139959091628545225221976413795645177291E-5Q, + 1.563900725721039825236927137885747138654E-3Q, + 1.355172364265825167113562519307194840307E-2Q, + 3.928058355906967977269780046844768588532E-2Q, + 1.107891967702173292405380993183694932208E-2Q, +}; +#define NQ16_ID 9 +static const __float128 Q16_ID[NQ16_ID + 1] = { + 3.199850952578356211091219295199301766718E-17Q, + 3.652601488020654842194486058637953363918E-14Q, + 1.620179741394865258354608590461839031281E-11Q, + 3.629359209474609630056463248923684371426E-9Q, + 4.473680923894354600193264347733477363305E-7Q, + 3.106368086644715743265603656011050476736E-5Q, + 1.198239259946770604954664925153424252622E-3Q, + 2.446041004004283102372887804475767568272E-2Q, + 2.403235525011860603014707768815113698768E-1Q, + 9.491006790682158612266270665136910927149E-1Q, + /* 1.000000000000000000000000000000000000000E0 */ + }; + +/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x), + Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2)) + Peak relative error 5.1e-36 + 0.0625 <= 1/x <= 0.125 */ +#define NQ8_16N 11 +static const __float128 Q8_16N[NQ8_16N + 1] = { + 1.001954266485599464105669390693597125904E-17Q, + 7.545499865295034556206475956620160007849E-15Q, + 2.267838684785673931024792538193202559922E-12Q, + 3.561909705814420373609574999542459912419E-10Q, + 3.216201422768092505214730633842924944671E-8Q, + 1.731194793857907454569364622452058554314E-6Q, + 5.576944613034537050396518509871004586039E-5Q, + 1.051787760316848982655967052985391418146E-3Q, + 1.102852974036687441600678598019883746959E-2Q, + 5.834647019292460494254225988766702933571E-2Q, + 1.290281921604364618912425380717127576529E-1Q, + 7.598886310387075708640370806458926458301E-2Q, +}; +#define NQ8_16D 11 +static const __float128 Q8_16D[NQ8_16D + 1] = { + 1.368001558508338469503329967729951830843E-16Q, + 1.034454121857542147020549303317348297289E-13Q, + 3.128109209247090744354764050629381674436E-11Q, + 4.957795214328501986562102573522064468671E-9Q, + 4.537872468606711261992676606899273588899E-7Q, + 2.493639207101727713192687060517509774182E-5Q, + 8.294957278145328349785532236663051405805E-4Q, + 1.646471258966713577374948205279380115839E-2Q, + 1.878910092770966718491814497982191447073E-1Q, + 1.152641605706170353727903052525652504075E0Q, + 3.383550240669773485412333679367792932235E0Q, + 3.823875252882035706910024716609908473970E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x), + Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2)) + Peak relative error 3.9e-35 + 0.125 <= 1/x <= 0.1875 */ +#define NQ5_8N 10 +static const __float128 Q5_8N[NQ5_8N + 1] = { + 1.750399094021293722243426623211733898747E-13Q, + 6.483426211748008735242909236490115050294E-11Q, + 9.279430665656575457141747875716899958373E-9Q, + 6.696634968526907231258534757736576340266E-7Q, + 2.666560823798895649685231292142838188061E-5Q, + 6.025087697259436271271562769707550594540E-4Q, + 7.652807734168613251901945778921336353485E-3Q, + 5.226269002589406461622551452343519078905E-2Q, + 1.748390159751117658969324896330142895079E-1Q, + 2.378188719097006494782174902213083589660E-1Q, + 8.383984859679804095463699702165659216831E-2Q, +}; +#define NQ5_8D 10 +static const __float128 Q5_8D[NQ5_8D + 1] = { + 2.389878229704327939008104855942987615715E-12Q, + 8.926142817142546018703814194987786425099E-10Q, + 1.294065862406745901206588525833274399038E-7Q, + 9.524139899457666250828752185212769682191E-6Q, + 3.908332488377770886091936221573123353489E-4Q, + 9.250427033957236609624199884089916836748E-3Q, + 1.263420066165922645975830877751588421451E-1Q, + 9.692527053860420229711317379861733180654E-1Q, + 3.937813834630430172221329298841520707954E0Q, + 7.603126427436356534498908111445191312181E0Q, + 5.670677653334105479259958485084550934305E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x), + Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2)) + Peak relative error 3.2e-35 + 0.1875 <= 1/x <= 0.25 */ +#define NQ4_5N 10 +static const __float128 Q4_5N[NQ4_5N + 1] = { + 2.233870042925895644234072357400122854086E-11Q, + 5.146223225761993222808463878999151699792E-9Q, + 4.459114531468296461688753521109797474523E-7Q, + 1.891397692931537975547242165291668056276E-5Q, + 4.279519145911541776938964806470674565504E-4Q, + 5.275239415656560634702073291768904783989E-3Q, + 3.468698403240744801278238473898432608887E-2Q, + 1.138773146337708415188856882915457888274E-1Q, + 1.622717518946443013587108598334636458955E-1Q, + 7.249040006390586123760992346453034628227E-2Q, + 1.941595365256460232175236758506411486667E-3Q, +}; +#define NQ4_5D 9 +static const __float128 Q4_5D[NQ4_5D + 1] = { + 3.049977232266999249626430127217988047453E-10Q, + 7.120883230531035857746096928889676144099E-8Q, + 6.301786064753734446784637919554359588859E-6Q, + 2.762010530095069598480766869426308077192E-4Q, + 6.572163250572867859316828886203406361251E-3Q, + 8.752566114841221958200215255461843397776E-2Q, + 6.487654992874805093499285311075289932664E-1Q, + 2.576550017826654579451615283022812801435E0Q, + 5.056392229924022835364779562707348096036E0Q, + 4.179770081068251464907531367859072157773E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x), + Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2)) + Peak relative error 1.4e-36 + 0.25 <= 1/x <= 0.3125 */ +#define NQ3r2_4N 10 +static const __float128 Q3r2_4N[NQ3r2_4N + 1] = { + 6.126167301024815034423262653066023684411E-10Q, + 1.043969327113173261820028225053598975128E-7Q, + 6.592927270288697027757438170153763220190E-6Q, + 2.009103660938497963095652951912071336730E-4Q, + 3.220543385492643525985862356352195896964E-3Q, + 2.774405975730545157543417650436941650990E-2Q, + 1.258114008023826384487378016636555041129E-1Q, + 2.811724258266902502344701449984698323860E-1Q, + 2.691837665193548059322831687432415014067E-1Q, + 7.949087384900985370683770525312735605034E-2Q, + 1.229509543620976530030153018986910810747E-3Q, +}; +#define NQ3r2_4D 9 +static const __float128 Q3r2_4D[NQ3r2_4D + 1] = { + 8.364260446128475461539941389210166156568E-9Q, + 1.451301850638956578622154585560759862764E-6Q, + 9.431830010924603664244578867057141839463E-5Q, + 3.004105101667433434196388593004526182741E-3Q, + 5.148157397848271739710011717102773780221E-2Q, + 4.901089301726939576055285374953887874895E-1Q, + 2.581760991981709901216967665934142240346E0Q, + 7.257105880775059281391729708630912791847E0Q, + 1.006014717326362868007913423810737369312E1Q, + 5.879416600465399514404064187445293212470E0Q, + /* 1.000000000000000000000000000000000000000E0*/ +}; + +/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x), + Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2)) + Peak relative error 3.8e-36 + 0.3125 <= 1/x <= 0.375 */ +#define NQ2r7_3r2N 9 +static const __float128 Q2r7_3r2N[NQ2r7_3r2N + 1] = { + 7.584861620402450302063691901886141875454E-8Q, + 9.300939338814216296064659459966041794591E-6Q, + 4.112108906197521696032158235392604947895E-4Q, + 8.515168851578898791897038357239630654431E-3Q, + 8.971286321017307400142720556749573229058E-2Q, + 4.885856732902956303343015636331874194498E-1Q, + 1.334506268733103291656253500506406045846E0Q, + 1.681207956863028164179042145803851824654E0Q, + 8.165042692571721959157677701625853772271E-1Q, + 9.805848115375053300608712721986235900715E-2Q, +}; +#define NQ2r7_3r2D 9 +static const __float128 Q2r7_3r2D[NQ2r7_3r2D + 1] = { + 1.035586492113036586458163971239438078160E-6Q, + 1.301999337731768381683593636500979713689E-4Q, + 5.993695702564527062553071126719088859654E-3Q, + 1.321184892887881883489141186815457808785E-1Q, + 1.528766555485015021144963194165165083312E0Q, + 9.561463309176490874525827051566494939295E0Q, + 3.203719484883967351729513662089163356911E1Q, + 5.497294687660930446641539152123568668447E1Q, + 4.391158169390578768508675452986948391118E1Q, + 1.347836630730048077907818943625789418378E1Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x), + Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2)) + Peak relative error 2.2e-35 + 0.375 <= 1/x <= 0.4375 */ +#define NQ2r3_2r7N 9 +static const __float128 Q2r3_2r7N[NQ2r3_2r7N + 1] = { + 4.455027774980750211349941766420190722088E-7Q, + 4.031998274578520170631601850866780366466E-5Q, + 1.273987274325947007856695677491340636339E-3Q, + 1.818754543377448509897226554179659122873E-2Q, + 1.266748858326568264126353051352269875352E-1Q, + 4.327578594728723821137731555139472880414E-1Q, + 6.892532471436503074928194969154192615359E-1Q, + 4.490775818438716873422163588640262036506E-1Q, + 8.649615949297322440032000346117031581572E-2Q, + 7.261345286655345047417257611469066147561E-4Q, +}; +#define NQ2r3_2r7D 8 +static const __float128 Q2r3_2r7D[NQ2r3_2r7D + 1] = { + 6.082600739680555266312417978064954793142E-6Q, + 5.693622538165494742945717226571441747567E-4Q, + 1.901625907009092204458328768129666975975E-2Q, + 2.958689532697857335456896889409923371570E-1Q, + 2.343124711045660081603809437993368799568E0Q, + 9.665894032187458293568704885528192804376E0Q, + 2.035273104990617136065743426322454881353E1Q, + 2.044102010478792896815088858740075165531E1Q, + 8.445937177863155827844146643468706599304E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x), + Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2)) + Peak relative error 3.1e-36 + 0.4375 <= 1/x <= 0.5 */ +#define NQ2_2r3N 9 +static const __float128 Q2_2r3N[NQ2_2r3N + 1] = { + 2.817566786579768804844367382809101929314E-6Q, + 2.122772176396691634147024348373539744935E-4Q, + 5.501378031780457828919593905395747517585E-3Q, + 6.355374424341762686099147452020466524659E-2Q, + 3.539652320122661637429658698954748337223E-1Q, + 9.571721066119617436343740541777014319695E-1Q, + 1.196258777828426399432550698612171955305E0Q, + 6.069388659458926158392384709893753793967E-1Q, + 9.026746127269713176512359976978248763621E-2Q, + 5.317668723070450235320878117210807236375E-4Q, +}; +#define NQ2_2r3D 8 +static const __float128 Q2_2r3D[NQ2_2r3D + 1] = { + 3.846924354014260866793741072933159380158E-5Q, + 3.017562820057704325510067178327449946763E-3Q, + 8.356305620686867949798885808540444210935E-2Q, + 1.068314930499906838814019619594424586273E0Q, + 6.900279623894821067017966573640732685233E0Q, + 2.307667390886377924509090271780839563141E1Q, + 3.921043465412723970791036825401273528513E1Q, + 3.167569478939719383241775717095729233436E1Q, + 1.051023841699200920276198346301543665909E1Q, + /* 1.000000000000000000000000000000000000000E0*/ +}; + + +/* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ + +static __float128 +neval (__float128 x, const __float128 *p, int n) +{ + __float128 y; + + p += n; + y = *p--; + do + { + y = y * x + *p--; + } + while (--n > 0); + return y; +} + + +/* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */ + +static __float128 +deval (__float128 x, const __float128 *p, int n) +{ + __float128 y; + + p += n; + y = x + *p--; + do + { + y = y * x + *p--; + } + while (--n > 0); + return y; +} + + +/* Bessel function of the first kind, order zero. */ + +__float128 +j0q (__float128 x) +{ + __float128 xx, xinv, z, p, q, c, s, cc, ss; + + if (! finiteq (x)) + { + if (x != x) + return x; + else + return 0.0Q; + } + if (x == 0.0Q) + return 1.0Q; + + xx = fabsq (x); + if (xx <= 2.0Q) + { + /* 0 <= x <= 2 */ + z = xx * xx; + p = z * z * neval (z, J0_2N, NJ0_2N) / deval (z, J0_2D, NJ0_2D); + p -= 0.25Q * z; + p += 1.0Q; + return p; + } + + xinv = 1.0Q / xx; + z = xinv * xinv; + if (xinv <= 0.25) + { + if (xinv <= 0.125) + { + if (xinv <= 0.0625) + { + p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID); + q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID); + } + else + { + p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D); + q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D); + } + } + else if (xinv <= 0.1875) + { + p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D); + q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D); + } + else + { + p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D); + q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D); + } + } /* .25 */ + else /* if (xinv <= 0.5) */ + { + if (xinv <= 0.375) + { + if (xinv <= 0.3125) + { + p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D); + q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D); + } + else + { + p = neval (z, P2r7_3r2N, NP2r7_3r2N) + / deval (z, P2r7_3r2D, NP2r7_3r2D); + q = neval (z, Q2r7_3r2N, NQ2r7_3r2N) + / deval (z, Q2r7_3r2D, NQ2r7_3r2D); + } + } + else if (xinv <= 0.4375) + { + p = neval (z, P2r3_2r7N, NP2r3_2r7N) + / deval (z, P2r3_2r7D, NP2r3_2r7D); + q = neval (z, Q2r3_2r7N, NQ2r3_2r7N) + / deval (z, Q2r3_2r7D, NQ2r3_2r7D); + } + else + { + p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D); + q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D); + } + } + p = 1.0Q + z * p; + q = z * xinv * q; + q = q - 0.125Q * xinv; + /* X = x - pi/4 + cos(X) = cos(x) cos(pi/4) + sin(x) sin(pi/4) + = 1/sqrt(2) * (cos(x) + sin(x)) + sin(X) = sin(x) cos(pi/4) - cos(x) sin(pi/4) + = 1/sqrt(2) * (sin(x) - cos(x)) + sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) + cf. Fdlibm. */ + sincosq (xx, &s, &c); + ss = s - c; + cc = s + c; + z = - cosq (xx + xx); + if ((s * c) < 0) + cc = z / ss; + else + ss = z / cc; + z = ONEOSQPI * (p * cc - q * ss) / sqrtq (xx); + return z; +} + + +/* Y0(x) = 2/pi * log(x) * J0(x) + R(x^2) + Peak absolute error 1.7e-36 (relative where Y0 > 1) + 0 <= x <= 2 */ +#define NY0_2N 7 +static __float128 Y0_2N[NY0_2N + 1] = { + -1.062023609591350692692296993537002558155E19Q, + 2.542000883190248639104127452714966858866E19Q, + -1.984190771278515324281415820316054696545E18Q, + 4.982586044371592942465373274440222033891E16Q, + -5.529326354780295177243773419090123407550E14Q, + 3.013431465522152289279088265336861140391E12Q, + -7.959436160727126750732203098982718347785E9Q, + 8.230845651379566339707130644134372793322E6Q, +}; +#define NY0_2D 7 +static __float128 Y0_2D[NY0_2D + 1] = { + 1.438972634353286978700329883122253752192E20Q, + 1.856409101981569254247700169486907405500E18Q, + 1.219693352678218589553725579802986255614E16Q, + 5.389428943282838648918475915779958097958E13Q, + 1.774125762108874864433872173544743051653E11Q, + 4.522104832545149534808218252434693007036E8Q, + 8.872187401232943927082914504125234454930E5Q, + 1.251945613186787532055610876304669413955E3Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + + +/* Bessel function of the second kind, order zero. */ + +__float128 +y0q (__float128 x) +{ + __float128 xx, xinv, z, p, q, c, s, cc, ss; + + if (! finiteq (x)) + { + if (x != x) + return x; + else + return 0.0Q; + } + if (x <= 0.0Q) + { + if (x < 0.0Q) + return (zero / (zero * x)); + return -HUGE_VALQ + x; + } + xx = fabsq (x); + if (xx <= 2.0Q) + { + /* 0 <= x <= 2 */ + z = xx * xx; + p = neval (z, Y0_2N, NY0_2N) / deval (z, Y0_2D, NY0_2D); + p = TWOOPI * logq (x) * j0q (x) + p; + return p; + } + + xinv = 1.0Q / xx; + z = xinv * xinv; + if (xinv <= 0.25) + { + if (xinv <= 0.125) + { + if (xinv <= 0.0625) + { + p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID); + q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID); + } + else + { + p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D); + q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D); + } + } + else if (xinv <= 0.1875) + { + p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D); + q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D); + } + else + { + p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D); + q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D); + } + } /* .25 */ + else /* if (xinv <= 0.5) */ + { + if (xinv <= 0.375) + { + if (xinv <= 0.3125) + { + p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D); + q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D); + } + else + { + p = neval (z, P2r7_3r2N, NP2r7_3r2N) + / deval (z, P2r7_3r2D, NP2r7_3r2D); + q = neval (z, Q2r7_3r2N, NQ2r7_3r2N) + / deval (z, Q2r7_3r2D, NQ2r7_3r2D); + } + } + else if (xinv <= 0.4375) + { + p = neval (z, P2r3_2r7N, NP2r3_2r7N) + / deval (z, P2r3_2r7D, NP2r3_2r7D); + q = neval (z, Q2r3_2r7N, NQ2r3_2r7N) + / deval (z, Q2r3_2r7D, NQ2r3_2r7D); + } + else + { + p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D); + q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D); + } + } + p = 1.0Q + z * p; + q = z * xinv * q; + q = q - 0.125Q * xinv; + /* X = x - pi/4 + cos(X) = cos(x) cos(pi/4) + sin(x) sin(pi/4) + = 1/sqrt(2) * (cos(x) + sin(x)) + sin(X) = sin(x) cos(pi/4) - cos(x) sin(pi/4) + = 1/sqrt(2) * (sin(x) - cos(x)) + sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) + cf. Fdlibm. */ + sincosq (x, &s, &c); + ss = s - c; + cc = s + c; + z = - cosq (x + x); + if ((s * c) < 0) + cc = z / ss; + else + ss = z / cc; + z = ONEOSQPI * (p * ss + q * cc) / sqrtq (x); + return z; +} diff --git a/libquadmath/math/j1q.c b/libquadmath/math/j1q.c new file mode 100644 index 000000000..f6bf2a256 --- /dev/null +++ b/libquadmath/math/j1q.c @@ -0,0 +1,926 @@ +/* j1l.c + * + * Bessel function of order one + * + * + * + * SYNOPSIS: + * + * long double x, y, j1l(); + * + * y = j1l( x ); + * + * + * + * DESCRIPTION: + * + * Returns Bessel function of first kind, order one of the argument. + * + * The domain is divided into two major intervals [0, 2] and + * (2, infinity). In the first interval the rational approximation is + * J1(x) = .5x + x x^2 R(x^2) + * + * The second interval is further partitioned into eight equal segments + * of 1/x. + * J1(x) = sqrt(2/(pi x)) (P1(x) cos(X) - Q1(x) sin(X)), + * X = x - 3 pi / 4, + * + * and the auxiliary functions are given by + * + * J1(x)cos(X) + Y1(x)sin(X) = sqrt( 2/(pi x)) P1(x), + * P1(x) = 1 + 1/x^2 R(1/x^2) + * + * Y1(x)cos(X) - J1(x)sin(X) = sqrt( 2/(pi x)) Q1(x), + * Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)). + * + * + * + * ACCURACY: + * + * Absolute error: + * arithmetic domain # trials peak rms + * IEEE 0, 30 100000 2.8e-34 2.7e-35 + * + * + */ + +/* y1l.c + * + * Bessel function of the second kind, order one + * + * + * + * SYNOPSIS: + * + * double x, y, y1l(); + * + * y = y1l( x ); + * + * + * + * DESCRIPTION: + * + * Returns Bessel function of the second kind, of order + * one, of the argument. + * + * The domain is divided into two major intervals [0, 2] and + * (2, infinity). In the first interval the rational approximation is + * Y1(x) = 2/pi * (log(x) * J1(x) - 1/x) + x R(x^2) . + * In the second interval the approximation is the same as for J1(x), and + * Y1(x) = sqrt(2/(pi x)) (P1(x) sin(X) + Q1(x) cos(X)), + * X = x - 3 pi / 4. + * + * ACCURACY: + * + * Absolute error, when y0(x) < 1; else relative error: + * + * arithmetic domain # trials peak rms + * IEEE 0, 30 100000 2.7e-34 2.9e-35 + * + */ + +/* Copyright 2001 by Stephen L. Moshier (moshier@na-net.onrl.gov). + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + +#include "quadmath-imp.h" + +/* 1 / sqrt(pi) */ +static const __float128 ONEOSQPI = 5.6418958354775628694807945156077258584405E-1Q; +/* 2 / pi */ +static const __float128 TWOOPI = 6.3661977236758134307553505349005744813784E-1Q; +static const __float128 zero = 0.0Q; + +/* J1(x) = .5x + x x^2 R(x^2) + Peak relative error 1.9e-35 + 0 <= x <= 2 */ +#define NJ0_2N 6 +static const __float128 J0_2N[NJ0_2N + 1] = { + -5.943799577386942855938508697619735179660E16Q, + 1.812087021305009192259946997014044074711E15Q, + -2.761698314264509665075127515729146460895E13Q, + 2.091089497823600978949389109350658815972E11Q, + -8.546413231387036372945453565654130054307E8Q, + 1.797229225249742247475464052741320612261E6Q, + -1.559552840946694171346552770008812083969E3Q +}; +#define NJ0_2D 6 +static const __float128 J0_2D[NJ0_2D + 1] = { + 9.510079323819108569501613916191477479397E17Q, + 1.063193817503280529676423936545854693915E16Q, + 5.934143516050192600795972192791775226920E13Q, + 2.168000911950620999091479265214368352883E11Q, + 5.673775894803172808323058205986256928794E8Q, + 1.080329960080981204840966206372671147224E6Q, + 1.411951256636576283942477881535283304912E3Q, + /* 1.000000000000000000000000000000000000000E0Q */ +}; + +/* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), + 0 <= 1/x <= .0625 + Peak relative error 3.6e-36 */ +#define NP16_IN 9 +static const __float128 P16_IN[NP16_IN + 1] = { + 5.143674369359646114999545149085139822905E-16Q, + 4.836645664124562546056389268546233577376E-13Q, + 1.730945562285804805325011561498453013673E-10Q, + 3.047976856147077889834905908605310585810E-8Q, + 2.855227609107969710407464739188141162386E-6Q, + 1.439362407936705484122143713643023998457E-4Q, + 3.774489768532936551500999699815873422073E-3Q, + 4.723962172984642566142399678920790598426E-2Q, + 2.359289678988743939925017240478818248735E-1Q, + 3.032580002220628812728954785118117124520E-1Q, +}; +#define NP16_ID 9 +static const __float128 P16_ID[NP16_ID + 1] = { + 4.389268795186898018132945193912677177553E-15Q, + 4.132671824807454334388868363256830961655E-12Q, + 1.482133328179508835835963635130894413136E-9Q, + 2.618941412861122118906353737117067376236E-7Q, + 2.467854246740858470815714426201888034270E-5Q, + 1.257192927368839847825938545925340230490E-3Q, + 3.362739031941574274949719324644120720341E-2Q, + 4.384458231338934105875343439265370178858E-1Q, + 2.412830809841095249170909628197264854651E0Q, + 4.176078204111348059102962617368214856874E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), + 0.0625 <= 1/x <= 0.125 + Peak relative error 1.9e-36 */ +#define NP8_16N 11 +static const __float128 P8_16N[NP8_16N + 1] = { + 2.984612480763362345647303274082071598135E-16Q, + 1.923651877544126103941232173085475682334E-13Q, + 4.881258879388869396043760693256024307743E-11Q, + 6.368866572475045408480898921866869811889E-9Q, + 4.684818344104910450523906967821090796737E-7Q, + 2.005177298271593587095982211091300382796E-5Q, + 4.979808067163957634120681477207147536182E-4Q, + 6.946005761642579085284689047091173581127E-3Q, + 5.074601112955765012750207555985299026204E-2Q, + 1.698599455896180893191766195194231825379E-1Q, + 1.957536905259237627737222775573623779638E-1Q, + 2.991314703282528370270179989044994319374E-2Q, +}; +#define NP8_16D 10 +static const __float128 P8_16D[NP8_16D + 1] = { + 2.546869316918069202079580939942463010937E-15Q, + 1.644650111942455804019788382157745229955E-12Q, + 4.185430770291694079925607420808011147173E-10Q, + 5.485331966975218025368698195861074143153E-8Q, + 4.062884421686912042335466327098932678905E-6Q, + 1.758139661060905948870523641319556816772E-4Q, + 4.445143889306356207566032244985607493096E-3Q, + 6.391901016293512632765621532571159071158E-2Q, + 4.933040207519900471177016015718145795434E-1Q, + 1.839144086168947712971630337250761842976E0Q, + 2.715120873995490920415616716916149586579E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), + 0.125 <= 1/x <= 0.1875 + Peak relative error 1.3e-36 */ +#define NP5_8N 10 +static const __float128 P5_8N[NP5_8N + 1] = { + 2.837678373978003452653763806968237227234E-12Q, + 9.726641165590364928442128579282742354806E-10Q, + 1.284408003604131382028112171490633956539E-7Q, + 8.524624695868291291250573339272194285008E-6Q, + 3.111516908953172249853673787748841282846E-4Q, + 6.423175156126364104172801983096596409176E-3Q, + 7.430220589989104581004416356260692450652E-2Q, + 4.608315409833682489016656279567605536619E-1Q, + 1.396870223510964882676225042258855977512E0Q, + 1.718500293904122365894630460672081526236E0Q, + 5.465927698800862172307352821870223855365E-1Q +}; +#define NP5_8D 10 +static const __float128 P5_8D[NP5_8D + 1] = { + 2.421485545794616609951168511612060482715E-11Q, + 8.329862750896452929030058039752327232310E-9Q, + 1.106137992233383429630592081375289010720E-6Q, + 7.405786153760681090127497796448503306939E-5Q, + 2.740364785433195322492093333127633465227E-3Q, + 5.781246470403095224872243564165254652198E-2Q, + 6.927711353039742469918754111511109983546E-1Q, + 4.558679283460430281188304515922826156690E0Q, + 1.534468499844879487013168065728837900009E1Q, + 2.313927430889218597919624843161569422745E1Q, + 1.194506341319498844336768473218382828637E1Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), + Peak relative error 1.4e-36 + 0.1875 <= 1/x <= 0.25 */ +#define NP4_5N 10 +static const __float128 P4_5N[NP4_5N + 1] = { + 1.846029078268368685834261260420933914621E-10Q, + 3.916295939611376119377869680335444207768E-8Q, + 3.122158792018920627984597530935323997312E-6Q, + 1.218073444893078303994045653603392272450E-4Q, + 2.536420827983485448140477159977981844883E-3Q, + 2.883011322006690823959367922241169171315E-2Q, + 1.755255190734902907438042414495469810830E-1Q, + 5.379317079922628599870898285488723736599E-1Q, + 7.284904050194300773890303361501726561938E-1Q, + 3.270110346613085348094396323925000362813E-1Q, + 1.804473805689725610052078464951722064757E-2Q, +}; +#define NP4_5D 9 +static const __float128 P4_5D[NP4_5D + 1] = { + 1.575278146806816970152174364308980863569E-9Q, + 3.361289173657099516191331123405675054321E-7Q, + 2.704692281550877810424745289838790693708E-5Q, + 1.070854930483999749316546199273521063543E-3Q, + 2.282373093495295842598097265627962125411E-2Q, + 2.692025460665354148328762368240343249830E-1Q, + 1.739892942593664447220951225734811133759E0Q, + 5.890727576752230385342377570386657229324E0Q, + 9.517442287057841500750256954117735128153E0Q, + 6.100616353935338240775363403030137736013E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), + Peak relative error 3.0e-36 + 0.25 <= 1/x <= 0.3125 */ +#define NP3r2_4N 9 +static const __float128 P3r2_4N[NP3r2_4N + 1] = { + 8.240803130988044478595580300846665863782E-8Q, + 1.179418958381961224222969866406483744580E-5Q, + 6.179787320956386624336959112503824397755E-4Q, + 1.540270833608687596420595830747166658383E-2Q, + 1.983904219491512618376375619598837355076E-1Q, + 1.341465722692038870390470651608301155565E0Q, + 4.617865326696612898792238245990854646057E0Q, + 7.435574801812346424460233180412308000587E0Q, + 4.671327027414635292514599201278557680420E0Q, + 7.299530852495776936690976966995187714739E-1Q, +}; +#define NP3r2_4D 9 +static const __float128 P3r2_4D[NP3r2_4D + 1] = { + 7.032152009675729604487575753279187576521E-7Q, + 1.015090352324577615777511269928856742848E-4Q, + 5.394262184808448484302067955186308730620E-3Q, + 1.375291438480256110455809354836988584325E-1Q, + 1.836247144461106304788160919310404376670E0Q, + 1.314378564254376655001094503090935880349E1Q, + 4.957184590465712006934452500894672343488E1Q, + 9.287394244300647738855415178790263465398E1Q, + 7.652563275535900609085229286020552768399E1Q, + 2.147042473003074533150718117770093209096E1Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), + Peak relative error 1.0e-35 + 0.3125 <= 1/x <= 0.375 */ +#define NP2r7_3r2N 9 +static const __float128 P2r7_3r2N[NP2r7_3r2N + 1] = { + 4.599033469240421554219816935160627085991E-7Q, + 4.665724440345003914596647144630893997284E-5Q, + 1.684348845667764271596142716944374892756E-3Q, + 2.802446446884455707845985913454440176223E-2Q, + 2.321937586453963310008279956042545173930E-1Q, + 9.640277413988055668692438709376437553804E-1Q, + 1.911021064710270904508663334033003246028E0Q, + 1.600811610164341450262992138893970224971E0Q, + 4.266299218652587901171386591543457861138E-1Q, + 1.316470424456061252962568223251247207325E-2Q, +}; +#define NP2r7_3r2D 8 +static const __float128 P2r7_3r2D[NP2r7_3r2D + 1] = { + 3.924508608545520758883457108453520099610E-6Q, + 4.029707889408829273226495756222078039823E-4Q, + 1.484629715787703260797886463307469600219E-2Q, + 2.553136379967180865331706538897231588685E-1Q, + 2.229457223891676394409880026887106228740E0Q, + 1.005708903856384091956550845198392117318E1Q, + 2.277082659664386953166629360352385889558E1Q, + 2.384726835193630788249826630376533988245E1Q, + 9.700989749041320895890113781610939632410E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), + Peak relative error 1.7e-36 + 0.3125 <= 1/x <= 0.4375 */ +#define NP2r3_2r7N 9 +static const __float128 P2r3_2r7N[NP2r3_2r7N + 1] = { + 3.916766777108274628543759603786857387402E-6Q, + 3.212176636756546217390661984304645137013E-4Q, + 9.255768488524816445220126081207248947118E-3Q, + 1.214853146369078277453080641911700735354E-1Q, + 7.855163309847214136198449861311404633665E-1Q, + 2.520058073282978403655488662066019816540E0Q, + 3.825136484837545257209234285382183711466E0Q, + 2.432569427554248006229715163865569506873E0Q, + 4.877934835018231178495030117729800489743E-1Q, + 1.109902737860249670981355149101343427885E-2Q, +}; +#define NP2r3_2r7D 8 +static const __float128 P2r3_2r7D[NP2r3_2r7D + 1] = { + 3.342307880794065640312646341190547184461E-5Q, + 2.782182891138893201544978009012096558265E-3Q, + 8.221304931614200702142049236141249929207E-2Q, + 1.123728246291165812392918571987858010949E0Q, + 7.740482453652715577233858317133423434590E0Q, + 2.737624677567945952953322566311201919139E1Q, + 4.837181477096062403118304137851260715475E1Q, + 3.941098643468580791437772701093795299274E1Q, + 1.245821247166544627558323920382547533630E1Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* J1(x)cosX + Y1(x)sinX = sqrt( 2/(pi x)) P1(x), P1(x) = 1 + 1/x^2 R(1/x^2), + Peak relative error 1.7e-35 + 0.4375 <= 1/x <= 0.5 */ +#define NP2_2r3N 8 +static const __float128 P2_2r3N[NP2_2r3N + 1] = { + 3.397930802851248553545191160608731940751E-4Q, + 2.104020902735482418784312825637833698217E-2Q, + 4.442291771608095963935342749477836181939E-1Q, + 4.131797328716583282869183304291833754967E0Q, + 1.819920169779026500146134832455189917589E1Q, + 3.781779616522937565300309684282401791291E1Q, + 3.459605449728864218972931220783543410347E1Q, + 1.173594248397603882049066603238568316561E1Q, + 9.455702270242780642835086549285560316461E-1Q, +}; +#define NP2_2r3D 8 +static const __float128 P2_2r3D[NP2_2r3D + 1] = { + 2.899568897241432883079888249845707400614E-3Q, + 1.831107138190848460767699919531132426356E-1Q, + 3.999350044057883839080258832758908825165E0Q, + 3.929041535867957938340569419874195303712E1Q, + 1.884245613422523323068802689915538908291E2Q, + 4.461469948819229734353852978424629815929E2Q, + 5.004998753999796821224085972610636347903E2Q, + 2.386342520092608513170837883757163414100E2Q, + 3.791322528149347975999851588922424189957E1Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), + Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), + Peak relative error 8.0e-36 + 0 <= 1/x <= .0625 */ +#define NQ16_IN 10 +static const __float128 Q16_IN[NQ16_IN + 1] = { + -3.917420835712508001321875734030357393421E-18Q, + -4.440311387483014485304387406538069930457E-15Q, + -1.951635424076926487780929645954007139616E-12Q, + -4.318256438421012555040546775651612810513E-10Q, + -5.231244131926180765270446557146989238020E-8Q, + -3.540072702902043752460711989234732357653E-6Q, + -1.311017536555269966928228052917534882984E-4Q, + -2.495184669674631806622008769674827575088E-3Q, + -2.141868222987209028118086708697998506716E-2Q, + -6.184031415202148901863605871197272650090E-2Q, + -1.922298704033332356899546792898156493887E-2Q, +}; +#define NQ16_ID 9 +static const __float128 Q16_ID[NQ16_ID + 1] = { + 3.820418034066293517479619763498400162314E-17Q, + 4.340702810799239909648911373329149354911E-14Q, + 1.914985356383416140706179933075303538524E-11Q, + 4.262333682610888819476498617261895474330E-9Q, + 5.213481314722233980346462747902942182792E-7Q, + 3.585741697694069399299005316809954590558E-5Q, + 1.366513429642842006385029778105539457546E-3Q, + 2.745282599850704662726337474371355160594E-2Q, + 2.637644521611867647651200098449903330074E-1Q, + 1.006953426110765984590782655598680488746E0Q, + /* 1.000000000000000000000000000000000000000E0 */ + }; + +/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), + Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), + Peak relative error 1.9e-36 + 0.0625 <= 1/x <= 0.125 */ +#define NQ8_16N 11 +static const __float128 Q8_16N[NQ8_16N + 1] = { + -2.028630366670228670781362543615221542291E-17Q, + -1.519634620380959966438130374006858864624E-14Q, + -4.540596528116104986388796594639405114524E-12Q, + -7.085151756671466559280490913558388648274E-10Q, + -6.351062671323970823761883833531546885452E-8Q, + -3.390817171111032905297982523519503522491E-6Q, + -1.082340897018886970282138836861233213972E-4Q, + -2.020120801187226444822977006648252379508E-3Q, + -2.093169910981725694937457070649605557555E-2Q, + -1.092176538874275712359269481414448063393E-1Q, + -2.374790947854765809203590474789108718733E-1Q, + -1.365364204556573800719985118029601401323E-1Q, +}; +#define NQ8_16D 11 +static const __float128 Q8_16D[NQ8_16D + 1] = { + 1.978397614733632533581207058069628242280E-16Q, + 1.487361156806202736877009608336766720560E-13Q, + 4.468041406888412086042576067133365913456E-11Q, + 7.027822074821007443672290507210594648877E-9Q, + 6.375740580686101224127290062867976007374E-7Q, + 3.466887658320002225888644977076410421940E-5Q, + 1.138625640905289601186353909213719596986E-3Q, + 2.224470799470414663443449818235008486439E-2Q, + 2.487052928527244907490589787691478482358E-1Q, + 1.483927406564349124649083853892380899217E0Q, + 4.182773513276056975777258788903489507705E0Q, + 4.419665392573449746043880892524360870944E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), + Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), + Peak relative error 1.5e-35 + 0.125 <= 1/x <= 0.1875 */ +#define NQ5_8N 10 +static const __float128 Q5_8N[NQ5_8N + 1] = { + -3.656082407740970534915918390488336879763E-13Q, + -1.344660308497244804752334556734121771023E-10Q, + -1.909765035234071738548629788698150760791E-8Q, + -1.366668038160120210269389551283666716453E-6Q, + -5.392327355984269366895210704976314135683E-5Q, + -1.206268245713024564674432357634540343884E-3Q, + -1.515456784370354374066417703736088291287E-2Q, + -1.022454301137286306933217746545237098518E-1Q, + -3.373438906472495080504907858424251082240E-1Q, + -4.510782522110845697262323973549178453405E-1Q, + -1.549000892545288676809660828213589804884E-1Q, +}; +#define NQ5_8D 10 +static const __float128 Q5_8D[NQ5_8D + 1] = { + 3.565550843359501079050699598913828460036E-12Q, + 1.321016015556560621591847454285330528045E-9Q, + 1.897542728662346479999969679234270605975E-7Q, + 1.381720283068706710298734234287456219474E-5Q, + 5.599248147286524662305325795203422873725E-4Q, + 1.305442352653121436697064782499122164843E-2Q, + 1.750234079626943298160445750078631894985E-1Q, + 1.311420542073436520965439883806946678491E0Q, + 5.162757689856842406744504211089724926650E0Q, + 9.527760296384704425618556332087850581308E0Q, + 6.604648207463236667912921642545100248584E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), + Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), + Peak relative error 1.3e-35 + 0.1875 <= 1/x <= 0.25 */ +#define NQ4_5N 10 +static const __float128 Q4_5N[NQ4_5N + 1] = { + -4.079513568708891749424783046520200903755E-11Q, + -9.326548104106791766891812583019664893311E-9Q, + -8.016795121318423066292906123815687003356E-7Q, + -3.372350544043594415609295225664186750995E-5Q, + -7.566238665947967882207277686375417983917E-4Q, + -9.248861580055565402130441618521591282617E-3Q, + -6.033106131055851432267702948850231270338E-2Q, + -1.966908754799996793730369265431584303447E-1Q, + -2.791062741179964150755788226623462207560E-1Q, + -1.255478605849190549914610121863534191666E-1Q, + -4.320429862021265463213168186061696944062E-3Q, +}; +#define NQ4_5D 9 +static const __float128 Q4_5D[NQ4_5D + 1] = { + 3.978497042580921479003851216297330701056E-10Q, + 9.203304163828145809278568906420772246666E-8Q, + 8.059685467088175644915010485174545743798E-6Q, + 3.490187375993956409171098277561669167446E-4Q, + 8.189109654456872150100501732073810028829E-3Q, + 1.072572867311023640958725265762483033769E-1Q, + 7.790606862409960053675717185714576937994E-1Q, + 3.016049768232011196434185423512777656328E0Q, + 5.722963851442769787733717162314477949360E0Q, + 4.510527838428473279647251350931380867663E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), + Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), + Peak relative error 2.1e-35 + 0.25 <= 1/x <= 0.3125 */ +#define NQ3r2_4N 9 +static const __float128 Q3r2_4N[NQ3r2_4N + 1] = { + -1.087480809271383885936921889040388133627E-8Q, + -1.690067828697463740906962973479310170932E-6Q, + -9.608064416995105532790745641974762550982E-5Q, + -2.594198839156517191858208513873961837410E-3Q, + -3.610954144421543968160459863048062977822E-2Q, + -2.629866798251843212210482269563961685666E-1Q, + -9.709186825881775885917984975685752956660E-1Q, + -1.667521829918185121727268867619982417317E0Q, + -1.109255082925540057138766105229900943501E0Q, + -1.812932453006641348145049323713469043328E-1Q, +}; +#define NQ3r2_4D 9 +static const __float128 Q3r2_4D[NQ3r2_4D + 1] = { + 1.060552717496912381388763753841473407026E-7Q, + 1.676928002024920520786883649102388708024E-5Q, + 9.803481712245420839301400601140812255737E-4Q, + 2.765559874262309494758505158089249012930E-2Q, + 4.117921827792571791298862613287549140706E-1Q, + 3.323769515244751267093378361930279161413E0Q, + 1.436602494405814164724810151689705353670E1Q, + 3.163087869617098638064881410646782408297E1Q, + 3.198181264977021649489103980298349589419E1Q, + 1.203649258862068431199471076202897823272E1Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), + Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), + Peak relative error 1.6e-36 + 0.3125 <= 1/x <= 0.375 */ +#define NQ2r7_3r2N 9 +static const __float128 Q2r7_3r2N[NQ2r7_3r2N + 1] = { + -1.723405393982209853244278760171643219530E-7Q, + -2.090508758514655456365709712333460087442E-5Q, + -9.140104013370974823232873472192719263019E-4Q, + -1.871349499990714843332742160292474780128E-2Q, + -1.948930738119938669637865956162512983416E-1Q, + -1.048764684978978127908439526343174139788E0Q, + -2.827714929925679500237476105843643064698E0Q, + -3.508761569156476114276988181329773987314E0Q, + -1.669332202790211090973255098624488308989E0Q, + -1.930796319299022954013840684651016077770E-1Q, +}; +#define NQ2r7_3r2D 9 +static const __float128 Q2r7_3r2D[NQ2r7_3r2D + 1] = { + 1.680730662300831976234547482334347983474E-6Q, + 2.084241442440551016475972218719621841120E-4Q, + 9.445316642108367479043541702688736295579E-3Q, + 2.044637889456631896650179477133252184672E-1Q, + 2.316091982244297350829522534435350078205E0Q, + 1.412031891783015085196708811890448488865E1Q, + 4.583830154673223384837091077279595496149E1Q, + 7.549520609270909439885998474045974122261E1Q, + 5.697605832808113367197494052388203310638E1Q, + 1.601496240876192444526383314589371686234E1Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), + Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), + Peak relative error 9.5e-36 + 0.375 <= 1/x <= 0.4375 */ +#define NQ2r3_2r7N 9 +static const __float128 Q2r3_2r7N[NQ2r3_2r7N + 1] = { + -8.603042076329122085722385914954878953775E-7Q, + -7.701746260451647874214968882605186675720E-5Q, + -2.407932004380727587382493696877569654271E-3Q, + -3.403434217607634279028110636919987224188E-2Q, + -2.348707332185238159192422084985713102877E-1Q, + -7.957498841538254916147095255700637463207E-1Q, + -1.258469078442635106431098063707934348577E0Q, + -8.162415474676345812459353639449971369890E-1Q, + -1.581783890269379690141513949609572806898E-1Q, + -1.890595651683552228232308756569450822905E-3Q, +}; +#define NQ2r3_2r7D 8 +static const __float128 Q2r3_2r7D[NQ2r3_2r7D + 1] = { + 8.390017524798316921170710533381568175665E-6Q, + 7.738148683730826286477254659973968763659E-4Q, + 2.541480810958665794368759558791634341779E-2Q, + 3.878879789711276799058486068562386244873E-1Q, + 3.003783779325811292142957336802456109333E0Q, + 1.206480374773322029883039064575464497400E1Q, + 2.458414064785315978408974662900438351782E1Q, + 2.367237826273668567199042088835448715228E1Q, + 9.231451197519171090875569102116321676763E0Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + +/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x), + Q1(x) = 1/x (.375 + 1/x^2 R(1/x^2)), + Peak relative error 1.4e-36 + 0.4375 <= 1/x <= 0.5 */ +#define NQ2_2r3N 9 +static const __float128 Q2_2r3N[NQ2_2r3N + 1] = { + -5.552507516089087822166822364590806076174E-6Q, + -4.135067659799500521040944087433752970297E-4Q, + -1.059928728869218962607068840646564457980E-2Q, + -1.212070036005832342565792241385459023801E-1Q, + -6.688350110633603958684302153362735625156E-1Q, + -1.793587878197360221340277951304429821582E0Q, + -2.225407682237197485644647380483725045326E0Q, + -1.123402135458940189438898496348239744403E0Q, + -1.679187241566347077204805190763597299805E-1Q, + -1.458550613639093752909985189067233504148E-3Q, +}; +#define NQ2_2r3D 8 +static const __float128 Q2_2r3D[NQ2_2r3D + 1] = { + 5.415024336507980465169023996403597916115E-5Q, + 4.179246497380453022046357404266022870788E-3Q, + 1.136306384261959483095442402929502368598E-1Q, + 1.422640343719842213484515445393284072830E0Q, + 8.968786703393158374728850922289204805764E0Q, + 2.914542473339246127533384118781216495934E1Q, + 4.781605421020380669870197378210457054685E1Q, + 3.693865837171883152382820584714795072937E1Q, + 1.153220502744204904763115556224395893076E1Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + + +/* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ + +static __float128 +neval (__float128 x, const __float128 *p, int n) +{ + __float128 y; + + p += n; + y = *p--; + do + { + y = y * x + *p--; + } + while (--n > 0); + return y; +} + + +/* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */ + +static __float128 +deval (__float128 x, const __float128 *p, int n) +{ + __float128 y; + + p += n; + y = x + *p--; + do + { + y = y * x + *p--; + } + while (--n > 0); + return y; +} + + +/* Bessel function of the first kind, order one. */ + +__float128 +j1q (__float128 x) +{ + __float128 xx, xinv, z, p, q, c, s, cc, ss; + + if (! finiteq (x)) + { + if (x != x) + return x; + else + return 0.0Q; + } + if (x == 0.0Q) + return x; + xx = fabsq (x); + if (xx <= 2.0Q) + { + /* 0 <= x <= 2 */ + z = xx * xx; + p = xx * z * neval (z, J0_2N, NJ0_2N) / deval (z, J0_2D, NJ0_2D); + p += 0.5Q * xx; + if (x < 0) + p = -p; + return p; + } + + xinv = 1.0Q / xx; + z = xinv * xinv; + if (xinv <= 0.25) + { + if (xinv <= 0.125) + { + if (xinv <= 0.0625) + { + p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID); + q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID); + } + else + { + p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D); + q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D); + } + } + else if (xinv <= 0.1875) + { + p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D); + q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D); + } + else + { + p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D); + q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D); + } + } /* .25 */ + else /* if (xinv <= 0.5) */ + { + if (xinv <= 0.375) + { + if (xinv <= 0.3125) + { + p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D); + q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D); + } + else + { + p = neval (z, P2r7_3r2N, NP2r7_3r2N) + / deval (z, P2r7_3r2D, NP2r7_3r2D); + q = neval (z, Q2r7_3r2N, NQ2r7_3r2N) + / deval (z, Q2r7_3r2D, NQ2r7_3r2D); + } + } + else if (xinv <= 0.4375) + { + p = neval (z, P2r3_2r7N, NP2r3_2r7N) + / deval (z, P2r3_2r7D, NP2r3_2r7D); + q = neval (z, Q2r3_2r7N, NQ2r3_2r7N) + / deval (z, Q2r3_2r7D, NQ2r3_2r7D); + } + else + { + p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D); + q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D); + } + } + p = 1.0Q + z * p; + q = z * q; + q = q * xinv + 0.375Q * xinv; + /* X = x - 3 pi/4 + cos(X) = cos(x) cos(3 pi/4) + sin(x) sin(3 pi/4) + = 1/sqrt(2) * (-cos(x) + sin(x)) + sin(X) = sin(x) cos(3 pi/4) - cos(x) sin(3 pi/4) + = -1/sqrt(2) * (sin(x) + cos(x)) + cf. Fdlibm. */ + sincosq (xx, &s, &c); + ss = -s - c; + cc = s - c; + z = cosq (xx + xx); + if ((s * c) > 0) + cc = z / ss; + else + ss = z / cc; + z = ONEOSQPI * (p * cc - q * ss) / sqrtq (xx); + if (x < 0) + z = -z; + return z; +} + + +/* Y1(x) = 2/pi * (log(x) * J1(x) - 1/x) + x R(x^2) + Peak relative error 6.2e-38 + 0 <= x <= 2 */ +#define NY0_2N 7 +static __float128 Y0_2N[NY0_2N + 1] = { + -6.804415404830253804408698161694720833249E19Q, + 1.805450517967019908027153056150465849237E19Q, + -8.065747497063694098810419456383006737312E17Q, + 1.401336667383028259295830955439028236299E16Q, + -1.171654432898137585000399489686629680230E14Q, + 5.061267920943853732895341125243428129150E11Q, + -1.096677850566094204586208610960870217970E9Q, + 9.541172044989995856117187515882879304461E5Q, +}; +#define NY0_2D 7 +static __float128 Y0_2D[NY0_2D + 1] = { + 3.470629591820267059538637461549677594549E20Q, + 4.120796439009916326855848107545425217219E18Q, + 2.477653371652018249749350657387030814542E16Q, + 9.954678543353888958177169349272167762797E13Q, + 2.957927997613630118216218290262851197754E11Q, + 6.748421382188864486018861197614025972118E8Q, + 1.173453425218010888004562071020305709319E6Q, + 1.450335662961034949894009554536003377187E3Q, + /* 1.000000000000000000000000000000000000000E0 */ +}; + + +/* Bessel function of the second kind, order one. */ + +__float128 +y1q (__float128 x) +{ + __float128 xx, xinv, z, p, q, c, s, cc, ss; + + if (! finiteq (x)) + { + if (x != x) + return x; + else + return 0.0Q; + } + if (x <= 0.0Q) + { + if (x < 0.0Q) + return (zero / (zero * x)); + return -HUGE_VALQ + x; + } + xx = fabsq (x); + if (xx <= 2.0Q) + { + /* 0 <= x <= 2 */ + z = xx * xx; + p = xx * neval (z, Y0_2N, NY0_2N) / deval (z, Y0_2D, NY0_2D); + p = -TWOOPI / xx + p; + p = TWOOPI * logq (x) * j1q (x) + p; + return p; + } + + xinv = 1.0Q / xx; + z = xinv * xinv; + if (xinv <= 0.25) + { + if (xinv <= 0.125) + { + if (xinv <= 0.0625) + { + p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID); + q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID); + } + else + { + p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D); + q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D); + } + } + else if (xinv <= 0.1875) + { + p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D); + q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D); + } + else + { + p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D); + q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D); + } + } /* .25 */ + else /* if (xinv <= 0.5) */ + { + if (xinv <= 0.375) + { + if (xinv <= 0.3125) + { + p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D); + q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D); + } + else + { + p = neval (z, P2r7_3r2N, NP2r7_3r2N) + / deval (z, P2r7_3r2D, NP2r7_3r2D); + q = neval (z, Q2r7_3r2N, NQ2r7_3r2N) + / deval (z, Q2r7_3r2D, NQ2r7_3r2D); + } + } + else if (xinv <= 0.4375) + { + p = neval (z, P2r3_2r7N, NP2r3_2r7N) + / deval (z, P2r3_2r7D, NP2r3_2r7D); + q = neval (z, Q2r3_2r7N, NQ2r3_2r7N) + / deval (z, Q2r3_2r7D, NQ2r3_2r7D); + } + else + { + p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D); + q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D); + } + } + p = 1.0Q + z * p; + q = z * q; + q = q * xinv + 0.375Q * xinv; + /* X = x - 3 pi/4 + cos(X) = cos(x) cos(3 pi/4) + sin(x) sin(3 pi/4) + = 1/sqrt(2) * (-cos(x) + sin(x)) + sin(X) = sin(x) cos(3 pi/4) - cos(x) sin(3 pi/4) + = -1/sqrt(2) * (sin(x) + cos(x)) + cf. Fdlibm. */ + sincosq (xx, &s, &c); + ss = -s - c; + cc = s - c; + z = cosq (xx + xx); + if ((s * c) > 0) + cc = z / ss; + else + ss = z / cc; + z = ONEOSQPI * (p * ss + q * cc) / sqrtq (xx); + return z; +} diff --git a/libquadmath/math/jnq.c b/libquadmath/math/jnq.c new file mode 100644 index 000000000..d82947a3c --- /dev/null +++ b/libquadmath/math/jnq.c @@ -0,0 +1,381 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* Modifications for 128-bit long double are + Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> + and are incorporated herein by permission of the author. The author + reserves the right to distribute this material elsewhere under different + copying permissions. These modifications are distributed here under + the following terms: + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + +/* + * __ieee754_jn(n, x), __ieee754_yn(n, x) + * floating point Bessel's function of the 1st and 2nd kind + * of order n + * + * Special cases: + * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; + * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. + * Note 2. About jn(n,x), yn(n,x) + * For n=0, j0(x) is called, + * for n=1, j1(x) is called, + * for n<x, forward recursion us used starting + * from values of j0(x) and j1(x). + * for n>x, a continued fraction approximation to + * j(n,x)/j(n-1,x) is evaluated and then backward + * recursion is used starting from a supposed value + * for j(n,x). The resulting value of j(0,x) is + * compared with the actual value to correct the + * supposed value of j(n,x). + * + * yn(n,x) is similar in all respects, except + * that forward recursion is used for all + * values of n>1. + * + */ + +#include "quadmath-imp.h" + +static const __float128 + invsqrtpi = 5.6418958354775628694807945156077258584405E-1Q, + two = 2.0e0Q, + one = 1.0e0Q, + zero = 0.0Q; + + +__float128 +jnq (int n, __float128 x) +{ + uint32_t se; + int32_t i, ix, sgn; + __float128 a, b, temp, di; + __float128 z, w; + ieee854_float128 u; + + + /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) + * Thus, J(-n,x) = J(n,-x) + */ + + u.value = x; + se = u.words32.w0; + ix = se & 0x7fffffff; + + /* if J(n,NaN) is NaN */ + if (ix >= 0x7fff0000) + { + if ((u.words32.w0 & 0xffff) | u.words32.w1 | u.words32.w2 | u.words32.w3) + return x + x; + } + + if (n < 0) + { + n = -n; + x = -x; + se ^= 0x80000000; + } + if (n == 0) + return (j0q (x)); + if (n == 1) + return (j1q (x)); + sgn = (n & 1) & (se >> 31); /* even n -- 0, odd n -- sign(x) */ + x = fabsq (x); + + if (x == 0.0Q || ix >= 0x7fff0000) /* if x is 0 or inf */ + b = zero; + else if ((__float128) n <= x) + { + /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ + if (ix >= 0x412D0000) + { /* x > 2**302 */ + + /* ??? Could use an expansion for large x here. */ + + /* (x >> n**2) + * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Let s=sin(x), c=cos(x), + * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then + * + * n sin(xn)*sqt2 cos(xn)*sqt2 + * ---------------------------------- + * 0 s-c c+s + * 1 -s-c -c+s + * 2 -s+c -c-s + * 3 s+c c-s + */ + __float128 s; + __float128 c; + sincosq (x, &s, &c); + switch (n & 3) + { + case 0: + temp = c + s; + break; + case 1: + temp = -c + s; + break; + case 2: + temp = -c - s; + break; + case 3: + temp = c - s; + break; + } + b = invsqrtpi * temp / sqrtq (x); + } + else + { + a = j0q (x); + b = j1q (x); + for (i = 1; i < n; i++) + { + temp = b; + b = b * ((__float128) (i + i) / x) - a; /* avoid underflow */ + a = temp; + } + } + } + else + { + if (ix < 0x3fc60000) + { /* x < 2**-57 */ + /* x is tiny, return the first Taylor expansion of J(n,x) + * J(n,x) = 1/n!*(x/2)^n - ... + */ + if (n >= 400) /* underflow, result < 10^-4952 */ + b = zero; + else + { + temp = x * 0.5; + b = temp; + for (a = one, i = 2; i <= n; i++) + { + a *= (__float128) i; /* a = n! */ + b *= temp; /* b = (x/2)^n */ + } + b = b / a; + } + } + else + { + /* use backward recurrence */ + /* x x^2 x^2 + * J(n,x)/J(n-1,x) = ---- ------ ------ ..... + * 2n - 2(n+1) - 2(n+2) + * + * 1 1 1 + * (for large x) = ---- ------ ------ ..... + * 2n 2(n+1) 2(n+2) + * -- - ------ - ------ - + * x x x + * + * Let w = 2n/x and h=2/x, then the above quotient + * is equal to the continued fraction: + * 1 + * = ----------------------- + * 1 + * w - ----------------- + * 1 + * w+h - --------- + * w+2h - ... + * + * To determine how many terms needed, let + * Q(0) = w, Q(1) = w(w+h) - 1, + * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), + * When Q(k) > 1e4 good for single + * When Q(k) > 1e9 good for double + * When Q(k) > 1e17 good for quadruple + */ + /* determine k */ + __float128 t, v; + __float128 q0, q1, h, tmp; + int32_t k, m; + w = (n + n) / (__float128) x; + h = 2.0Q / (__float128) x; + q0 = w; + z = w + h; + q1 = w * z - 1.0Q; + k = 1; + while (q1 < 1.0e17Q) + { + k += 1; + z += h; + tmp = z * q1 - q0; + q0 = q1; + q1 = tmp; + } + m = n + n; + for (t = zero, i = 2 * (n + k); i >= m; i -= 2) + t = one / (i / x - t); + a = t; + b = one; + /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) + * Hence, if n*(log(2n/x)) > ... + * single 8.8722839355e+01 + * double 7.09782712893383973096e+02 + * __float128 1.1356523406294143949491931077970765006170e+04 + * then recurrent value may overflow and the result is + * likely underflow to zero + */ + tmp = n; + v = two / x; + tmp = tmp * logq (fabsq (v * tmp)); + + if (tmp < 1.1356523406294143949491931077970765006170e+04Q) + { + for (i = n - 1, di = (__float128) (i + i); i > 0; i--) + { + temp = b; + b *= di; + b = b / x - a; + a = temp; + di -= two; + } + } + else + { + for (i = n - 1, di = (__float128) (i + i); i > 0; i--) + { + temp = b; + b *= di; + b = b / x - a; + a = temp; + di -= two; + /* scale b to avoid spurious overflow */ + if (b > 1e100Q) + { + a /= b; + t /= b; + b = one; + } + } + } + b = (t * j0q (x) / b); + } + } + if (sgn == 1) + return -b; + else + return b; +} + +__float128 +ynq (int n, __float128 x) +{ + uint32_t se; + int32_t i, ix; + int32_t sign; + __float128 a, b, temp; + ieee854_float128 u; + + u.value = x; + se = u.words32.w0; + ix = se & 0x7fffffff; + + /* if Y(n,NaN) is NaN */ + if (ix >= 0x7fff0000) + { + if ((u.words32.w0 & 0xffff) | u.words32.w1 | u.words32.w2 | u.words32.w3) + return x + x; + } + if (x <= 0.0Q) + { + if (x == 0.0Q) + return -HUGE_VALQ + x; + if (se & 0x80000000) + return zero / (zero * x); + } + sign = 1; + if (n < 0) + { + n = -n; + sign = 1 - ((n & 1) << 1); + } + if (n == 0) + return (y0q (x)); + if (n == 1) + return (sign * y1q (x)); + if (ix >= 0x7fff0000) + return zero; + if (ix >= 0x412D0000) + { /* x > 2**302 */ + + /* ??? See comment above on the possible futility of this. */ + + /* (x >> n**2) + * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Let s=sin(x), c=cos(x), + * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then + * + * n sin(xn)*sqt2 cos(xn)*sqt2 + * ---------------------------------- + * 0 s-c c+s + * 1 -s-c -c+s + * 2 -s+c -c-s + * 3 s+c c-s + */ + __float128 s; + __float128 c; + sincosq (x, &s, &c); + switch (n & 3) + { + case 0: + temp = s - c; + break; + case 1: + temp = -s - c; + break; + case 2: + temp = -s + c; + break; + case 3: + temp = s + c; + break; + } + b = invsqrtpi * temp / sqrtq (x); + } + else + { + a = y0q (x); + b = y1q (x); + /* quit if b is -inf */ + u.value = b; + se = u.words32.w0 & 0xffff0000; + for (i = 1; i < n && se != 0xffff0000; i++) + { + temp = b; + b = ((__float128) (i + i) / x) * b - a; + u.value = b; + se = u.words32.w0 & 0xffff0000; + a = temp; + } + } + if (sign > 0) + return b; + else + return -b; +} diff --git a/libquadmath/math/ldexpq.c b/libquadmath/math/ldexpq.c new file mode 100644 index 000000000..394d4590c --- /dev/null +++ b/libquadmath/math/ldexpq.c @@ -0,0 +1,27 @@ +/* s_ldexpl.c -- long double version of s_ldexp.c. + * Conversion to long double by Ulrich Drepper, + * Cygnus Support, drepper@cygnus.com. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include <errno.h> +#include "quadmath-imp.h" + +__float128 +ldexpq (__float128 value, int exp) +{ + if(!finiteq(value)||value==0.0Q) return value; + value = scalbnq(value,exp); + if(!finiteq(value)||value==0.0Q) errno = ERANGE; + return value; +} diff --git a/libquadmath/math/lgammaq.c b/libquadmath/math/lgammaq.c new file mode 100644 index 000000000..6e7697ac6 --- /dev/null +++ b/libquadmath/math/lgammaq.c @@ -0,0 +1,1034 @@ +/* lgammal + * + * Natural logarithm of gamma function + * + * + * + * SYNOPSIS: + * + * __float128 x, y, lgammal(); + * extern int sgngam; + * + * y = lgammal(x); + * + * + * + * DESCRIPTION: + * + * Returns the base e (2.718...) logarithm of the absolute + * value of the gamma function of the argument. + * The sign (+1 or -1) of the gamma function is returned in a + * global (extern) variable named sgngam. + * + * The positive domain is partitioned into numerous segments for approximation. + * For x > 10, + * log gamma(x) = (x - 0.5) log(x) - x + log sqrt(2 pi) + 1/x R(1/x^2) + * Near the minimum at x = x0 = 1.46... the approximation is + * log gamma(x0 + z) = log gamma(x0) + z^2 P(z)/Q(z) + * for small z. + * Elsewhere between 0 and 10, + * log gamma(n + z) = log gamma(n) + z P(z)/Q(z) + * for various selected n and small z. + * + * The cosecant reflection formula is employed for negative arguments. + * + * + * + * ACCURACY: + * + * + * arithmetic domain # trials peak rms + * Relative error: + * IEEE 10, 30 100000 3.9e-34 9.8e-35 + * IEEE 0, 10 100000 3.8e-34 5.3e-35 + * Absolute error: + * IEEE -10, 0 100000 8.0e-34 8.0e-35 + * IEEE -30, -10 100000 4.4e-34 1.0e-34 + * IEEE -100, 100 100000 1.0e-34 + * + * The absolute error criterion is the same as relative error + * when the function magnitude is greater than one but it is absolute + * when the magnitude is less than one. + * + */ + +/* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov> + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + +#include "quadmath-imp.h" + +static const __float128 PIQ = 3.1415926535897932384626433832795028841972E0Q; +static const __float128 MAXLGM = 1.0485738685148938358098967157129705071571E4928Q; +static const __float128 one = 1.0Q; +static const __float128 zero = 0.0Q; +static const __float128 huge = 1.0e4000Q; + +/* log gamma(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x P(1/x^2) + 1/x <= 0.0741 (x >= 13.495...) + Peak relative error 1.5e-36 */ +static const __float128 ls2pi = 9.1893853320467274178032973640561763986140E-1Q; +#define NRASY 12 +static const __float128 RASY[NRASY + 1] = +{ + 8.333333333333333333333333333310437112111E-2Q, + -2.777777777777777777777774789556228296902E-3Q, + 7.936507936507936507795933938448586499183E-4Q, + -5.952380952380952041799269756378148574045E-4Q, + 8.417508417507928904209891117498524452523E-4Q, + -1.917526917481263997778542329739806086290E-3Q, + 6.410256381217852504446848671499409919280E-3Q, + -2.955064066900961649768101034477363301626E-2Q, + 1.796402955865634243663453415388336954675E-1Q, + -1.391522089007758553455753477688592767741E0Q, + 1.326130089598399157988112385013829305510E1Q, + -1.420412699593782497803472576479997819149E2Q, + 1.218058922427762808938869872528846787020E3Q +}; + + +/* log gamma(x+13) = log gamma(13) + x P(x)/Q(x) + -0.5 <= x <= 0.5 + 12.5 <= x+13 <= 13.5 + Peak relative error 1.1e-36 */ +static const __float128 lgam13a = 1.9987213134765625E1Q; +static const __float128 lgam13b = 1.3608962611495173623870550785125024484248E-6Q; +#define NRN13 7 +static const __float128 RN13[NRN13 + 1] = +{ + 8.591478354823578150238226576156275285700E11Q, + 2.347931159756482741018258864137297157668E11Q, + 2.555408396679352028680662433943000804616E10Q, + 1.408581709264464345480765758902967123937E9Q, + 4.126759849752613822953004114044451046321E7Q, + 6.133298899622688505854211579222889943778E5Q, + 3.929248056293651597987893340755876578072E3Q, + 6.850783280018706668924952057996075215223E0Q +}; +#define NRD13 6 +static const __float128 RD13[NRD13 + 1] = +{ + 3.401225382297342302296607039352935541669E11Q, + 8.756765276918037910363513243563234551784E10Q, + 8.873913342866613213078554180987647243903E9Q, + 4.483797255342763263361893016049310017973E8Q, + 1.178186288833066430952276702931512870676E7Q, + 1.519928623743264797939103740132278337476E5Q, + 7.989298844938119228411117593338850892311E2Q + /* 1.0E0Q */ +}; + + +/* log gamma(x+12) = log gamma(12) + x P(x)/Q(x) + -0.5 <= x <= 0.5 + 11.5 <= x+12 <= 12.5 + Peak relative error 4.1e-36 */ +static const __float128 lgam12a = 1.75023040771484375E1Q; +static const __float128 lgam12b = 3.7687254483392876529072161996717039575982E-6Q; +#define NRN12 7 +static const __float128 RN12[NRN12 + 1] = +{ + 4.709859662695606986110997348630997559137E11Q, + 1.398713878079497115037857470168777995230E11Q, + 1.654654931821564315970930093932954900867E10Q, + 9.916279414876676861193649489207282144036E8Q, + 3.159604070526036074112008954113411389879E7Q, + 5.109099197547205212294747623977502492861E5Q, + 3.563054878276102790183396740969279826988E3Q, + 6.769610657004672719224614163196946862747E0Q +}; +#define NRD12 6 +static const __float128 RD12[NRD12 + 1] = +{ + 1.928167007860968063912467318985802726613E11Q, + 5.383198282277806237247492369072266389233E10Q, + 5.915693215338294477444809323037871058363E9Q, + 3.241438287570196713148310560147925781342E8Q, + 9.236680081763754597872713592701048455890E6Q, + 1.292246897881650919242713651166596478850E5Q, + 7.366532445427159272584194816076600211171E2Q + /* 1.0E0Q */ +}; + + +/* log gamma(x+11) = log gamma(11) + x P(x)/Q(x) + -0.5 <= x <= 0.5 + 10.5 <= x+11 <= 11.5 + Peak relative error 1.8e-35 */ +static const __float128 lgam11a = 1.5104400634765625E1Q; +static const __float128 lgam11b = 1.1938309890295225709329251070371882250744E-5Q; +#define NRN11 7 +static const __float128 RN11[NRN11 + 1] = +{ + 2.446960438029415837384622675816736622795E11Q, + 7.955444974446413315803799763901729640350E10Q, + 1.030555327949159293591618473447420338444E10Q, + 6.765022131195302709153994345470493334946E8Q, + 2.361892792609204855279723576041468347494E7Q, + 4.186623629779479136428005806072176490125E5Q, + 3.202506022088912768601325534149383594049E3Q, + 6.681356101133728289358838690666225691363E0Q +}; +#define NRD11 6 +static const __float128 RD11[NRD11 + 1] = +{ + 1.040483786179428590683912396379079477432E11Q, + 3.172251138489229497223696648369823779729E10Q, + 3.806961885984850433709295832245848084614E9Q, + 2.278070344022934913730015420611609620171E8Q, + 7.089478198662651683977290023829391596481E6Q, + 1.083246385105903533237139380509590158658E5Q, + 6.744420991491385145885727942219463243597E2Q + /* 1.0E0Q */ +}; + + +/* log gamma(x+10) = log gamma(10) + x P(x)/Q(x) + -0.5 <= x <= 0.5 + 9.5 <= x+10 <= 10.5 + Peak relative error 5.4e-37 */ +static const __float128 lgam10a = 1.280181884765625E1Q; +static const __float128 lgam10b = 8.6324252196112077178745667061642811492557E-6Q; +#define NRN10 7 +static const __float128 RN10[NRN10 + 1] = +{ + -1.239059737177249934158597996648808363783E14Q, + -4.725899566371458992365624673357356908719E13Q, + -7.283906268647083312042059082837754850808E12Q, + -5.802855515464011422171165179767478794637E11Q, + -2.532349691157548788382820303182745897298E10Q, + -5.884260178023777312587193693477072061820E8Q, + -6.437774864512125749845840472131829114906E6Q, + -2.350975266781548931856017239843273049384E4Q +}; +#define NRD10 7 +static const __float128 RD10[NRD10 + 1] = +{ + -5.502645997581822567468347817182347679552E13Q, + -1.970266640239849804162284805400136473801E13Q, + -2.819677689615038489384974042561531409392E12Q, + -2.056105863694742752589691183194061265094E11Q, + -8.053670086493258693186307810815819662078E9Q, + -1.632090155573373286153427982504851867131E8Q, + -1.483575879240631280658077826889223634921E6Q, + -4.002806669713232271615885826373550502510E3Q + /* 1.0E0Q */ +}; + + +/* log gamma(x+9) = log gamma(9) + x P(x)/Q(x) + -0.5 <= x <= 0.5 + 8.5 <= x+9 <= 9.5 + Peak relative error 3.6e-36 */ +static const __float128 lgam9a = 1.06045989990234375E1Q; +static const __float128 lgam9b = 3.9037218127284172274007216547549861681400E-6Q; +#define NRN9 7 +static const __float128 RN9[NRN9 + 1] = +{ + -4.936332264202687973364500998984608306189E13Q, + -2.101372682623700967335206138517766274855E13Q, + -3.615893404644823888655732817505129444195E12Q, + -3.217104993800878891194322691860075472926E11Q, + -1.568465330337375725685439173603032921399E10Q, + -4.073317518162025744377629219101510217761E8Q, + -4.983232096406156139324846656819246974500E6Q, + -2.036280038903695980912289722995505277253E4Q +}; +#define NRD9 7 +static const __float128 RD9[NRD9 + 1] = +{ + -2.306006080437656357167128541231915480393E13Q, + -9.183606842453274924895648863832233799950E12Q, + -1.461857965935942962087907301194381010380E12Q, + -1.185728254682789754150068652663124298303E11Q, + -5.166285094703468567389566085480783070037E9Q, + -1.164573656694603024184768200787835094317E8Q, + -1.177343939483908678474886454113163527909E6Q, + -3.529391059783109732159524500029157638736E3Q + /* 1.0E0Q */ +}; + + +/* log gamma(x+8) = log gamma(8) + x P(x)/Q(x) + -0.5 <= x <= 0.5 + 7.5 <= x+8 <= 8.5 + Peak relative error 2.4e-37 */ +static const __float128 lgam8a = 8.525146484375E0Q; +static const __float128 lgam8b = 1.4876690414300165531036347125050759667737E-5Q; +#define NRN8 8 +static const __float128 RN8[NRN8 + 1] = +{ + 6.600775438203423546565361176829139703289E11Q, + 3.406361267593790705240802723914281025800E11Q, + 7.222460928505293914746983300555538432830E10Q, + 8.102984106025088123058747466840656458342E9Q, + 5.157620015986282905232150979772409345927E8Q, + 1.851445288272645829028129389609068641517E7Q, + 3.489261702223124354745894067468953756656E5Q, + 2.892095396706665774434217489775617756014E3Q, + 6.596977510622195827183948478627058738034E0Q +}; +#define NRD8 7 +static const __float128 RD8[NRD8 + 1] = +{ + 3.274776546520735414638114828622673016920E11Q, + 1.581811207929065544043963828487733970107E11Q, + 3.108725655667825188135393076860104546416E10Q, + 3.193055010502912617128480163681842165730E9Q, + 1.830871482669835106357529710116211541839E8Q, + 5.790862854275238129848491555068073485086E6Q, + 9.305213264307921522842678835618803553589E4Q, + 6.216974105861848386918949336819572333622E2Q + /* 1.0E0Q */ +}; + + +/* log gamma(x+7) = log gamma(7) + x P(x)/Q(x) + -0.5 <= x <= 0.5 + 6.5 <= x+7 <= 7.5 + Peak relative error 3.2e-36 */ +static const __float128 lgam7a = 6.5792388916015625E0Q; +static const __float128 lgam7b = 1.2320408538495060178292903945321122583007E-5Q; +#define NRN7 8 +static const __float128 RN7[NRN7 + 1] = +{ + 2.065019306969459407636744543358209942213E11Q, + 1.226919919023736909889724951708796532847E11Q, + 2.996157990374348596472241776917953749106E10Q, + 3.873001919306801037344727168434909521030E9Q, + 2.841575255593761593270885753992732145094E8Q, + 1.176342515359431913664715324652399565551E7Q, + 2.558097039684188723597519300356028511547E5Q, + 2.448525238332609439023786244782810774702E3Q, + 6.460280377802030953041566617300902020435E0Q +}; +#define NRD7 7 +static const __float128 RD7[NRD7 + 1] = +{ + 1.102646614598516998880874785339049304483E11Q, + 6.099297512712715445879759589407189290040E10Q, + 1.372898136289611312713283201112060238351E10Q, + 1.615306270420293159907951633566635172343E9Q, + 1.061114435798489135996614242842561967459E8Q, + 3.845638971184305248268608902030718674691E6Q, + 7.081730675423444975703917836972720495507E4Q, + 5.423122582741398226693137276201344096370E2Q + /* 1.0E0Q */ +}; + + +/* log gamma(x+6) = log gamma(6) + x P(x)/Q(x) + -0.5 <= x <= 0.5 + 5.5 <= x+6 <= 6.5 + Peak relative error 6.2e-37 */ +static const __float128 lgam6a = 4.7874908447265625E0Q; +static const __float128 lgam6b = 8.9805548349424770093452324304839959231517E-7Q; +#define NRN6 8 +static const __float128 RN6[NRN6 + 1] = +{ + -3.538412754670746879119162116819571823643E13Q, + -2.613432593406849155765698121483394257148E13Q, + -8.020670732770461579558867891923784753062E12Q, + -1.322227822931250045347591780332435433420E12Q, + -1.262809382777272476572558806855377129513E11Q, + -7.015006277027660872284922325741197022467E9Q, + -2.149320689089020841076532186783055727299E8Q, + -3.167210585700002703820077565539658995316E6Q, + -1.576834867378554185210279285358586385266E4Q +}; +#define NRD6 8 +static const __float128 RD6[NRD6 + 1] = +{ + -2.073955870771283609792355579558899389085E13Q, + -1.421592856111673959642750863283919318175E13Q, + -4.012134994918353924219048850264207074949E12Q, + -6.013361045800992316498238470888523722431E11Q, + -5.145382510136622274784240527039643430628E10Q, + -2.510575820013409711678540476918249524123E9Q, + -6.564058379709759600836745035871373240904E7Q, + -7.861511116647120540275354855221373571536E5Q, + -2.821943442729620524365661338459579270561E3Q + /* 1.0E0Q */ +}; + + +/* log gamma(x+5) = log gamma(5) + x P(x)/Q(x) + -0.5 <= x <= 0.5 + 4.5 <= x+5 <= 5.5 + Peak relative error 3.4e-37 */ +static const __float128 lgam5a = 3.17803955078125E0Q; +static const __float128 lgam5b = 1.4279566695619646941601297055408873990961E-5Q; +#define NRN5 9 +static const __float128 RN5[NRN5 + 1] = +{ + 2.010952885441805899580403215533972172098E11Q, + 1.916132681242540921354921906708215338584E11Q, + 7.679102403710581712903937970163206882492E10Q, + 1.680514903671382470108010973615268125169E10Q, + 2.181011222911537259440775283277711588410E9Q, + 1.705361119398837808244780667539728356096E8Q, + 7.792391565652481864976147945997033946360E6Q, + 1.910741381027985291688667214472560023819E5Q, + 2.088138241893612679762260077783794329559E3Q, + 6.330318119566998299106803922739066556550E0Q +}; +#define NRD5 8 +static const __float128 RD5[NRD5 + 1] = +{ + 1.335189758138651840605141370223112376176E11Q, + 1.174130445739492885895466097516530211283E11Q, + 4.308006619274572338118732154886328519910E10Q, + 8.547402888692578655814445003283720677468E9Q, + 9.934628078575618309542580800421370730906E8Q, + 6.847107420092173812998096295422311820672E7Q, + 2.698552646016599923609773122139463150403E6Q, + 5.526516251532464176412113632726150253215E4Q, + 4.772343321713697385780533022595450486932E2Q + /* 1.0E0Q */ +}; + + +/* log gamma(x+4) = log gamma(4) + x P(x)/Q(x) + -0.5 <= x <= 0.5 + 3.5 <= x+4 <= 4.5 + Peak relative error 6.7e-37 */ +static const __float128 lgam4a = 1.791748046875E0Q; +static const __float128 lgam4b = 1.1422353055000812477358380702272722990692E-5Q; +#define NRN4 9 +static const __float128 RN4[NRN4 + 1] = +{ + -1.026583408246155508572442242188887829208E13Q, + -1.306476685384622809290193031208776258809E13Q, + -7.051088602207062164232806511992978915508E12Q, + -2.100849457735620004967624442027793656108E12Q, + -3.767473790774546963588549871673843260569E11Q, + -4.156387497364909963498394522336575984206E10Q, + -2.764021460668011732047778992419118757746E9Q, + -1.036617204107109779944986471142938641399E8Q, + -1.895730886640349026257780896972598305443E6Q, + -1.180509051468390914200720003907727988201E4Q +}; +#define NRD4 9 +static const __float128 RD4[NRD4 + 1] = +{ + -8.172669122056002077809119378047536240889E12Q, + -9.477592426087986751343695251801814226960E12Q, + -4.629448850139318158743900253637212801682E12Q, + -1.237965465892012573255370078308035272942E12Q, + -1.971624313506929845158062177061297598956E11Q, + -1.905434843346570533229942397763361493610E10Q, + -1.089409357680461419743730978512856675984E9Q, + -3.416703082301143192939774401370222822430E7Q, + -4.981791914177103793218433195857635265295E5Q, + -2.192507743896742751483055798411231453733E3Q + /* 1.0E0Q */ +}; + + +/* log gamma(x+3) = log gamma(3) + x P(x)/Q(x) + -0.25 <= x <= 0.5 + 2.75 <= x+3 <= 3.5 + Peak relative error 6.0e-37 */ +static const __float128 lgam3a = 6.93145751953125E-1Q; +static const __float128 lgam3b = 1.4286068203094172321214581765680755001344E-6Q; + +#define NRN3 9 +static const __float128 RN3[NRN3 + 1] = +{ + -4.813901815114776281494823863935820876670E11Q, + -8.425592975288250400493910291066881992620E11Q, + -6.228685507402467503655405482985516909157E11Q, + -2.531972054436786351403749276956707260499E11Q, + -6.170200796658926701311867484296426831687E10Q, + -9.211477458528156048231908798456365081135E9Q, + -8.251806236175037114064561038908691305583E8Q, + -4.147886355917831049939930101151160447495E7Q, + -1.010851868928346082547075956946476932162E6Q, + -8.333374463411801009783402800801201603736E3Q +}; +#define NRD3 9 +static const __float128 RD3[NRD3 + 1] = +{ + -5.216713843111675050627304523368029262450E11Q, + -8.014292925418308759369583419234079164391E11Q, + -5.180106858220030014546267824392678611990E11Q, + -1.830406975497439003897734969120997840011E11Q, + -3.845274631904879621945745960119924118925E10Q, + -4.891033385370523863288908070309417710903E9Q, + -3.670172254411328640353855768698287474282E8Q, + -1.505316381525727713026364396635522516989E7Q, + -2.856327162923716881454613540575964890347E5Q, + -1.622140448015769906847567212766206894547E3Q + /* 1.0E0Q */ +}; + + +/* log gamma(x+2.5) = log gamma(2.5) + x P(x)/Q(x) + -0.125 <= x <= 0.25 + 2.375 <= x+2.5 <= 2.75 */ +static const __float128 lgam2r5a = 2.8466796875E-1Q; +static const __float128 lgam2r5b = 1.4901722919159632494669682701924320137696E-5Q; +#define NRN2r5 8 +static const __float128 RN2r5[NRN2r5 + 1] = +{ + -4.676454313888335499356699817678862233205E9Q, + -9.361888347911187924389905984624216340639E9Q, + -7.695353600835685037920815799526540237703E9Q, + -3.364370100981509060441853085968900734521E9Q, + -8.449902011848163568670361316804900559863E8Q, + -1.225249050950801905108001246436783022179E8Q, + -9.732972931077110161639900388121650470926E6Q, + -3.695711763932153505623248207576425983573E5Q, + -4.717341584067827676530426007495274711306E3Q +}; +#define NRD2r5 8 +static const __float128 RD2r5[NRD2r5 + 1] = +{ + -6.650657966618993679456019224416926875619E9Q, + -1.099511409330635807899718829033488771623E10Q, + -7.482546968307837168164311101447116903148E9Q, + -2.702967190056506495988922973755870557217E9Q, + -5.570008176482922704972943389590409280950E8Q, + -6.536934032192792470926310043166993233231E7Q, + -4.101991193844953082400035444146067511725E6Q, + -1.174082735875715802334430481065526664020E5Q, + -9.932840389994157592102947657277692978511E2Q + /* 1.0E0Q */ +}; + + +/* log gamma(x+2) = x P(x)/Q(x) + -0.125 <= x <= +0.375 + 1.875 <= x+2 <= 2.375 + Peak relative error 4.6e-36 */ +#define NRN2 9 +static const __float128 RN2[NRN2 + 1] = +{ + -3.716661929737318153526921358113793421524E9Q, + -1.138816715030710406922819131397532331321E10Q, + -1.421017419363526524544402598734013569950E10Q, + -9.510432842542519665483662502132010331451E9Q, + -3.747528562099410197957514973274474767329E9Q, + -8.923565763363912474488712255317033616626E8Q, + -1.261396653700237624185350402781338231697E8Q, + -9.918402520255661797735331317081425749014E6Q, + -3.753996255897143855113273724233104768831E5Q, + -4.778761333044147141559311805999540765612E3Q +}; +#define NRD2 9 +static const __float128 RD2[NRD2 + 1] = +{ + -8.790916836764308497770359421351673950111E9Q, + -2.023108608053212516399197678553737477486E10Q, + -1.958067901852022239294231785363504458367E10Q, + -1.035515043621003101254252481625188704529E10Q, + -3.253884432621336737640841276619272224476E9Q, + -6.186383531162456814954947669274235815544E8Q, + -6.932557847749518463038934953605969951466E7Q, + -4.240731768287359608773351626528479703758E6Q, + -1.197343995089189188078944689846348116630E5Q, + -1.004622911670588064824904487064114090920E3Q +/* 1.0E0 */ +}; + + +/* log gamma(x+1.75) = log gamma(1.75) + x P(x)/Q(x) + -0.125 <= x <= +0.125 + 1.625 <= x+1.75 <= 1.875 + Peak relative error 9.2e-37 */ +static const __float128 lgam1r75a = -8.441162109375E-2Q; +static const __float128 lgam1r75b = 1.0500073264444042213965868602268256157604E-5Q; +#define NRN1r75 8 +static const __float128 RN1r75[NRN1r75 + 1] = +{ + -5.221061693929833937710891646275798251513E7Q, + -2.052466337474314812817883030472496436993E8Q, + -2.952718275974940270675670705084125640069E8Q, + -2.132294039648116684922965964126389017840E8Q, + -8.554103077186505960591321962207519908489E7Q, + -1.940250901348870867323943119132071960050E7Q, + -2.379394147112756860769336400290402208435E6Q, + -1.384060879999526222029386539622255797389E5Q, + -2.698453601378319296159355612094598695530E3Q +}; +#define NRD1r75 8 +static const __float128 RD1r75[NRD1r75 + 1] = +{ + -2.109754689501705828789976311354395393605E8Q, + -5.036651829232895725959911504899241062286E8Q, + -4.954234699418689764943486770327295098084E8Q, + -2.589558042412676610775157783898195339410E8Q, + -7.731476117252958268044969614034776883031E7Q, + -1.316721702252481296030801191240867486965E7Q, + -1.201296501404876774861190604303728810836E6Q, + -5.007966406976106636109459072523610273928E4Q, + -6.155817990560743422008969155276229018209E2Q + /* 1.0E0Q */ +}; + + +/* log gamma(x+x0) = y0 + x^2 P(x)/Q(x) + -0.0867 <= x <= +0.1634 + 1.374932... <= x+x0 <= 1.625032... + Peak relative error 4.0e-36 */ +static const __float128 x0a = 1.4616241455078125Q; +static const __float128 x0b = 7.9994605498412626595423257213002588621246E-6Q; +static const __float128 y0a = -1.21490478515625E-1Q; +static const __float128 y0b = 4.1879797753919044854428223084178486438269E-6Q; +#define NRN1r5 8 +static const __float128 RN1r5[NRN1r5 + 1] = +{ + 6.827103657233705798067415468881313128066E5Q, + 1.910041815932269464714909706705242148108E6Q, + 2.194344176925978377083808566251427771951E6Q, + 1.332921400100891472195055269688876427962E6Q, + 4.589080973377307211815655093824787123508E5Q, + 8.900334161263456942727083580232613796141E4Q, + 9.053840838306019753209127312097612455236E3Q, + 4.053367147553353374151852319743594873771E2Q, + 5.040631576303952022968949605613514584950E0Q +}; +#define NRD1r5 8 +static const __float128 RD1r5[NRD1r5 + 1] = +{ + 1.411036368843183477558773688484699813355E6Q, + 4.378121767236251950226362443134306184849E6Q, + 5.682322855631723455425929877581697918168E6Q, + 3.999065731556977782435009349967042222375E6Q, + 1.653651390456781293163585493620758410333E6Q, + 4.067774359067489605179546964969435858311E5Q, + 5.741463295366557346748361781768833633256E4Q, + 4.226404539738182992856094681115746692030E3Q, + 1.316980975410327975566999780608618774469E2Q, + /* 1.0E0Q */ +}; + + +/* log gamma(x+1.25) = log gamma(1.25) + x P(x)/Q(x) + -.125 <= x <= +.125 + 1.125 <= x+1.25 <= 1.375 + Peak relative error = 4.9e-36 */ +static const __float128 lgam1r25a = -9.82818603515625E-2Q; +static const __float128 lgam1r25b = 1.0023929749338536146197303364159774377296E-5Q; +#define NRN1r25 9 +static const __float128 RN1r25[NRN1r25 + 1] = +{ + -9.054787275312026472896002240379580536760E4Q, + -8.685076892989927640126560802094680794471E4Q, + 2.797898965448019916967849727279076547109E5Q, + 6.175520827134342734546868356396008898299E5Q, + 5.179626599589134831538516906517372619641E5Q, + 2.253076616239043944538380039205558242161E5Q, + 5.312653119599957228630544772499197307195E4Q, + 6.434329437514083776052669599834938898255E3Q, + 3.385414416983114598582554037612347549220E2Q, + 4.907821957946273805080625052510832015792E0Q +}; +#define NRD1r25 8 +static const __float128 RD1r25[NRD1r25 + 1] = +{ + 3.980939377333448005389084785896660309000E5Q, + 1.429634893085231519692365775184490465542E6Q, + 2.145438946455476062850151428438668234336E6Q, + 1.743786661358280837020848127465970357893E6Q, + 8.316364251289743923178092656080441655273E5Q, + 2.355732939106812496699621491135458324294E5Q, + 3.822267399625696880571810137601310855419E4Q, + 3.228463206479133236028576845538387620856E3Q, + 1.152133170470059555646301189220117965514E2Q + /* 1.0E0Q */ +}; + + +/* log gamma(x + 1) = x P(x)/Q(x) + 0.0 <= x <= +0.125 + 1.0 <= x+1 <= 1.125 + Peak relative error 1.1e-35 */ +#define NRN1 8 +static const __float128 RN1[NRN1 + 1] = +{ + -9.987560186094800756471055681088744738818E3Q, + -2.506039379419574361949680225279376329742E4Q, + -1.386770737662176516403363873617457652991E4Q, + 1.439445846078103202928677244188837130744E4Q, + 2.159612048879650471489449668295139990693E4Q, + 1.047439813638144485276023138173676047079E4Q, + 2.250316398054332592560412486630769139961E3Q, + 1.958510425467720733041971651126443864041E2Q, + 4.516830313569454663374271993200291219855E0Q +}; +#define NRD1 7 +static const __float128 RD1[NRD1 + 1] = +{ + 1.730299573175751778863269333703788214547E4Q, + 6.807080914851328611903744668028014678148E4Q, + 1.090071629101496938655806063184092302439E5Q, + 9.124354356415154289343303999616003884080E4Q, + 4.262071638655772404431164427024003253954E4Q, + 1.096981664067373953673982635805821283581E4Q, + 1.431229503796575892151252708527595787588E3Q, + 7.734110684303689320830401788262295992921E1Q + /* 1.0E0 */ +}; + + +/* log gamma(x + 1) = x P(x)/Q(x) + -0.125 <= x <= 0 + 0.875 <= x+1 <= 1.0 + Peak relative error 7.0e-37 */ +#define NRNr9 8 +static const __float128 RNr9[NRNr9 + 1] = +{ + 4.441379198241760069548832023257571176884E5Q, + 1.273072988367176540909122090089580368732E6Q, + 9.732422305818501557502584486510048387724E5Q, + -5.040539994443998275271644292272870348684E5Q, + -1.208719055525609446357448132109723786736E6Q, + -7.434275365370936547146540554419058907156E5Q, + -2.075642969983377738209203358199008185741E5Q, + -2.565534860781128618589288075109372218042E4Q, + -1.032901669542994124131223797515913955938E3Q, +}; +#define NRDr9 8 +static const __float128 RDr9[NRDr9 + 1] = +{ + -7.694488331323118759486182246005193998007E5Q, + -3.301918855321234414232308938454112213751E6Q, + -5.856830900232338906742924836032279404702E6Q, + -5.540672519616151584486240871424021377540E6Q, + -3.006530901041386626148342989181721176919E6Q, + -9.350378280513062139466966374330795935163E5Q, + -1.566179100031063346901755685375732739511E5Q, + -1.205016539620260779274902967231510804992E4Q, + -2.724583156305709733221564484006088794284E2Q +/* 1.0E0 */ +}; + + +/* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ + +static __float128 +neval (__float128 x, const __float128 *p, int n) +{ + __float128 y; + + p += n; + y = *p--; + do + { + y = y * x + *p--; + } + while (--n > 0); + return y; +} + + +/* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */ + +static __float128 +deval (__float128 x, const __float128 *p, int n) +{ + __float128 y; + + p += n; + y = x + *p--; + do + { + y = y * x + *p--; + } + while (--n > 0); + return y; +} + + +__float128 +lgammaq (__float128 x) +{ + __float128 p, q, w, z, nx; + int i, nn, sign; + + sign = 1; + + if (! finiteq (x)) + return x * x; + + if (x == 0.0Q) + { + if (signbitq (x)) + sign = -1; + } + + if (x < 0.0Q) + { + q = -x; + p = floorq (q); + if (p == q) + return (one / (p - p)); + i = p; + if ((i & 1) == 0) + sign = -1; + else + sign = 1; + z = q - p; + if (z > 0.5Q) + { + p += 1.0Q; + z = p - q; + } + z = q * sinq (PIQ * z); + if (z == 0.0Q) + return (sign * huge * huge); + w = lgammaq (q); + z = logq (PIQ / z) - w; + return (z); + } + + if (x < 13.5Q) + { + p = 0.0Q; + nx = floorq (x + 0.5Q); + nn = nx; + switch (nn) + { + case 0: + /* log gamma (x + 1) = log(x) + log gamma(x) */ + if (x <= 0.125) + { + p = x * neval (x, RN1, NRN1) / deval (x, RD1, NRD1); + } + else if (x <= 0.375) + { + z = x - 0.25Q; + p = z * neval (z, RN1r25, NRN1r25) / deval (z, RD1r25, NRD1r25); + p += lgam1r25b; + p += lgam1r25a; + } + else if (x <= 0.625) + { + z = x + (1.0Q - x0a); + z = z - x0b; + p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5); + p = p * z * z; + p = p + y0b; + p = p + y0a; + } + else if (x <= 0.875) + { + z = x - 0.75Q; + p = z * neval (z, RN1r75, NRN1r75) / deval (z, RD1r75, NRD1r75); + p += lgam1r75b; + p += lgam1r75a; + } + else + { + z = x - 1.0Q; + p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2); + } + p = p - logq (x); + break; + + case 1: + if (x < 0.875Q) + { + if (x <= 0.625) + { + z = x + (1.0Q - x0a); + z = z - x0b; + p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5); + p = p * z * z; + p = p + y0b; + p = p + y0a; + } + else if (x <= 0.875) + { + z = x - 0.75Q; + p = z * neval (z, RN1r75, NRN1r75) + / deval (z, RD1r75, NRD1r75); + p += lgam1r75b; + p += lgam1r75a; + } + else + { + z = x - 1.0Q; + p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2); + } + p = p - logq (x); + } + else if (x < 1.0Q) + { + z = x - 1.0Q; + p = z * neval (z, RNr9, NRNr9) / deval (z, RDr9, NRDr9); + } + else if (x == 1.0Q) + p = 0.0Q; + else if (x <= 1.125Q) + { + z = x - 1.0Q; + p = z * neval (z, RN1, NRN1) / deval (z, RD1, NRD1); + } + else if (x <= 1.375) + { + z = x - 1.25Q; + p = z * neval (z, RN1r25, NRN1r25) / deval (z, RD1r25, NRD1r25); + p += lgam1r25b; + p += lgam1r25a; + } + else + { + /* 1.375 <= x+x0 <= 1.625 */ + z = x - x0a; + z = z - x0b; + p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5); + p = p * z * z; + p = p + y0b; + p = p + y0a; + } + break; + + case 2: + if (x < 1.625Q) + { + z = x - x0a; + z = z - x0b; + p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5); + p = p * z * z; + p = p + y0b; + p = p + y0a; + } + else if (x < 1.875Q) + { + z = x - 1.75Q; + p = z * neval (z, RN1r75, NRN1r75) / deval (z, RD1r75, NRD1r75); + p += lgam1r75b; + p += lgam1r75a; + } + else if (x == 2.0Q) + p = 0.0Q; + else if (x < 2.375Q) + { + z = x - 2.0Q; + p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2); + } + else + { + z = x - 2.5Q; + p = z * neval (z, RN2r5, NRN2r5) / deval (z, RD2r5, NRD2r5); + p += lgam2r5b; + p += lgam2r5a; + } + break; + + case 3: + if (x < 2.75) + { + z = x - 2.5Q; + p = z * neval (z, RN2r5, NRN2r5) / deval (z, RD2r5, NRD2r5); + p += lgam2r5b; + p += lgam2r5a; + } + else + { + z = x - 3.0Q; + p = z * neval (z, RN3, NRN3) / deval (z, RD3, NRD3); + p += lgam3b; + p += lgam3a; + } + break; + + case 4: + z = x - 4.0Q; + p = z * neval (z, RN4, NRN4) / deval (z, RD4, NRD4); + p += lgam4b; + p += lgam4a; + break; + + case 5: + z = x - 5.0Q; + p = z * neval (z, RN5, NRN5) / deval (z, RD5, NRD5); + p += lgam5b; + p += lgam5a; + break; + + case 6: + z = x - 6.0Q; + p = z * neval (z, RN6, NRN6) / deval (z, RD6, NRD6); + p += lgam6b; + p += lgam6a; + break; + + case 7: + z = x - 7.0Q; + p = z * neval (z, RN7, NRN7) / deval (z, RD7, NRD7); + p += lgam7b; + p += lgam7a; + break; + + case 8: + z = x - 8.0Q; + p = z * neval (z, RN8, NRN8) / deval (z, RD8, NRD8); + p += lgam8b; + p += lgam8a; + break; + + case 9: + z = x - 9.0Q; + p = z * neval (z, RN9, NRN9) / deval (z, RD9, NRD9); + p += lgam9b; + p += lgam9a; + break; + + case 10: + z = x - 10.0Q; + p = z * neval (z, RN10, NRN10) / deval (z, RD10, NRD10); + p += lgam10b; + p += lgam10a; + break; + + case 11: + z = x - 11.0Q; + p = z * neval (z, RN11, NRN11) / deval (z, RD11, NRD11); + p += lgam11b; + p += lgam11a; + break; + + case 12: + z = x - 12.0Q; + p = z * neval (z, RN12, NRN12) / deval (z, RD12, NRD12); + p += lgam12b; + p += lgam12a; + break; + + case 13: + z = x - 13.0Q; + p = z * neval (z, RN13, NRN13) / deval (z, RD13, NRD13); + p += lgam13b; + p += lgam13a; + break; + } + return p; + } + + if (x > MAXLGM) + return (sign * huge * huge); + + q = ls2pi - x; + q = (x - 0.5Q) * logq (x) + q; + if (x > 1.0e18Q) + return (q); + + p = 1.0Q / (x * x); + q += neval (p, RASY, NRASY) / x; + return (q); +} diff --git a/libquadmath/math/llrintq.c b/libquadmath/math/llrintq.c new file mode 100644 index 000000000..eef31d823 --- /dev/null +++ b/libquadmath/math/llrintq.c @@ -0,0 +1,71 @@ +/* Round argument to nearest integral value according to current rounding + direction. + Copyright (C) 1997, 1999, 2006 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997 and + Jakub Jelinek <jj@ultra.linux.cz>, 1999. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + +static const __float128 two112[2] = +{ + 5.19229685853482762853049632922009600E+33Q, /* 0x406F000000000000, 0 */ + -5.19229685853482762853049632922009600E+33Q /* 0xC06F000000000000, 0 */ +}; + +long long int +llrintq (__float128 x) +{ + int32_t j0; + uint64_t i0,i1; + volatile __float128 w; + __float128 t; + long long int result; + int sx; + + GET_FLT128_WORDS64 (i0, i1, x); + j0 = ((i0 >> 48) & 0x7fff) - 0x3fff; + sx = i0 >> 63; + i0 &= 0x0000ffffffffffffLL; + i0 |= 0x0001000000000000LL; + + if (j0 < (int32_t) (8 * sizeof (long long int)) - 1) + { + w = two112[sx] + x; + t = w - two112[sx]; + GET_FLT128_WORDS64 (i0, i1, t); + j0 = ((i0 >> 48) & 0x7fff) - 0x3fff; + i0 &= 0x0000ffffffffffffLL; + i0 |= 0x0001000000000000LL; + + if (j0 < 0) + result = 0; + else if (j0 <= 48) + result = i0 >> (48 - j0); + else + result = ((long long int) i0 << (j0 - 48)) | (i1 >> (112 - j0)); + } + else + { + /* The number is too large. It is left implementation defined + what happens. */ + return (long long int) x; + } + + return sx ? -result : result; +} diff --git a/libquadmath/math/llroundq.c b/libquadmath/math/llroundq.c new file mode 100644 index 000000000..c108e7ad1 --- /dev/null +++ b/libquadmath/math/llroundq.c @@ -0,0 +1,73 @@ +/* Round long double value to long long int. + Copyright (C) 1997, 1999, 2004 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997 and + Jakub Jelinek <jj@ultra.linux.cz>, 1999. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + + +long long int +llroundq (__float128 x) +{ + int64_t j0; + uint64_t i1, i0; + long long int result; + int sign; + + GET_FLT128_WORDS64 (i0, i1, x); + j0 = ((i0 >> 48) & 0x7fff) - 0x3fff; + sign = (i0 & 0x8000000000000000ULL) != 0 ? -1 : 1; + i0 &= 0x0000ffffffffffffLL; + i0 |= 0x0001000000000000LL; + + if (j0 < 48) + { + if (j0 < 0) + return j0 < -1 ? 0 : sign; + else + { + i0 += 0x0000800000000000LL >> j0; + result = i0 >> (48 - j0); + } + } + else if (j0 < (int32_t) (8 * sizeof (long long int)) - 1) + { + if (j0 >= 112) + result = ((long long int) i0 << (j0 - 48)) | (i1 << (j0 - 112)); + else + { + uint64_t j = i1 + (0x8000000000000000ULL >> (j0 - 48)); + if (j < i1) + ++i0; + + if (j0 == 48) + result = (long long int) i0; + else + result = ((long long int) i0 << (j0 - 48)) | (j >> (112 - j0)); + } + } + else + { + /* The number is too large. It is left implementation defined + what happens. */ + return (long long int) x; + } + + return sign * result; +} diff --git a/libquadmath/math/log10q.c b/libquadmath/math/log10q.c new file mode 100644 index 000000000..50caf18b7 --- /dev/null +++ b/libquadmath/math/log10q.c @@ -0,0 +1,256 @@ +/* log10l.c + * + * Common logarithm, 128-bit long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, log10l(); + * + * y = log10l( x ); + * + * + * + * DESCRIPTION: + * + * Returns the base 10 logarithm of x. + * + * The argument is separated into its exponent and fractional + * parts. If the exponent is between -1 and +1, the logarithm + * of the fraction is approximated by + * + * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x). + * + * Otherwise, setting z = 2(x-1)/x+1), + * + * log(x) = z + z^3 P(z)/Q(z). + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0.5, 2.0 30000 2.3e-34 4.9e-35 + * IEEE exp(+-10000) 30000 1.0e-34 4.1e-35 + * + * In the tests over the interval exp(+-10000), the logarithms + * of the random arguments were uniformly distributed over + * [-10000, +10000]. + * + */ + +/* + Cephes Math Library Release 2.2: January, 1991 + Copyright 1984, 1991 by Stephen L. Moshier + Adapted for glibc November, 2001 + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + + */ + +#include "quadmath-imp.h" + +/* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x) + * 1/sqrt(2) <= x < sqrt(2) + * Theoretical peak relative error = 5.3e-37, + * relative peak error spread = 2.3e-14 + */ +static const __float128 P[13] = +{ + 1.313572404063446165910279910527789794488E4Q, + 7.771154681358524243729929227226708890930E4Q, + 2.014652742082537582487669938141683759923E5Q, + 3.007007295140399532324943111654767187848E5Q, + 2.854829159639697837788887080758954924001E5Q, + 1.797628303815655343403735250238293741397E5Q, + 7.594356839258970405033155585486712125861E4Q, + 2.128857716871515081352991964243375186031E4Q, + 3.824952356185897735160588078446136783779E3Q, + 4.114517881637811823002128927449878962058E2Q, + 2.321125933898420063925789532045674660756E1Q, + 4.998469661968096229986658302195402690910E-1Q, + 1.538612243596254322971797716843006400388E-6Q +}; +static const __float128 Q[12] = +{ + 3.940717212190338497730839731583397586124E4Q, + 2.626900195321832660448791748036714883242E5Q, + 7.777690340007566932935753241556479363645E5Q, + 1.347518538384329112529391120390701166528E6Q, + 1.514882452993549494932585972882995548426E6Q, + 1.158019977462989115839826904108208787040E6Q, + 6.132189329546557743179177159925690841200E5Q, + 2.248234257620569139969141618556349415120E5Q, + 5.605842085972455027590989944010492125825E4Q, + 9.147150349299596453976674231612674085381E3Q, + 9.104928120962988414618126155557301584078E2Q, + 4.839208193348159620282142911143429644326E1Q +/* 1.000000000000000000000000000000000000000E0Q, */ +}; + +/* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), + * where z = 2(x-1)/(x+1) + * 1/sqrt(2) <= x < sqrt(2) + * Theoretical peak relative error = 1.1e-35, + * relative peak error spread 1.1e-9 + */ +static const __float128 R[6] = +{ + 1.418134209872192732479751274970992665513E5Q, + -8.977257995689735303686582344659576526998E4Q, + 2.048819892795278657810231591630928516206E4Q, + -2.024301798136027039250415126250455056397E3Q, + 8.057002716646055371965756206836056074715E1Q, + -8.828896441624934385266096344596648080902E-1Q +}; +static const __float128 S[6] = +{ + 1.701761051846631278975701529965589676574E6Q, + -1.332535117259762928288745111081235577029E6Q, + 4.001557694070773974936904547424676279307E5Q, + -5.748542087379434595104154610899551484314E4Q, + 3.998526750980007367835804959888064681098E3Q, + -1.186359407982897997337150403816839480438E2Q +/* 1.000000000000000000000000000000000000000E0Q, */ +}; + +static const __float128 +/* log10(2) */ +L102A = 0.3125Q, +L102B = -1.14700043360188047862611052755069732318101185E-2Q, +/* log10(e) */ +L10EA = 0.5Q, +L10EB = -6.570551809674817234887108108339491770560299E-2Q, +/* sqrt(2)/2 */ +SQRTH = 7.071067811865475244008443621048490392848359E-1Q; + + + +/* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ + +static __float128 +neval (__float128 x, const __float128 *p, int n) +{ + __float128 y; + + p += n; + y = *p--; + do + { + y = y * x + *p--; + } + while (--n > 0); + return y; +} + + +/* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */ + +static __float128 +deval (__float128 x, const __float128 *p, int n) +{ + __float128 y; + + p += n; + y = x + *p--; + do + { + y = y * x + *p--; + } + while (--n > 0); + return y; +} + + + +__float128 +log10q (__float128 x) +{ + __float128 z; + __float128 y; + int e; + int64_t hx, lx; + +/* Test for domain */ + GET_FLT128_WORDS64 (hx, lx, x); + if (((hx & 0x7fffffffffffffffLL) | lx) == 0) + return (-1.0Q / (x - x)); + if (hx < 0) + return (x - x) / (x - x); + if (hx >= 0x7fff000000000000LL) + return (x + x); + +/* separate mantissa from exponent */ + +/* Note, frexp is used so that denormal numbers + * will be handled properly. + */ + x = frexpq (x, &e); + + +/* logarithm using log(x) = z + z**3 P(z)/Q(z), + * where z = 2(x-1)/x+1) + */ + if ((e > 2) || (e < -2)) + { + if (x < SQRTH) + { /* 2( 2x-1 )/( 2x+1 ) */ + e -= 1; + z = x - 0.5Q; + y = 0.5Q * z + 0.5Q; + } + else + { /* 2 (x-1)/(x+1) */ + z = x - 0.5Q; + z -= 0.5Q; + y = 0.5Q * x + 0.5Q; + } + x = z / y; + z = x * x; + y = x * (z * neval (z, R, 5) / deval (z, S, 5)); + goto done; + } + + +/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ + + if (x < SQRTH) + { + e -= 1; + x = 2.0 * x - 1.0Q; /* 2x - 1 */ + } + else + { + x = x - 1.0Q; + } + z = x * x; + y = x * (z * neval (x, P, 12) / deval (x, Q, 11)); + y = y - 0.5 * z; + +done: + + /* Multiply log of fraction by log10(e) + * and base 2 exponent by log10(2). + */ + z = y * L10EB; + z += x * L10EB; + z += e * L102B; + z += y * L10EA; + z += x * L10EA; + z += e * L102A; + return (z); +} diff --git a/libquadmath/math/log1pq.c b/libquadmath/math/log1pq.c new file mode 100644 index 000000000..a466dc892 --- /dev/null +++ b/libquadmath/math/log1pq.c @@ -0,0 +1,244 @@ +/* log1pl.c + * + * Relative error logarithm + * Natural logarithm of 1+x, 128-bit long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, log1pl(); + * + * y = log1pl( x ); + * + * + * + * DESCRIPTION: + * + * Returns the base e (2.718...) logarithm of 1+x. + * + * The argument 1+x is separated into its exponent and fractional + * parts. If the exponent is between -1 and +1, the logarithm + * of the fraction is approximated by + * + * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x). + * + * Otherwise, setting z = 2(w-1)/(w+1), + * + * log(w) = z + z^3 P(z)/Q(z). + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -1, 8 100000 1.9e-34 4.3e-35 + */ + +/* Copyright 2001 by Stephen L. Moshier + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + + +#include "quadmath-imp.h" + +/* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x) + * 1/sqrt(2) <= 1+x < sqrt(2) + * Theoretical peak relative error = 5.3e-37, + * relative peak error spread = 2.3e-14 + */ +static const __float128 + P12 = 1.538612243596254322971797716843006400388E-6Q, + P11 = 4.998469661968096229986658302195402690910E-1Q, + P10 = 2.321125933898420063925789532045674660756E1Q, + P9 = 4.114517881637811823002128927449878962058E2Q, + P8 = 3.824952356185897735160588078446136783779E3Q, + P7 = 2.128857716871515081352991964243375186031E4Q, + P6 = 7.594356839258970405033155585486712125861E4Q, + P5 = 1.797628303815655343403735250238293741397E5Q, + P4 = 2.854829159639697837788887080758954924001E5Q, + P3 = 3.007007295140399532324943111654767187848E5Q, + P2 = 2.014652742082537582487669938141683759923E5Q, + P1 = 7.771154681358524243729929227226708890930E4Q, + P0 = 1.313572404063446165910279910527789794488E4Q, + /* Q12 = 1.000000000000000000000000000000000000000E0Q, */ + Q11 = 4.839208193348159620282142911143429644326E1Q, + Q10 = 9.104928120962988414618126155557301584078E2Q, + Q9 = 9.147150349299596453976674231612674085381E3Q, + Q8 = 5.605842085972455027590989944010492125825E4Q, + Q7 = 2.248234257620569139969141618556349415120E5Q, + Q6 = 6.132189329546557743179177159925690841200E5Q, + Q5 = 1.158019977462989115839826904108208787040E6Q, + Q4 = 1.514882452993549494932585972882995548426E6Q, + Q3 = 1.347518538384329112529391120390701166528E6Q, + Q2 = 7.777690340007566932935753241556479363645E5Q, + Q1 = 2.626900195321832660448791748036714883242E5Q, + Q0 = 3.940717212190338497730839731583397586124E4Q; + +/* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), + * where z = 2(x-1)/(x+1) + * 1/sqrt(2) <= x < sqrt(2) + * Theoretical peak relative error = 1.1e-35, + * relative peak error spread 1.1e-9 + */ +static const __float128 + R5 = -8.828896441624934385266096344596648080902E-1Q, + R4 = 8.057002716646055371965756206836056074715E1Q, + R3 = -2.024301798136027039250415126250455056397E3Q, + R2 = 2.048819892795278657810231591630928516206E4Q, + R1 = -8.977257995689735303686582344659576526998E4Q, + R0 = 1.418134209872192732479751274970992665513E5Q, + /* S6 = 1.000000000000000000000000000000000000000E0Q, */ + S5 = -1.186359407982897997337150403816839480438E2Q, + S4 = 3.998526750980007367835804959888064681098E3Q, + S3 = -5.748542087379434595104154610899551484314E4Q, + S2 = 4.001557694070773974936904547424676279307E5Q, + S1 = -1.332535117259762928288745111081235577029E6Q, + S0 = 1.701761051846631278975701529965589676574E6Q; + +/* C1 + C2 = ln 2 */ +static const __float128 C1 = 6.93145751953125E-1Q; +static const __float128 C2 = 1.428606820309417232121458176568075500134E-6Q; + +static const __float128 sqrth = 0.7071067811865475244008443621048490392848Q; +static const __float128 zero = 0.0Q; + + +__float128 +log1pq (__float128 xm1) +{ + __float128 x, y, z, r, s; + ieee854_float128 u; + int32_t hx; + int e; + + /* Test for NaN or infinity input. */ + u.value = xm1; + hx = u.words32.w0; + if (hx >= 0x7fff0000) + return xm1; + + /* log1p(+- 0) = +- 0. */ + if (((hx & 0x7fffffff) == 0) + && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0) + return xm1; + + x = xm1 + 1.0Q; + + /* log1p(-1) = -inf */ + if (x <= 0.0Q) + { + if (x == 0.0Q) + return (-1.0Q / (x - x)); + else + return (zero / (x - x)); + } + + /* Separate mantissa from exponent. */ + + /* Use frexp used so that denormal numbers will be handled properly. */ + x = frexpq (x, &e); + + /* Logarithm using log(x) = z + z^3 P(z^2)/Q(z^2), + where z = 2(x-1)/x+1). */ + if ((e > 2) || (e < -2)) + { + if (x < sqrth) + { /* 2( 2x-1 )/( 2x+1 ) */ + e -= 1; + z = x - 0.5Q; + y = 0.5Q * z + 0.5Q; + } + else + { /* 2 (x-1)/(x+1) */ + z = x - 0.5Q; + z -= 0.5Q; + y = 0.5Q * x + 0.5Q; + } + x = z / y; + z = x * x; + r = ((((R5 * z + + R4) * z + + R3) * z + + R2) * z + + R1) * z + + R0; + s = (((((z + + S5) * z + + S4) * z + + S3) * z + + S2) * z + + S1) * z + + S0; + z = x * (z * r / s); + z = z + e * C2; + z = z + x; + z = z + e * C1; + return (z); + } + + + /* Logarithm using log(1+x) = x - .5x^2 + x^3 P(x)/Q(x). */ + + if (x < sqrth) + { + e -= 1; + if (e != 0) + x = 2.0Q * x - 1.0Q; /* 2x - 1 */ + else + x = xm1; + } + else + { + if (e != 0) + x = x - 1.0Q; + else + x = xm1; + } + z = x * x; + r = (((((((((((P12 * x + + P11) * x + + P10) * x + + P9) * x + + P8) * x + + P7) * x + + P6) * x + + P5) * x + + P4) * x + + P3) * x + + P2) * x + + P1) * x + + P0; + s = (((((((((((x + + Q11) * x + + Q10) * x + + Q9) * x + + Q8) * x + + Q7) * x + + Q6) * x + + Q5) * x + + Q4) * x + + Q3) * x + + Q2) * x + + Q1) * x + + Q0; + y = x * (z * r / s); + y = y + e * C2; + z = y - 0.5Q * z; + z = z + x; + z = z + e * C1; + return (z); +} diff --git a/libquadmath/math/log2q.c b/libquadmath/math/log2q.c new file mode 100644 index 000000000..963b38c84 --- /dev/null +++ b/libquadmath/math/log2q.c @@ -0,0 +1,248 @@ +/* log2l.c + * Base 2 logarithm, 128-bit long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, log2l(); + * + * y = log2l( x ); + * + * + * + * DESCRIPTION: + * + * Returns the base 2 logarithm of x. + * + * The argument is separated into its exponent and fractional + * parts. If the exponent is between -1 and +1, the (natural) + * logarithm of the fraction is approximated by + * + * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x). + * + * Otherwise, setting z = 2(x-1)/x+1), + * + * log(x) = z + z^3 P(z)/Q(z). + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0.5, 2.0 100,000 2.6e-34 4.9e-35 + * IEEE exp(+-10000) 100,000 9.6e-35 4.0e-35 + * + * In the tests over the interval exp(+-10000), the logarithms + * of the random arguments were uniformly distributed over + * [-10000, +10000]. + * + */ + +/* + Cephes Math Library Release 2.2: January, 1991 + Copyright 1984, 1991 by Stephen L. Moshier + Adapted for glibc November, 2001 + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + */ + +#include "quadmath-imp.h" + +/* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x) + * 1/sqrt(2) <= x < sqrt(2) + * Theoretical peak relative error = 5.3e-37, + * relative peak error spread = 2.3e-14 + */ +static const __float128 P[13] = +{ + 1.313572404063446165910279910527789794488E4Q, + 7.771154681358524243729929227226708890930E4Q, + 2.014652742082537582487669938141683759923E5Q, + 3.007007295140399532324943111654767187848E5Q, + 2.854829159639697837788887080758954924001E5Q, + 1.797628303815655343403735250238293741397E5Q, + 7.594356839258970405033155585486712125861E4Q, + 2.128857716871515081352991964243375186031E4Q, + 3.824952356185897735160588078446136783779E3Q, + 4.114517881637811823002128927449878962058E2Q, + 2.321125933898420063925789532045674660756E1Q, + 4.998469661968096229986658302195402690910E-1Q, + 1.538612243596254322971797716843006400388E-6Q +}; +static const __float128 Q[12] = +{ + 3.940717212190338497730839731583397586124E4Q, + 2.626900195321832660448791748036714883242E5Q, + 7.777690340007566932935753241556479363645E5Q, + 1.347518538384329112529391120390701166528E6Q, + 1.514882452993549494932585972882995548426E6Q, + 1.158019977462989115839826904108208787040E6Q, + 6.132189329546557743179177159925690841200E5Q, + 2.248234257620569139969141618556349415120E5Q, + 5.605842085972455027590989944010492125825E4Q, + 9.147150349299596453976674231612674085381E3Q, + 9.104928120962988414618126155557301584078E2Q, + 4.839208193348159620282142911143429644326E1Q +/* 1.000000000000000000000000000000000000000E0Q, */ +}; + +/* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), + * where z = 2(x-1)/(x+1) + * 1/sqrt(2) <= x < sqrt(2) + * Theoretical peak relative error = 1.1e-35, + * relative peak error spread 1.1e-9 + */ +static const __float128 R[6] = +{ + 1.418134209872192732479751274970992665513E5Q, + -8.977257995689735303686582344659576526998E4Q, + 2.048819892795278657810231591630928516206E4Q, + -2.024301798136027039250415126250455056397E3Q, + 8.057002716646055371965756206836056074715E1Q, + -8.828896441624934385266096344596648080902E-1Q +}; +static const __float128 S[6] = +{ + 1.701761051846631278975701529965589676574E6Q, + -1.332535117259762928288745111081235577029E6Q, + 4.001557694070773974936904547424676279307E5Q, + -5.748542087379434595104154610899551484314E4Q, + 3.998526750980007367835804959888064681098E3Q, + -1.186359407982897997337150403816839480438E2Q +/* 1.000000000000000000000000000000000000000E0Q, */ +}; + +static const __float128 +/* log2(e) - 1 */ +LOG2EA = 4.4269504088896340735992468100189213742664595E-1Q, +/* sqrt(2)/2 */ +SQRTH = 7.071067811865475244008443621048490392848359E-1Q; + + +/* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ + +static __float128 +neval (__float128 x, const __float128 *p, int n) +{ + __float128 y; + + p += n; + y = *p--; + do + { + y = y * x + *p--; + } + while (--n > 0); + return y; +} + + +/* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */ + +static __float128 +deval (__float128 x, const __float128 *p, int n) +{ + __float128 y; + + p += n; + y = x + *p--; + do + { + y = y * x + *p--; + } + while (--n > 0); + return y; +} + + + +__float128 +log2q (__float128 x) +{ + __float128 z; + __float128 y; + int e; + int64_t hx, lx; + +/* Test for domain */ + GET_FLT128_WORDS64 (hx, lx, x); + if (((hx & 0x7fffffffffffffffLL) | lx) == 0) + return (-1.0Q / (x - x)); + if (hx < 0) + return (x - x) / (x - x); + if (hx >= 0x7fff000000000000LL) + return (x + x); + +/* separate mantissa from exponent */ + +/* Note, frexp is used so that denormal numbers + * will be handled properly. + */ + x = frexpq (x, &e); + + +/* logarithm using log(x) = z + z**3 P(z)/Q(z), + * where z = 2(x-1)/x+1) + */ + if ((e > 2) || (e < -2)) + { + if (x < SQRTH) + { /* 2( 2x-1 )/( 2x+1 ) */ + e -= 1; + z = x - 0.5Q; + y = 0.5Q * z + 0.5Q; + } + else + { /* 2 (x-1)/(x+1) */ + z = x - 0.5Q; + z -= 0.5Q; + y = 0.5Q * x + 0.5Q; + } + x = z / y; + z = x * x; + y = x * (z * neval (z, R, 5) / deval (z, S, 5)); + goto done; + } + + +/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ + + if (x < SQRTH) + { + e -= 1; + x = 2.0 * x - 1.0Q; /* 2x - 1 */ + } + else + { + x = x - 1.0Q; + } + z = x * x; + y = x * (z * neval (x, P, 12) / deval (x, Q, 11)); + y = y - 0.5 * z; + +done: + +/* Multiply log of fraction by log2(e) + * and base 2 exponent by 1 + */ + z = y * LOG2EA; + z += x * LOG2EA; + z += y; + z += x; + z += e; + return (z); +} diff --git a/libquadmath/math/logq.c b/libquadmath/math/logq.c new file mode 100644 index 000000000..cd1a48631 --- /dev/null +++ b/libquadmath/math/logq.c @@ -0,0 +1,279 @@ +/* logll.c + * + * Natural logarithm for 128-bit long double precision. + * + * + * + * SYNOPSIS: + * + * long double x, y, logl(); + * + * y = logl( x ); + * + * + * + * DESCRIPTION: + * + * Returns the base e (2.718...) logarithm of x. + * + * The argument is separated into its exponent and fractional + * parts. Use of a lookup table increases the speed of the routine. + * The program uses logarithms tabulated at intervals of 1/128 to + * cover the domain from approximately 0.7 to 1.4. + * + * On the interval [-1/128, +1/128] the logarithm of 1+x is approximated by + * log(1+x) = x - 0.5 x^2 + x^3 P(x) . + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0.875, 1.125 100000 1.2e-34 4.1e-35 + * IEEE 0.125, 8 100000 1.2e-34 4.1e-35 + * + * + * WARNING: + * + * This program uses integer operations on bit fields of floating-point + * numbers. It does not work with data structures other than the + * structure assumed. + * + */ + +/* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov> + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + +#include "quadmath-imp.h" + +/* log(1+x) = x - .5 x^2 + x^3 l(x) + -.0078125 <= x <= +.0078125 + peak relative error 1.2e-37 */ +static const __float128 +l3 = 3.333333333333333333333333333333336096926E-1Q, +l4 = -2.499999999999999999999999999486853077002E-1Q, +l5 = 1.999999999999999999999999998515277861905E-1Q, +l6 = -1.666666666666666666666798448356171665678E-1Q, +l7 = 1.428571428571428571428808945895490721564E-1Q, +l8 = -1.249999999999999987884655626377588149000E-1Q, +l9 = 1.111111111111111093947834982832456459186E-1Q, +l10 = -1.000000000000532974938900317952530453248E-1Q, +l11 = 9.090909090915566247008015301349979892689E-2Q, +l12 = -8.333333211818065121250921925397567745734E-2Q, +l13 = 7.692307559897661630807048686258659316091E-2Q, +l14 = -7.144242754190814657241902218399056829264E-2Q, +l15 = 6.668057591071739754844678883223432347481E-2Q; + +/* Lookup table of ln(t) - (t-1) + t = 0.5 + (k+26)/128) + k = 0, ..., 91 */ +static const __float128 logtbl[92] = { +-5.5345593589352099112142921677820359632418E-2Q, +-5.2108257402767124761784665198737642086148E-2Q, +-4.8991686870576856279407775480686721935120E-2Q, +-4.5993270766361228596215288742353061431071E-2Q, +-4.3110481649613269682442058976885699556950E-2Q, +-4.0340872319076331310838085093194799765520E-2Q, +-3.7682072451780927439219005993827431503510E-2Q, +-3.5131785416234343803903228503274262719586E-2Q, +-3.2687785249045246292687241862699949178831E-2Q, +-3.0347913785027239068190798397055267411813E-2Q, +-2.8110077931525797884641940838507561326298E-2Q, +-2.5972247078357715036426583294246819637618E-2Q, +-2.3932450635346084858612873953407168217307E-2Q, +-2.1988775689981395152022535153795155900240E-2Q, +-2.0139364778244501615441044267387667496733E-2Q, +-1.8382413762093794819267536615342902718324E-2Q, +-1.6716169807550022358923589720001638093023E-2Q, +-1.5138929457710992616226033183958974965355E-2Q, +-1.3649036795397472900424896523305726435029E-2Q, +-1.2244881690473465543308397998034325468152E-2Q, +-1.0924898127200937840689817557742469105693E-2Q, +-9.6875626072830301572839422532631079809328E-3Q, +-8.5313926245226231463436209313499745894157E-3Q, +-7.4549452072765973384933565912143044991706E-3Q, +-6.4568155251217050991200599386801665681310E-3Q, +-5.5356355563671005131126851708522185605193E-3Q, +-4.6900728132525199028885749289712348829878E-3Q, +-3.9188291218610470766469347968659624282519E-3Q, +-3.2206394539524058873423550293617843896540E-3Q, +-2.5942708080877805657374888909297113032132E-3Q, +-2.0385211375711716729239156839929281289086E-3Q, +-1.5522183228760777967376942769773768850872E-3Q, +-1.1342191863606077520036253234446621373191E-3Q, +-7.8340854719967065861624024730268350459991E-4Q, +-4.9869831458030115699628274852562992756174E-4Q, +-2.7902661731604211834685052867305795169688E-4Q, +-1.2335696813916860754951146082826952093496E-4Q, +-3.0677461025892873184042490943581654591817E-5Q, +#define ZERO logtbl[38] + 0.0000000000000000000000000000000000000000E0Q, +-3.0359557945051052537099938863236321874198E-5Q, +-1.2081346403474584914595395755316412213151E-4Q, +-2.7044071846562177120083903771008342059094E-4Q, +-4.7834133324631162897179240322783590830326E-4Q, +-7.4363569786340080624467487620270965403695E-4Q, +-1.0654639687057968333207323853366578860679E-3Q, +-1.4429854811877171341298062134712230604279E-3Q, +-1.8753781835651574193938679595797367137975E-3Q, +-2.3618380914922506054347222273705859653658E-3Q, +-2.9015787624124743013946600163375853631299E-3Q, +-3.4938307889254087318399313316921940859043E-3Q, +-4.1378413103128673800485306215154712148146E-3Q, +-4.8328735414488877044289435125365629849599E-3Q, +-5.5782063183564351739381962360253116934243E-3Q, +-6.3731336597098858051938306767880719015261E-3Q, +-7.2169643436165454612058905294782949315193E-3Q, +-8.1090214990427641365934846191367315083867E-3Q, +-9.0486422112807274112838713105168375482480E-3Q, +-1.0035177140880864314674126398350812606841E-2Q, +-1.1067990155502102718064936259435676477423E-2Q, +-1.2146457974158024928196575103115488672416E-2Q, +-1.3269969823361415906628825374158424754308E-2Q, +-1.4437927104692837124388550722759686270765E-2Q, +-1.5649743073340777659901053944852735064621E-2Q, +-1.6904842527181702880599758489058031645317E-2Q, +-1.8202661505988007336096407340750378994209E-2Q, +-1.9542647000370545390701192438691126552961E-2Q, +-2.0924256670080119637427928803038530924742E-2Q, +-2.2346958571309108496179613803760727786257E-2Q, +-2.3810230892650362330447187267648486279460E-2Q, +-2.5313561699385640380910474255652501521033E-2Q, +-2.6856448685790244233704909690165496625399E-2Q, +-2.8438398935154170008519274953860128449036E-2Q, +-3.0058928687233090922411781058956589863039E-2Q, +-3.1717563112854831855692484086486099896614E-2Q, +-3.3413836095418743219397234253475252001090E-2Q, +-3.5147290019036555862676702093393332533702E-2Q, +-3.6917475563073933027920505457688955423688E-2Q, +-3.8723951502862058660874073462456610731178E-2Q, +-4.0566284516358241168330505467000838017425E-2Q, +-4.2444048996543693813649967076598766917965E-2Q, +-4.4356826869355401653098777649745233339196E-2Q, +-4.6304207416957323121106944474331029996141E-2Q, +-4.8285787106164123613318093945035804818364E-2Q, +-5.0301169421838218987124461766244507342648E-2Q, +-5.2349964705088137924875459464622098310997E-2Q, +-5.4431789996103111613753440311680967840214E-2Q, +-5.6546268881465384189752786409400404404794E-2Q, +-5.8693031345788023909329239565012647817664E-2Q, +-6.0871713627532018185577188079210189048340E-2Q, +-6.3081958078862169742820420185833800925568E-2Q, +-6.5323413029406789694910800219643791556918E-2Q, +-6.7595732653791419081537811574227049288168E-2Q +}; + +/* ln(2) = ln2a + ln2b with extended precision. */ +static const __float128 + ln2a = 6.93145751953125e-1Q, + ln2b = 1.4286068203094172321214581765680755001344E-6Q; + +__float128 +logq (__float128 x) +{ + __float128 z, y, w; + ieee854_float128 u, t; + unsigned int m; + int k, e; + + u.value = x; + m = u.words32.w0; + + /* Check for IEEE special cases. */ + k = m & 0x7fffffff; + /* log(0) = -infinity. */ + if ((k | u.words32.w1 | u.words32.w2 | u.words32.w3) == 0) + { + return -0.5Q / ZERO; + } + /* log ( x < 0 ) = NaN */ + if (m & 0x80000000) + { + return (x - x) / ZERO; + } + /* log (infinity or NaN) */ + if (k >= 0x7fff0000) + { + return x + x; + } + + /* Extract exponent and reduce domain to 0.703125 <= u < 1.40625 */ + e = (int) (m >> 16) - (int) 0x3ffe; + m &= 0xffff; + u.words32.w0 = m | 0x3ffe0000; + m |= 0x10000; + /* Find lookup table index k from high order bits of the significand. */ + if (m < 0x16800) + { + k = (m - 0xff00) >> 9; + /* t is the argument 0.5 + (k+26)/128 + of the nearest item to u in the lookup table. */ + t.words32.w0 = 0x3fff0000 + (k << 9); + t.words32.w1 = 0; + t.words32.w2 = 0; + t.words32.w3 = 0; + u.words32.w0 += 0x10000; + e -= 1; + k += 64; + } + else + { + k = (m - 0xfe00) >> 10; + t.words32.w0 = 0x3ffe0000 + (k << 10); + t.words32.w1 = 0; + t.words32.w2 = 0; + t.words32.w3 = 0; + } + /* On this interval the table is not used due to cancellation error. */ + if ((x <= 1.0078125Q) && (x >= 0.9921875Q)) + { + z = x - 1.0Q; + k = 64; + t.value = 1.0Q; + e = 0; + } + else + { + /* log(u) = log( t u/t ) = log(t) + log(u/t) + log(t) is tabulated in the lookup table. + Express log(u/t) = log(1+z), where z = u/t - 1 = (u-t)/t. + cf. Cody & Waite. */ + z = (u.value - t.value) / t.value; + } + /* Series expansion of log(1+z). */ + w = z * z; + y = ((((((((((((l15 * z + + l14) * z + + l13) * z + + l12) * z + + l11) * z + + l10) * z + + l9) * z + + l8) * z + + l7) * z + + l6) * z + + l5) * z + + l4) * z + + l3) * z * w; + y -= 0.5 * w; + y += e * ln2b; /* Base 2 exponent offset times ln(2). */ + y += z; + y += logtbl[k-26]; /* log(t) - (t-1) */ + y += (t.value - 1.0Q); + y += e * ln2a; + return y; +} diff --git a/libquadmath/math/lrintq.c b/libquadmath/math/lrintq.c new file mode 100644 index 000000000..d1497ae38 --- /dev/null +++ b/libquadmath/math/lrintq.c @@ -0,0 +1,85 @@ +/* Round argument to nearest integral value according to current rounding + direction. + Copyright (C) 1997, 1999, 2004, 2006 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997 and + Jakub Jelinek <jj@ultra.linux.cz>, 1999. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + +static const __float128 two112[2] = +{ + 5.19229685853482762853049632922009600E+33Q, /* 0x406F000000000000, 0 */ + -5.19229685853482762853049632922009600E+33Q /* 0xC06F000000000000, 0 */ +}; + +long int +lrintq (__float128 x) +{ + int32_t j0; + uint64_t i0,i1; + volatile __float128 w; + __float128 t; + long int result; + int sx; + + GET_FLT128_WORDS64 (i0, i1, x); + j0 = ((i0 >> 48) & 0x7fff) - 0x3fff; + sx = i0 >> 63; + i0 &= 0x0000ffffffffffffLL; + i0 |= 0x0001000000000000LL; + + if (j0 < 48) + { + w = two112[sx] + x; + t = w - two112[sx]; + GET_FLT128_WORDS64 (i0, i1, t); + j0 = ((i0 >> 48) & 0x7fff) - 0x3fff; + i0 &= 0x0000ffffffffffffLL; + i0 |= 0x0001000000000000LL; + + result = (j0 < 0 ? 0 : i0 >> (48 - j0)); + } + else if (j0 < (int32_t) (8 * sizeof (long int)) - 1) + { + if (j0 >= 112) + result = ((long int) i0 << (j0 - 48)) | (i1 << (j0 - 112)); + else + { + w = two112[sx] + x; + t = w - two112[sx]; + GET_FLT128_WORDS64 (i0, i1, t); + j0 = ((i0 >> 48) & 0x7fff) - 0x3fff; + i0 &= 0x0000ffffffffffffLL; + i0 |= 0x0001000000000000LL; + + if (j0 == 48) + result = (long int) i0; + else + result = ((long int) i0 << (j0 - 48)) | (i1 >> (112 - j0)); + } + } + else + { + /* The number is too large. It is left implementation defined + what happens. */ + return (long int) x; + } + + return sx ? -result : result; +} diff --git a/libquadmath/math/lroundq.c b/libquadmath/math/lroundq.c new file mode 100644 index 000000000..4b12d2452 --- /dev/null +++ b/libquadmath/math/lroundq.c @@ -0,0 +1,73 @@ +/* Round long double value to long int. + Copyright (C) 1997, 1999, 2004 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997 and + Jakub Jelinek <jj@ultra.linux.cz>, 1999. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + + +long int +lroundq (__float128 x) +{ + int64_t j0; + uint64_t i1, i0; + long int result; + int sign; + + GET_FLT128_WORDS64 (i0, i1, x); + j0 = ((i0 >> 48) & 0x7fff) - 0x3fff; + sign = (i0 & 0x8000000000000000ULL) != 0 ? -1 : 1; + i0 &= 0x0000ffffffffffffLL; + i0 |= 0x0001000000000000LL; + + if (j0 < 48) + { + if (j0 < 0) + return j0 < -1 ? 0 : sign; + else + { + i0 += 0x0000800000000000LL >> j0; + result = i0 >> (48 - j0); + } + } + else if (j0 < (int32_t) (8 * sizeof (long int)) - 1) + { + if (j0 >= 112) + result = ((long int) i0 << (j0 - 48)) | (i1 << (j0 - 112)); + else + { + uint64_t j = i1 + (0x8000000000000000ULL >> (j0 - 48)); + if (j < i1) + ++i0; + + if (j0 == 48) + result = (long int) i0; + else + result = ((long int) i0 << (j0 - 48)) | (j >> (112 - j0)); + } + } + else + { + /* The number is too large. It is left implementation defined + what happens. */ + return (long int) x; + } + + return sign * result; +} diff --git a/libquadmath/math/modfq.c b/libquadmath/math/modfq.c new file mode 100644 index 000000000..dc564fe35 --- /dev/null +++ b/libquadmath/math/modfq.c @@ -0,0 +1,64 @@ +/* s_modfl.c -- long double version of s_modf.c. + * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include "quadmath-imp.h" + +static const __float128 one = 1.0; + +__float128 +modfq (__float128 x, __float128 *iptr) +{ + int64_t i0,i1,j0; + uint64_t i; + GET_FLT128_WORDS64(i0,i1,x); + j0 = ((i0>>48)&0x7fff)-0x3fff; /* exponent of x */ + if(j0<48) { /* integer part in high x */ + if(j0<0) { /* |x|<1 */ + /* *iptr = +-0 */ + SET_FLT128_WORDS64(*iptr,i0&0x8000000000000000ULL,0); + return x; + } else { + i = (0x0000ffffffffffffLL)>>j0; + if(((i0&i)|i1)==0) { /* x is integral */ + *iptr = x; + /* return +-0 */ + SET_FLT128_WORDS64(x,i0&0x8000000000000000ULL,0); + return x; + } else { + SET_FLT128_WORDS64(*iptr,i0&(~i),0); + return x - *iptr; + } + } + } else if (j0>111) { /* no fraction part */ + *iptr = x*one; + /* We must handle NaNs separately. */ + if (j0 == 0x4000 && ((i0 & 0x0000ffffffffffffLL) | i1)) + return x*one; + /* return +-0 */ + SET_FLT128_WORDS64(x,i0&0x8000000000000000ULL,0); + return x; + } else { /* fraction part in low x */ + i = -1ULL>>(j0-48); + if((i1&i)==0) { /* x is integral */ + *iptr = x; + /* return +-0 */ + SET_FLT128_WORDS64(x,i0&0x8000000000000000ULL,0); + return x; + } else { + SET_FLT128_WORDS64(*iptr,i0,i1&(~i)); + return x - *iptr; + } + } +} diff --git a/libquadmath/math/nanq.c b/libquadmath/math/nanq.c new file mode 100644 index 000000000..bace47064 --- /dev/null +++ b/libquadmath/math/nanq.c @@ -0,0 +1,11 @@ +#include "quadmath-imp.h" + +__float128 +nanq (const char *tagp __attribute__ ((unused))) +{ + // FIXME -- we should use the argument + ieee854_float128 f; + f.ieee.exponent = 0x7fff; + f.ieee.mant_high = 0x1; + return f.value; +} diff --git a/libquadmath/math/nearbyintq.c b/libquadmath/math/nearbyintq.c new file mode 100644 index 000000000..8e92c5afd --- /dev/null +++ b/libquadmath/math/nearbyintq.c @@ -0,0 +1,98 @@ +/* nearbyintq.c -- __float128 version of s_nearbyint.c. + * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* + * nearbyintq(x) + * Return x rounded to integral value according to the prevailing + * rounding mode. + * Method: + * Using floating addition. + * Exception: + * Inexact flag raised if x not equal to rintq(x). + */ + +#include "quadmath-imp.h" +#ifdef HAVE_FENV_H +# include <fenv.h> +# if defined HAVE_FEHOLDEXCEPT && defined HAVE_FESETENV +# define USE_FENV_H +# endif +#endif + +static const __float128 +TWO112[2]={ + 5.19229685853482762853049632922009600E+33Q, /* 0x406F000000000000, 0 */ + -5.19229685853482762853049632922009600E+33Q /* 0xC06F000000000000, 0 */ +}; + +__float128 +nearbyintq(__float128 x) +{ +#ifdef USE_FENV_H + fenv_t env; +#endif + int64_t i0,j0,sx; + uint64_t i,i1; + __float128 w,t; + GET_FLT128_WORDS64(i0,i1,x); + sx = (((uint64_t)i0)>>63); + j0 = ((i0>>48)&0x7fff)-0x3fff; + if(j0<48) { + if(j0<0) { + if(((i0&0x7fffffffffffffffLL)|i1)==0) return x; + i1 |= (i0&0x0000ffffffffffffLL); + i0 &= 0xffffe00000000000ULL; + i0 |= ((i1|-i1)>>16)&0x0000800000000000LL; + SET_FLT128_MSW64(x,i0); +#ifdef USE_FENV_H + feholdexcept (&env); +#endif + w = TWO112[sx]+x; + t = w-TWO112[sx]; +#ifdef USE_FENV_H + fesetenv (&env); +#endif + GET_FLT128_MSW64(i0,t); + SET_FLT128_MSW64(t,(i0&0x7fffffffffffffffLL)|(sx<<63)); + return t; + } else { + i = (0x0000ffffffffffffLL)>>j0; + if(((i0&i)|i1)==0) return x; /* x is integral */ + i>>=1; + if(((i0&i)|i1)!=0) { + if(j0==47) i1 = 0x4000000000000000ULL; else + i0 = (i0&(~i))|((0x0000200000000000LL)>>j0); + } + } + } else if (j0>111) { + if(j0==0x4000) return x+x; /* inf or NaN */ + else return x; /* x is integral */ + } else { + i = -1ULL>>(j0-48); + if((i1&i)==0) return x; /* x is integral */ + i>>=1; + if((i1&i)!=0) i1 = (i1&(~i))|((0x4000000000000000LL)>>(j0-48)); + } + SET_FLT128_WORDS64(x,i0,i1); +#ifdef USE_FENV_H + feholdexcept (&env); +#endif + w = TWO112[sx]+x; + t = w-TWO112[sx]; +#ifdef USE_FENV_H + fesetenv (&env); +#endif + return t; +} diff --git a/libquadmath/math/nextafterq.c b/libquadmath/math/nextafterq.c new file mode 100644 index 000000000..01bfa6579 --- /dev/null +++ b/libquadmath/math/nextafterq.c @@ -0,0 +1,63 @@ +/* s_nextafterl.c -- long double version of s_nextafter.c. + * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include "quadmath-imp.h" + +__float128 +nextafterq (__float128 x, __float128 y) +{ + int64_t hx,hy,ix,iy; + uint64_t lx,ly; + + GET_FLT128_WORDS64(hx,lx,x); + GET_FLT128_WORDS64(hy,ly,y); + ix = hx&0x7fffffffffffffffLL; /* |x| */ + iy = hy&0x7fffffffffffffffLL; /* |y| */ + + if(((ix>=0x7fff000000000000LL)&&((ix-0x7fff000000000000LL)|lx)!=0) || /* x is nan */ + ((iy>=0x7fff000000000000LL)&&((iy-0x7fff000000000000LL)|ly)!=0)) /* y is nan */ + return x+y; + if(x==y) return y; /* x=y, return y */ + if((ix|lx)==0) { /* x == 0 */ + SET_FLT128_WORDS64(x,hy&0x8000000000000000ULL,1);/* return +-minsubnormal */ + + /* here we should raise an underflow flag */ + return x; + } + if(hx>=0) { /* x > 0 */ + if(hx>hy||((hx==hy)&&(lx>ly))) { /* x > y, x -= ulp */ + if(lx==0) hx--; + lx--; + } else { /* x < y, x += ulp */ + lx++; + if(lx==0) hx++; + } + } else { /* x < 0 */ + if(hy>=0||hx>hy||((hx==hy)&&(lx>ly))){/* x < y, x -= ulp */ + if(lx==0) hx--; + lx--; + } else { /* x > y, x += ulp */ + lx++; + if(lx==0) hx++; + } + } + hy = hx&0x7fff000000000000LL; + if(hy==0x7fff000000000000LL) return x+x;/* overflow */ + if(hy==0) { + /* here we should raise an underflow flag */ + } + SET_FLT128_WORDS64(x,hx,lx); + return x; +} diff --git a/libquadmath/math/powq.c b/libquadmath/math/powq.c new file mode 100644 index 000000000..d38632438 --- /dev/null +++ b/libquadmath/math/powq.c @@ -0,0 +1,440 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* Expansions and modifications for 128-bit long double are + Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> + and are incorporated herein by permission of the author. The author + reserves the right to distribute this material elsewhere under different + copying permissions. These modifications are distributed here under + the following terms: + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + +/* __ieee754_powl(x,y) return x**y + * + * n + * Method: Let x = 2 * (1+f) + * 1. Compute and return log2(x) in two pieces: + * log2(x) = w1 + w2, + * where w1 has 113-53 = 60 bit trailing zeros. + * 2. Perform y*log2(x) = n+y' by simulating muti-precision + * arithmetic, where |y'|<=0.5. + * 3. Return x**y = 2**n*exp(y'*log2) + * + * Special cases: + * 1. (anything) ** 0 is 1 + * 2. (anything) ** 1 is itself + * 3. (anything) ** NAN is NAN + * 4. NAN ** (anything except 0) is NAN + * 5. +-(|x| > 1) ** +INF is +INF + * 6. +-(|x| > 1) ** -INF is +0 + * 7. +-(|x| < 1) ** +INF is +0 + * 8. +-(|x| < 1) ** -INF is +INF + * 9. +-1 ** +-INF is NAN + * 10. +0 ** (+anything except 0, NAN) is +0 + * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 + * 12. +0 ** (-anything except 0, NAN) is +INF + * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF + * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) + * 15. +INF ** (+anything except 0,NAN) is +INF + * 16. +INF ** (-anything except 0,NAN) is +0 + * 17. -INF ** (anything) = -0 ** (-anything) + * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) + * 19. (-anything except 0 and inf) ** (non-integer) is NAN + * + */ + +#include "quadmath-imp.h" + +static const __float128 bp[] = { + 1.0Q, + 1.5Q, +}; + +/* log_2(1.5) */ +static const __float128 dp_h[] = { + 0.0, + 5.8496250072115607565592654282227158546448E-1Q +}; + +/* Low part of log_2(1.5) */ +static const __float128 dp_l[] = { + 0.0, + 1.0579781240112554492329533686862998106046E-16Q +}; + +static const __float128 zero = 0.0Q, + one = 1.0Q, + two = 2.0Q, + two113 = 1.0384593717069655257060992658440192E34Q, + huge = 1.0e3000Q, + tiny = 1.0e-3000Q; + +/* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2)) + z = (x-1)/(x+1) + 1 <= x <= 1.25 + Peak relative error 2.3e-37 */ +static const __float128 LN[] = +{ + -3.0779177200290054398792536829702930623200E1Q, + 6.5135778082209159921251824580292116201640E1Q, + -4.6312921812152436921591152809994014413540E1Q, + 1.2510208195629420304615674658258363295208E1Q, + -9.9266909031921425609179910128531667336670E-1Q +}; +static const __float128 LD[] = +{ + -5.129862866715009066465422805058933131960E1Q, + 1.452015077564081884387441590064272782044E2Q, + -1.524043275549860505277434040464085593165E2Q, + 7.236063513651544224319663428634139768808E1Q, + -1.494198912340228235853027849917095580053E1Q + /* 1.0E0 */ +}; + +/* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2))) + 0 <= x <= 0.5 + Peak relative error 5.7e-38 */ +static const __float128 PN[] = +{ + 5.081801691915377692446852383385968225675E8Q, + 9.360895299872484512023336636427675327355E6Q, + 4.213701282274196030811629773097579432957E4Q, + 5.201006511142748908655720086041570288182E1Q, + 9.088368420359444263703202925095675982530E-3Q, +}; +static const __float128 PD[] = +{ + 3.049081015149226615468111430031590411682E9Q, + 1.069833887183886839966085436512368982758E8Q, + 8.259257717868875207333991924545445705394E5Q, + 1.872583833284143212651746812884298360922E3Q, + /* 1.0E0 */ +}; + +static const __float128 + /* ln 2 */ + lg2 = 6.9314718055994530941723212145817656807550E-1Q, + lg2_h = 6.9314718055994528622676398299518041312695E-1Q, + lg2_l = 2.3190468138462996154948554638754786504121E-17Q, + ovt = 8.0085662595372944372e-0017Q, + /* 2/(3*log(2)) */ + cp = 9.6179669392597560490661645400126142495110E-1Q, + cp_h = 9.6179669392597555432899980587535537779331E-1Q, + cp_l = 5.0577616648125906047157785230014751039424E-17Q; + +__float128 +powq (__float128 x, __float128 y) +{ + __float128 z, ax, z_h, z_l, p_h, p_l; + __float128 y1, t1, t2, r, s, t, u, v, w; + __float128 s2, s_h, s_l, t_h, t_l; + int32_t i, j, k, yisint, n; + uint32_t ix, iy; + int32_t hx, hy; + ieee854_float128 o, p, q; + + p.value = x; + hx = p.words32.w0; + ix = hx & 0x7fffffff; + + q.value = y; + hy = q.words32.w0; + iy = hy & 0x7fffffff; + + + /* y==zero: x**0 = 1 */ + if ((iy | q.words32.w1 | q.words32.w2 | q.words32.w3) == 0) + return one; + + /* 1.0**y = 1; -1.0**+-Inf = 1 */ + if (x == one) + return one; + if (x == -1.0Q && iy == 0x7fff0000 + && (q.words32.w1 | q.words32.w2 | q.words32.w3) == 0) + return one; + + /* +-NaN return x+y */ + if ((ix > 0x7fff0000) + || ((ix == 0x7fff0000) + && ((p.words32.w1 | p.words32.w2 | p.words32.w3) != 0)) + || (iy > 0x7fff0000) + || ((iy == 0x7fff0000) + && ((q.words32.w1 | q.words32.w2 | q.words32.w3) != 0))) + return x + y; + + /* determine if y is an odd int when x < 0 + * yisint = 0 ... y is not an integer + * yisint = 1 ... y is an odd int + * yisint = 2 ... y is an even int + */ + yisint = 0; + if (hx < 0) + { + if (iy >= 0x40700000) /* 2^113 */ + yisint = 2; /* even integer y */ + else if (iy >= 0x3fff0000) /* 1.0 */ + { + if (floorq (y) == y) + { + z = 0.5 * y; + if (floorq (z) == z) + yisint = 2; + else + yisint = 1; + } + } + } + + /* special value of y */ + if ((q.words32.w1 | q.words32.w2 | q.words32.w3) == 0) + { + if (iy == 0x7fff0000) /* y is +-inf */ + { + if (((ix - 0x3fff0000) | p.words32.w1 | p.words32.w2 | p.words32.w3) + == 0) + return y - y; /* +-1**inf is NaN */ + else if (ix >= 0x3fff0000) /* (|x|>1)**+-inf = inf,0 */ + return (hy >= 0) ? y : zero; + else /* (|x|<1)**-,+inf = inf,0 */ + return (hy < 0) ? -y : zero; + } + if (iy == 0x3fff0000) + { /* y is +-1 */ + if (hy < 0) + return one / x; + else + return x; + } + if (hy == 0x40000000) + return x * x; /* y is 2 */ + if (hy == 0x3ffe0000) + { /* y is 0.5 */ + if (hx >= 0) /* x >= +0 */ + return sqrtq (x); + } + } + + ax = fabsq (x); + /* special value of x */ + if ((p.words32.w1 | p.words32.w2 | p.words32.w3) == 0) + { + if (ix == 0x7fff0000 || ix == 0 || ix == 0x3fff0000) + { + z = ax; /*x is +-0,+-inf,+-1 */ + if (hy < 0) + z = one / z; /* z = (1/|x|) */ + if (hx < 0) + { + if (((ix - 0x3fff0000) | yisint) == 0) + { + z = (z - z) / (z - z); /* (-1)**non-int is NaN */ + } + else if (yisint == 1) + z = -z; /* (x<0)**odd = -(|x|**odd) */ + } + return z; + } + } + + /* (x<0)**(non-int) is NaN */ + if (((((uint32_t) hx >> 31) - 1) | yisint) == 0) + return (x - x) / (x - x); + + /* |y| is huge. + 2^-16495 = 1/2 of smallest representable value. + If (1 - 1/131072)^y underflows, y > 1.4986e9 */ + if (iy > 0x401d654b) + { + /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */ + if (iy > 0x407d654b) + { + if (ix <= 0x3ffeffff) + return (hy < 0) ? huge * huge : tiny * tiny; + if (ix >= 0x3fff0000) + return (hy > 0) ? huge * huge : tiny * tiny; + } + /* over/underflow if x is not close to one */ + if (ix < 0x3ffeffff) + return (hy < 0) ? huge * huge : tiny * tiny; + if (ix > 0x3fff0000) + return (hy > 0) ? huge * huge : tiny * tiny; + } + + n = 0; + /* take care subnormal number */ + if (ix < 0x00010000) + { + ax *= two113; + n -= 113; + o.value = ax; + ix = o.words32.w0; + } + n += ((ix) >> 16) - 0x3fff; + j = ix & 0x0000ffff; + /* determine interval */ + ix = j | 0x3fff0000; /* normalize ix */ + if (j <= 0x3988) + k = 0; /* |x|<sqrt(3/2) */ + else if (j < 0xbb67) + k = 1; /* |x|<sqrt(3) */ + else + { + k = 0; + n += 1; + ix -= 0x00010000; + } + + o.value = ax; + o.words32.w0 = ix; + ax = o.value; + + /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ + u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ + v = one / (ax + bp[k]); + s = u * v; + s_h = s; + + o.value = s_h; + o.words32.w3 = 0; + o.words32.w2 &= 0xf8000000; + s_h = o.value; + /* t_h=ax+bp[k] High */ + t_h = ax + bp[k]; + o.value = t_h; + o.words32.w3 = 0; + o.words32.w2 &= 0xf8000000; + t_h = o.value; + t_l = ax - (t_h - bp[k]); + s_l = v * ((u - s_h * t_h) - s_h * t_l); + /* compute log(ax) */ + s2 = s * s; + u = LN[0] + s2 * (LN[1] + s2 * (LN[2] + s2 * (LN[3] + s2 * LN[4]))); + v = LD[0] + s2 * (LD[1] + s2 * (LD[2] + s2 * (LD[3] + s2 * (LD[4] + s2)))); + r = s2 * s2 * u / v; + r += s_l * (s_h + s); + s2 = s_h * s_h; + t_h = 3.0 + s2 + r; + o.value = t_h; + o.words32.w3 = 0; + o.words32.w2 &= 0xf8000000; + t_h = o.value; + t_l = r - ((t_h - 3.0) - s2); + /* u+v = s*(1+...) */ + u = s_h * t_h; + v = s_l * t_h + t_l * s; + /* 2/(3log2)*(s+...) */ + p_h = u + v; + o.value = p_h; + o.words32.w3 = 0; + o.words32.w2 &= 0xf8000000; + p_h = o.value; + p_l = v - (p_h - u); + z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */ + z_l = cp_l * p_h + p_l * cp + dp_l[k]; + /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ + t = (__float128) n; + t1 = (((z_h + z_l) + dp_h[k]) + t); + o.value = t1; + o.words32.w3 = 0; + o.words32.w2 &= 0xf8000000; + t1 = o.value; + t2 = z_l - (((t1 - t) - dp_h[k]) - z_h); + + /* s (sign of result -ve**odd) = -1 else = 1 */ + s = one; + if (((((uint32_t) hx >> 31) - 1) | (yisint - 1)) == 0) + s = -one; /* (-ve)**(odd int) */ + + /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ + y1 = y; + o.value = y1; + o.words32.w3 = 0; + o.words32.w2 &= 0xf8000000; + y1 = o.value; + p_l = (y - y1) * t1 + y * t2; + p_h = y1 * t1; + z = p_l + p_h; + o.value = z; + j = o.words32.w0; + if (j >= 0x400d0000) /* z >= 16384 */ + { + /* if z > 16384 */ + if (((j - 0x400d0000) | o.words32.w1 | o.words32.w2 | o.words32.w3) != 0) + return s * huge * huge; /* overflow */ + else + { + if (p_l + ovt > z - p_h) + return s * huge * huge; /* overflow */ + } + } + else if ((j & 0x7fffffff) >= 0x400d01b9) /* z <= -16495 */ + { + /* z < -16495 */ + if (((j - 0xc00d01bc) | o.words32.w1 | o.words32.w2 | o.words32.w3) + != 0) + return s * tiny * tiny; /* underflow */ + else + { + if (p_l <= z - p_h) + return s * tiny * tiny; /* underflow */ + } + } + /* compute 2**(p_h+p_l) */ + i = j & 0x7fffffff; + k = (i >> 16) - 0x3fff; + n = 0; + if (i > 0x3ffe0000) + { /* if |z| > 0.5, set n = [z+0.5] */ + n = floorq (z + 0.5Q); + t = n; + p_h -= t; + } + t = p_l + p_h; + o.value = t; + o.words32.w3 = 0; + o.words32.w2 &= 0xf8000000; + t = o.value; + u = t * lg2_h; + v = (p_l - (t - p_h)) * lg2 + t * lg2_l; + z = u + v; + w = v - (z - u); + /* exp(z) */ + t = z * z; + u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4]))); + v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t))); + t1 = z - t * u / v; + r = (z * t1) / (t1 - two) - (w + z * w); + z = one - (r - z); + o.value = z; + j = o.words32.w0; + j += (n << 16); + if ((j >> 16) <= 0) + z = scalbnq (z, n); /* subnormal output */ + else + { + o.words32.w0 = j; + z = o.value; + } + return s * z; +} diff --git a/libquadmath/math/rem_pio2q.c b/libquadmath/math/rem_pio2q.c new file mode 100644 index 000000000..47ee8ef20 --- /dev/null +++ b/libquadmath/math/rem_pio2q.c @@ -0,0 +1,587 @@ +#include "quadmath-imp.h" +#include <math.h> + + +/* @(#)k_rem_pio2.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* + * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) + * double x[],y[]; int e0,nx,prec; int ipio2[]; + * + * __kernel_rem_pio2 return the last three digits of N with + * y = x - N*pi/2 + * so that |y| < pi/2. + * + * The method is to compute the integer (mod 8) and fraction parts of + * (2/pi)*x without doing the full multiplication. In general we + * skip the part of the product that are known to be a huge integer ( + * more accurately, = 0 mod 8 ). Thus the number of operations are + * independent of the exponent of the input. + * + * (2/pi) is represented by an array of 24-bit integers in ipio2[]. + * + * Input parameters: + * x[] The input value (must be positive) is broken into nx + * pieces of 24-bit integers in double precision format. + * x[i] will be the i-th 24 bit of x. The scaled exponent + * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 + * match x's up to 24 bits. + * + * Example of breaking a double positive z into x[0]+x[1]+x[2]: + * e0 = ilogb(z)-23 + * z = scalbn(z,-e0) + * for i = 0,1,2 + * x[i] = floor(z) + * z = (z-x[i])*2**24 + * + * + * y[] ouput result in an array of double precision numbers. + * The dimension of y[] is: + * 24-bit precision 1 + * 53-bit precision 2 + * 64-bit precision 2 + * 113-bit precision 3 + * The actual value is the sum of them. Thus for 113-bit + * precision, one may have to do something like: + * + * long double t,w,r_head, r_tail; + * t = (long double)y[2] + (long double)y[1]; + * w = (long double)y[0]; + * r_head = t+w; + * r_tail = w - (r_head - t); + * + * e0 The exponent of x[0] + * + * nx dimension of x[] + * + * prec an integer indicating the precision: + * 0 24 bits (single) + * 1 53 bits (double) + * 2 64 bits (extended) + * 3 113 bits (quad) + * + * ipio2[] + * integer array, contains the (24*i)-th to (24*i+23)-th + * bit of 2/pi after binary point. The corresponding + * floating value is + * + * ipio2[i] * 2^(-24(i+1)). + * + * External function: + * double scalbn(), floor(); + * + * + * Here is the description of some local variables: + * + * jk jk+1 is the initial number of terms of ipio2[] needed + * in the computation. The recommended value is 2,3,4, + * 6 for single, double, extended,and quad. + * + * jz local integer variable indicating the number of + * terms of ipio2[] used. + * + * jx nx - 1 + * + * jv index for pointing to the suitable ipio2[] for the + * computation. In general, we want + * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 + * is an integer. Thus + * e0-3-24*jv >= 0 or (e0-3)/24 >= jv + * Hence jv = max(0,(e0-3)/24). + * + * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. + * + * q[] double array with integral value, representing the + * 24-bits chunk of the product of x and 2/pi. + * + * q0 the corresponding exponent of q[0]. Note that the + * exponent for q[i] would be q0-24*i. + * + * PIo2[] double precision array, obtained by cutting pi/2 + * into 24 bits chunks. + * + * f[] ipio2[] in floating point + * + * iq[] integer array by breaking up q[] in 24-bits chunk. + * + * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] + * + * ih integer. If >0 it indicates q[] is >= 0.5, hence + * it also indicates the *sign* of the result. + * + */ + +/* + * Constants: + * The hexadecimal values are the intended ones for the following + * constants. The decimal values may be used, provided that the + * compiler will convert from decimal to binary accurately enough + * to produce the hexadecimal values shown. + */ + + +static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ + +static const double PIo2[] = { + 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ + 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ + 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ + 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ + 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ + 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ + 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ + 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ +}; + +static const double + zero = 0.0, + one = 1.0, + two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ + twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ + + +static int +__quadmath_kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2) +{ + int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; + double z,fw,f[20],fq[20],q[20]; + + /* initialize jk*/ + jk = init_jk[prec]; + jp = jk; + + /* determine jx,jv,q0, note that 3>q0 */ + jx = nx-1; + jv = (e0-3)/24; if(jv<0) jv=0; + q0 = e0-24*(jv+1); + + /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ + j = jv-jx; m = jx+jk; + for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j]; + + /* compute q[0],q[1],...q[jk] */ + for (i=0;i<=jk;i++) { + for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; + } + + jz = jk; +recompute: + /* distill q[] into iq[] reversingly */ + for(i=0,j=jz,z=q[jz];j>0;i++,j--) { + fw = (double)((int32_t)(twon24* z)); + iq[i] = (int32_t)(z-two24*fw); + z = q[j-1]+fw; + } + + /* compute n */ + z = scalbn(z,q0); /* actual value of z */ + z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ + n = (int32_t) z; + z -= (double)n; + ih = 0; + if(q0>0) { /* need iq[jz-1] to determine n */ + i = (iq[jz-1]>>(24-q0)); n += i; + iq[jz-1] -= i<<(24-q0); + ih = iq[jz-1]>>(23-q0); + } + else if(q0==0) ih = iq[jz-1]>>23; + else if(z>=0.5) ih=2; + + if(ih>0) { /* q > 0.5 */ + n += 1; carry = 0; + for(i=0;i<jz ;i++) { /* compute 1-q */ + j = iq[i]; + if(carry==0) { + if(j!=0) { + carry = 1; iq[i] = 0x1000000- j; + } + } else iq[i] = 0xffffff - j; + } + if(q0>0) { /* rare case: chance is 1 in 12 */ + switch(q0) { + case 1: + iq[jz-1] &= 0x7fffff; break; + case 2: + iq[jz-1] &= 0x3fffff; break; + } + } + if(ih==2) { + z = one - z; + if(carry!=0) z -= scalbn(one,q0); + } + } + + /* check if recomputation is needed */ + if(z==zero) { + j = 0; + for (i=jz-1;i>=jk;i--) j |= iq[i]; + if(j==0) { /* need recomputation */ + for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ + + for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ + f[jx+i] = (double) ipio2[jv+i]; + for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; + q[i] = fw; + } + jz += k; + goto recompute; + } + } + + /* chop off zero terms */ + if(z==0.0) { + jz -= 1; q0 -= 24; + while(iq[jz]==0) { jz--; q0-=24;} + } else { /* break z into 24-bit if necessary */ + z = scalbn(z,-q0); + if(z>=two24) { + fw = (double)((int32_t)(twon24*z)); + iq[jz] = (int32_t)(z-two24*fw); + jz += 1; q0 += 24; + iq[jz] = (int32_t) fw; + } else iq[jz] = (int32_t) z ; + } + + /* convert integer "bit" chunk to floating-point value */ + fw = scalbn(one,q0); + for(i=jz;i>=0;i--) { + q[i] = fw*(double)iq[i]; fw*=twon24; + } + + /* compute PIo2[0,...,jp]*q[jz,...,0] */ + for(i=jz;i>=0;i--) { + for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; + fq[jz-i] = fw; + } + + /* compress fq[] into y[] */ + switch(prec) { + case 0: + fw = 0.0; + for (i=jz;i>=0;i--) fw += fq[i]; + y[0] = (ih==0)? fw: -fw; + break; + case 1: + case 2: + fw = 0.0; + for (i=jz;i>=0;i--) fw += fq[i]; + y[0] = (ih==0)? fw: -fw; + fw = fq[0]-fw; + for (i=1;i<=jz;i++) fw += fq[i]; + y[1] = (ih==0)? fw: -fw; + break; + case 3: /* painful */ + for (i=jz;i>0;i--) { +#if __FLT_EVAL_METHOD__ != 0 + volatile +#endif + double fv = (double)(fq[i-1]+fq[i]); + fq[i] += fq[i-1]-fv; + fq[i-1] = fv; + } + for (i=jz;i>1;i--) { +#if __FLT_EVAL_METHOD__ != 0 + volatile +#endif + double fv = (double)(fq[i-1]+fq[i]); + fq[i] += fq[i-1]-fv; + fq[i-1] = fv; + } + for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; + if(ih==0) { + y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; + } else { + y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; + } + } + return n&7; +} + + + + + +/* Quad-precision floating point argument reduction. + Copyright (C) 1999 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Jakub Jelinek <jj@ultra.linux.cz> + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +/* + * Table of constants for 2/pi, 5628 hexadecimal digits of 2/pi + */ +static const int32_t two_over_pi[] = { +0xa2f983, 0x6e4e44, 0x1529fc, 0x2757d1, 0xf534dd, 0xc0db62, +0x95993c, 0x439041, 0xfe5163, 0xabdebb, 0xc561b7, 0x246e3a, +0x424dd2, 0xe00649, 0x2eea09, 0xd1921c, 0xfe1deb, 0x1cb129, +0xa73ee8, 0x8235f5, 0x2ebb44, 0x84e99c, 0x7026b4, 0x5f7e41, +0x3991d6, 0x398353, 0x39f49c, 0x845f8b, 0xbdf928, 0x3b1ff8, +0x97ffde, 0x05980f, 0xef2f11, 0x8b5a0a, 0x6d1f6d, 0x367ecf, +0x27cb09, 0xb74f46, 0x3f669e, 0x5fea2d, 0x7527ba, 0xc7ebe5, +0xf17b3d, 0x0739f7, 0x8a5292, 0xea6bfb, 0x5fb11f, 0x8d5d08, +0x560330, 0x46fc7b, 0x6babf0, 0xcfbc20, 0x9af436, 0x1da9e3, +0x91615e, 0xe61b08, 0x659985, 0x5f14a0, 0x68408d, 0xffd880, +0x4d7327, 0x310606, 0x1556ca, 0x73a8c9, 0x60e27b, 0xc08c6b, +0x47c419, 0xc367cd, 0xdce809, 0x2a8359, 0xc4768b, 0x961ca6, +0xddaf44, 0xd15719, 0x053ea5, 0xff0705, 0x3f7e33, 0xe832c2, +0xde4f98, 0x327dbb, 0xc33d26, 0xef6b1e, 0x5ef89f, 0x3a1f35, +0xcaf27f, 0x1d87f1, 0x21907c, 0x7c246a, 0xfa6ed5, 0x772d30, +0x433b15, 0xc614b5, 0x9d19c3, 0xc2c4ad, 0x414d2c, 0x5d000c, +0x467d86, 0x2d71e3, 0x9ac69b, 0x006233, 0x7cd2b4, 0x97a7b4, +0xd55537, 0xf63ed7, 0x1810a3, 0xfc764d, 0x2a9d64, 0xabd770, +0xf87c63, 0x57b07a, 0xe71517, 0x5649c0, 0xd9d63b, 0x3884a7, +0xcb2324, 0x778ad6, 0x23545a, 0xb91f00, 0x1b0af1, 0xdfce19, +0xff319f, 0x6a1e66, 0x615799, 0x47fbac, 0xd87f7e, 0xb76522, +0x89e832, 0x60bfe6, 0xcdc4ef, 0x09366c, 0xd43f5d, 0xd7de16, +0xde3b58, 0x929bde, 0x2822d2, 0xe88628, 0x4d58e2, 0x32cac6, +0x16e308, 0xcb7de0, 0x50c017, 0xa71df3, 0x5be018, 0x34132e, +0x621283, 0x014883, 0x5b8ef5, 0x7fb0ad, 0xf2e91e, 0x434a48, +0xd36710, 0xd8ddaa, 0x425fae, 0xce616a, 0xa4280a, 0xb499d3, +0xf2a606, 0x7f775c, 0x83c2a3, 0x883c61, 0x78738a, 0x5a8caf, +0xbdd76f, 0x63a62d, 0xcbbff4, 0xef818d, 0x67c126, 0x45ca55, +0x36d9ca, 0xd2a828, 0x8d61c2, 0x77c912, 0x142604, 0x9b4612, +0xc459c4, 0x44c5c8, 0x91b24d, 0xf31700, 0xad43d4, 0xe54929, +0x10d5fd, 0xfcbe00, 0xcc941e, 0xeece70, 0xf53e13, 0x80f1ec, +0xc3e7b3, 0x28f8c7, 0x940593, 0x3e71c1, 0xb3092e, 0xf3450b, +0x9c1288, 0x7b20ab, 0x9fb52e, 0xc29247, 0x2f327b, 0x6d550c, +0x90a772, 0x1fe76b, 0x96cb31, 0x4a1679, 0xe27941, 0x89dff4, +0x9794e8, 0x84e6e2, 0x973199, 0x6bed88, 0x365f5f, 0x0efdbb, +0xb49a48, 0x6ca467, 0x427271, 0x325d8d, 0xb8159f, 0x09e5bc, +0x25318d, 0x3974f7, 0x1c0530, 0x010c0d, 0x68084b, 0x58ee2c, +0x90aa47, 0x02e774, 0x24d6bd, 0xa67df7, 0x72486e, 0xef169f, +0xa6948e, 0xf691b4, 0x5153d1, 0xf20acf, 0x339820, 0x7e4bf5, +0x6863b2, 0x5f3edd, 0x035d40, 0x7f8985, 0x295255, 0xc06437, +0x10d86d, 0x324832, 0x754c5b, 0xd4714e, 0x6e5445, 0xc1090b, +0x69f52a, 0xd56614, 0x9d0727, 0x50045d, 0xdb3bb4, 0xc576ea, +0x17f987, 0x7d6b49, 0xba271d, 0x296996, 0xacccc6, 0x5414ad, +0x6ae290, 0x89d988, 0x50722c, 0xbea404, 0x940777, 0x7030f3, +0x27fc00, 0xa871ea, 0x49c266, 0x3de064, 0x83dd97, 0x973fa3, +0xfd9443, 0x8c860d, 0xde4131, 0x9d3992, 0x8c70dd, 0xe7b717, +0x3bdf08, 0x2b3715, 0xa0805c, 0x93805a, 0x921110, 0xd8e80f, +0xaf806c, 0x4bffdb, 0x0f9038, 0x761859, 0x15a562, 0xbbcb61, +0xb989c7, 0xbd4010, 0x04f2d2, 0x277549, 0xf6b6eb, 0xbb22db, +0xaa140a, 0x2f2689, 0x768364, 0x333b09, 0x1a940e, 0xaa3a51, +0xc2a31d, 0xaeedaf, 0x12265c, 0x4dc26d, 0x9c7a2d, 0x9756c0, +0x833f03, 0xf6f009, 0x8c402b, 0x99316d, 0x07b439, 0x15200c, +0x5bc3d8, 0xc492f5, 0x4badc6, 0xa5ca4e, 0xcd37a7, 0x36a9e6, +0x9492ab, 0x6842dd, 0xde6319, 0xef8c76, 0x528b68, 0x37dbfc, +0xaba1ae, 0x3115df, 0xa1ae00, 0xdafb0c, 0x664d64, 0xb705ed, +0x306529, 0xbf5657, 0x3aff47, 0xb9f96a, 0xf3be75, 0xdf9328, +0x3080ab, 0xf68c66, 0x15cb04, 0x0622fa, 0x1de4d9, 0xa4b33d, +0x8f1b57, 0x09cd36, 0xe9424e, 0xa4be13, 0xb52333, 0x1aaaf0, +0xa8654f, 0xa5c1d2, 0x0f3f0b, 0xcd785b, 0x76f923, 0x048b7b, +0x721789, 0x53a6c6, 0xe26e6f, 0x00ebef, 0x584a9b, 0xb7dac4, +0xba66aa, 0xcfcf76, 0x1d02d1, 0x2df1b1, 0xc1998c, 0x77adc3, +0xda4886, 0xa05df7, 0xf480c6, 0x2ff0ac, 0x9aecdd, 0xbc5c3f, +0x6dded0, 0x1fc790, 0xb6db2a, 0x3a25a3, 0x9aaf00, 0x9353ad, +0x0457b6, 0xb42d29, 0x7e804b, 0xa707da, 0x0eaa76, 0xa1597b, +0x2a1216, 0x2db7dc, 0xfde5fa, 0xfedb89, 0xfdbe89, 0x6c76e4, +0xfca906, 0x70803e, 0x156e85, 0xff87fd, 0x073e28, 0x336761, +0x86182a, 0xeabd4d, 0xafe7b3, 0x6e6d8f, 0x396795, 0x5bbf31, +0x48d784, 0x16df30, 0x432dc7, 0x356125, 0xce70c9, 0xb8cb30, +0xfd6cbf, 0xa200a4, 0xe46c05, 0xa0dd5a, 0x476f21, 0xd21262, +0x845cb9, 0x496170, 0xe0566b, 0x015299, 0x375550, 0xb7d51e, +0xc4f133, 0x5f6e13, 0xe4305d, 0xa92e85, 0xc3b21d, 0x3632a1, +0xa4b708, 0xd4b1ea, 0x21f716, 0xe4698f, 0x77ff27, 0x80030c, +0x2d408d, 0xa0cd4f, 0x99a520, 0xd3a2b3, 0x0a5d2f, 0x42f9b4, +0xcbda11, 0xd0be7d, 0xc1db9b, 0xbd17ab, 0x81a2ca, 0x5c6a08, +0x17552e, 0x550027, 0xf0147f, 0x8607e1, 0x640b14, 0x8d4196, +0xdebe87, 0x2afdda, 0xb6256b, 0x34897b, 0xfef305, 0x9ebfb9, +0x4f6a68, 0xa82a4a, 0x5ac44f, 0xbcf82d, 0x985ad7, 0x95c7f4, +0x8d4d0d, 0xa63a20, 0x5f57a4, 0xb13f14, 0x953880, 0x0120cc, +0x86dd71, 0xb6dec9, 0xf560bf, 0x11654d, 0x6b0701, 0xacb08c, +0xd0c0b2, 0x485551, 0x0efb1e, 0xc37295, 0x3b06a3, 0x3540c0, +0x7bdc06, 0xcc45e0, 0xfa294e, 0xc8cad6, 0x41f3e8, 0xde647c, +0xd8649b, 0x31bed9, 0xc397a4, 0xd45877, 0xc5e369, 0x13daf0, +0x3c3aba, 0x461846, 0x5f7555, 0xf5bdd2, 0xc6926e, 0x5d2eac, +0xed440e, 0x423e1c, 0x87c461, 0xe9fd29, 0xf3d6e7, 0xca7c22, +0x35916f, 0xc5e008, 0x8dd7ff, 0xe26a6e, 0xc6fdb0, 0xc10893, +0x745d7c, 0xb2ad6b, 0x9d6ecd, 0x7b723e, 0x6a11c6, 0xa9cff7, +0xdf7329, 0xbac9b5, 0x5100b7, 0x0db2e2, 0x24ba74, 0x607de5, +0x8ad874, 0x2c150d, 0x0c1881, 0x94667e, 0x162901, 0x767a9f, +0xbefdfd, 0xef4556, 0x367ed9, 0x13d9ec, 0xb9ba8b, 0xfc97c4, +0x27a831, 0xc36ef1, 0x36c594, 0x56a8d8, 0xb5a8b4, 0x0ecccf, +0x2d8912, 0x34576f, 0x89562c, 0xe3ce99, 0xb920d6, 0xaa5e6b, +0x9c2a3e, 0xcc5f11, 0x4a0bfd, 0xfbf4e1, 0x6d3b8e, 0x2c86e2, +0x84d4e9, 0xa9b4fc, 0xd1eeef, 0xc9352e, 0x61392f, 0x442138, +0xc8d91b, 0x0afc81, 0x6a4afb, 0xd81c2f, 0x84b453, 0x8c994e, +0xcc2254, 0xdc552a, 0xd6c6c0, 0x96190b, 0xb8701a, 0x649569, +0x605a26, 0xee523f, 0x0f117f, 0x11b5f4, 0xf5cbfc, 0x2dbc34, +0xeebc34, 0xcc5de8, 0x605edd, 0x9b8e67, 0xef3392, 0xb817c9, +0x9b5861, 0xbc57e1, 0xc68351, 0x103ed8, 0x4871dd, 0xdd1c2d, +0xa118af, 0x462c21, 0xd7f359, 0x987ad9, 0xc0549e, 0xfa864f, +0xfc0656, 0xae79e5, 0x362289, 0x22ad38, 0xdc9367, 0xaae855, +0x382682, 0x9be7ca, 0xa40d51, 0xb13399, 0x0ed7a9, 0x480569, +0xf0b265, 0xa7887f, 0x974c88, 0x36d1f9, 0xb39221, 0x4a827b, +0x21cf98, 0xdc9f40, 0x5547dc, 0x3a74e1, 0x42eb67, 0xdf9dfe, +0x5fd45e, 0xa4677b, 0x7aacba, 0xa2f655, 0x23882b, 0x55ba41, +0x086e59, 0x862a21, 0x834739, 0xe6e389, 0xd49ee5, 0x40fb49, +0xe956ff, 0xca0f1c, 0x8a59c5, 0x2bfa94, 0xc5c1d3, 0xcfc50f, +0xae5adb, 0x86c547, 0x624385, 0x3b8621, 0x94792c, 0x876110, +0x7b4c2a, 0x1a2c80, 0x12bf43, 0x902688, 0x893c78, 0xe4c4a8, +0x7bdbe5, 0xc23ac4, 0xeaf426, 0x8a67f7, 0xbf920d, 0x2ba365, +0xb1933d, 0x0b7cbd, 0xdc51a4, 0x63dd27, 0xdde169, 0x19949a, +0x9529a8, 0x28ce68, 0xb4ed09, 0x209f44, 0xca984e, 0x638270, +0x237c7e, 0x32b90f, 0x8ef5a7, 0xe75614, 0x08f121, 0x2a9db5, +0x4d7e6f, 0x5119a5, 0xabf9b5, 0xd6df82, 0x61dd96, 0x023616, +0x9f3ac4, 0xa1a283, 0x6ded72, 0x7a8d39, 0xa9b882, 0x5c326b, +0x5b2746, 0xed3400, 0x7700d2, 0x55f4fc, 0x4d5901, 0x8071e0, +0xe13f89, 0xb295f3, 0x64a8f1, 0xaea74b, 0x38fc4c, 0xeab2bb, +0x47270b, 0xabc3a7, 0x34ba60, 0x52dd34, 0xf8563a, 0xeb7e8a, +0x31bb36, 0x5895b7, 0x47f7a9, 0x94c3aa, 0xd39225, 0x1e7f3e, +0xd8974e, 0xbba94f, 0xd8ae01, 0xe661b4, 0x393d8e, 0xa523aa, +0x33068e, 0x1633b5, 0x3bb188, 0x1d3a9d, 0x4013d0, 0xcc1be5, +0xf862e7, 0x3bf28f, 0x39b5bf, 0x0bc235, 0x22747e, 0xa247c0, +0xd52d1f, 0x19add3, 0x9094df, 0x9311d0, 0xb42b25, 0x496db2, +0xe264b2, 0x5ef135, 0x3bc6a4, 0x1a4ad0, 0xaac92e, 0x64e886, +0x573091, 0x982cfb, 0x311b1a, 0x08728b, 0xbdcee1, 0x60e142, +0xeb641d, 0xd0bba3, 0xe559d4, 0x597b8c, 0x2a4483, 0xf332ba, +0xf84867, 0x2c8d1b, 0x2fa9b0, 0x50f3dd, 0xf9f573, 0xdb61b4, +0xfe233e, 0x6c41a6, 0xeea318, 0x775a26, 0xbc5e5c, 0xcea708, +0x94dc57, 0xe20196, 0xf1e839, 0xbe4851, 0x5d2d2f, 0x4e9555, +0xd96ec2, 0xe7d755, 0x6304e0, 0xc02e0e, 0xfc40a0, 0xbbf9b3, +0x7125a7, 0x222dfb, 0xf619d8, 0x838c1c, 0x6619e6, 0xb20d55, +0xbb5137, 0x79e809, 0xaf9149, 0x0d73de, 0x0b0da5, 0xce7f58, +0xac1934, 0x724667, 0x7a1a13, 0x9e26bc, 0x4555e7, 0x585cb5, +0x711d14, 0x486991, 0x480d60, 0x56adab, 0xd62f64, 0x96ee0c, +0x212ff3, 0x5d6d88, 0xa67684, 0x95651e, 0xab9e0a, 0x4ddefe, +0x571010, 0x836a39, 0xf8ea31, 0x9e381d, 0xeac8b1, 0xcac96b, +0x37f21e, 0xd505e9, 0x984743, 0x9fc56c, 0x0331b7, 0x3b8bf8, +0x86e56a, 0x8dc343, 0x6230e7, 0x93cfd5, 0x6a8f2d, 0x733005, +0x1af021, 0xa09fcb, 0x7415a1, 0xd56b23, 0x6ff725, 0x2f4bc7, +0xb8a591, 0x7fac59, 0x5c55de, 0x212c38, 0xb13296, 0x5cff50, +0x366262, 0xfa7b16, 0xf4d9a6, 0x2acfe7, 0xf07403, 0xd4d604, +0x6fd916, 0x31b1bf, 0xcbb450, 0x5bd7c8, 0x0ce194, 0x6bd643, +0x4fd91c, 0xdf4543, 0x5f3453, 0xe2b5aa, 0xc9aec8, 0x131485, +0xf9d2bf, 0xbadb9e, 0x76f5b9, 0xaf15cf, 0xca3182, 0x14b56d, +0xe9fe4d, 0x50fc35, 0xf5aed5, 0xa2d0c1, 0xc96057, 0x192eb6, +0xe91d92, 0x07d144, 0xaea3c6, 0x343566, 0x26d5b4, 0x3161e2, +0x37f1a2, 0x209eff, 0x958e23, 0x493798, 0x35f4a6, 0x4bdc02, +0xc2be13, 0xbe80a0, 0x0b72a3, 0x115c5f, 0x1e1bd1, 0x0db4d3, +0x869e85, 0x96976b, 0x2ac91f, 0x8a26c2, 0x3070f0, 0x041412, +0xfc9fa5, 0xf72a38, 0x9c6878, 0xe2aa76, 0x50cfe1, 0x559274, +0x934e38, 0x0a92f7, 0x5533f0, 0xa63db4, 0x399971, 0xe2b755, +0xa98a7c, 0x008f19, 0xac54d2, 0x2ea0b4, 0xf5f3e0, 0x60c849, +0xffd269, 0xae52ce, 0x7a5fdd, 0xe9ce06, 0xfb0ae8, 0xa50cce, +0xea9d3e, 0x3766dd, 0xb834f5, 0x0da090, 0x846f88, 0x4ae3d5, +0x099a03, 0x2eae2d, 0xfcb40a, 0xfb9b33, 0xe281dd, 0x1b16ba, +0xd8c0af, 0xd96b97, 0xb52dc9, 0x9c277f, 0x5951d5, 0x21ccd6, +0xb6496b, 0x584562, 0xb3baf2, 0xa1a5c4, 0x7ca2cf, 0xa9b93d, +0x7b7b89, 0x483d38, +}; + +static const __float128 c[] = { +/* 93 bits of pi/2 */ +#define PI_2_1 c[0] + 1.57079632679489661923132169155131424e+00Q, /* 3fff921fb54442d18469898cc5100000 */ + +/* pi/2 - PI_2_1 */ +#define PI_2_1t c[1] + 8.84372056613570112025531863263659260e-29Q, /* 3fa1c06e0e68948127044533e63a0106 */ +}; + + +int32_t +__quadmath_rem_pio2q (__float128 x, __float128 *y) +{ + __float128 z, w, t; + double tx[8]; + int64_t exp, n, ix, hx; + uint64_t lx; + + GET_FLT128_WORDS64 (hx, lx, x); + ix = hx & 0x7fffffffffffffffLL; + if (ix <= 0x3ffe921fb54442d1LL) /* x in <-pi/4, pi/4> */ + { + y[0] = x; + y[1] = 0; + return 0; + } + + if (ix < 0x40002d97c7f3321dLL) /* |x| in <pi/4, 3pi/4) */ + { + if (hx > 0) + { + /* 113 + 93 bit PI is ok */ + z = x - PI_2_1; + y[0] = z - PI_2_1t; + y[1] = (z - y[0]) - PI_2_1t; + return 1; + } + else + { + /* 113 + 93 bit PI is ok */ + z = x + PI_2_1; + y[0] = z + PI_2_1t; + y[1] = (z - y[0]) + PI_2_1t; + return -1; + } + } + + if (ix >= 0x7fff000000000000LL) /* x is +=oo or NaN */ + { + y[0] = x - x; + y[1] = y[0]; + return 0; + } + + /* Handle large arguments. + We split the 113 bits of the mantissa into 5 24bit integers + stored in a double array. */ + exp = (ix >> 48) - 16383 - 23; + + /* This is faster than doing this in floating point, because we + have to convert it to integers anyway and like this we can keep + both integer and floating point units busy. */ + tx [0] = (double)(((ix >> 25) & 0x7fffff) | 0x800000); + tx [1] = (double)((ix >> 1) & 0xffffff); + tx [2] = (double)(((ix << 23) | (lx >> 41)) & 0xffffff); + tx [3] = (double)((lx >> 17) & 0xffffff); + tx [4] = (double)((lx << 7) & 0xffffff); + + n = __quadmath_kernel_rem_pio2 (tx, tx + 5, exp, + ((lx << 7) & 0xffffff) ? 5 : 4, + 3, two_over_pi); + + /* The result is now stored in 3 double values, we need to convert it into + two __float128 values. */ + t = (__float128) tx [6] + (__float128) tx [7]; + w = (__float128) tx [5]; + + if (hx >= 0) + { + y[0] = w + t; + y[1] = t - (y[0] - w); + return n; + } + else + { + y[0] = -(w + t); + y[1] = -t - (y[0] + w); + return -n; + } +} diff --git a/libquadmath/math/remainderq.c b/libquadmath/math/remainderq.c new file mode 100644 index 000000000..421f728ff --- /dev/null +++ b/libquadmath/math/remainderq.c @@ -0,0 +1,67 @@ +/* e_fmodl.c -- long double version of e_fmod.c. + * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz. + */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* remainderq(x,p) + * Return : + * returns x REM p = x - [x/p]*p as if in infinite + * precise arithmetic, where [x/p] is the (infinite bit) + * integer nearest x/p (in half way case choose the even one). + * Method : + * Based on fmodq() return x-[x/p]chopped*p exactlp. + */ + +#include "quadmath-imp.h" + +static const __float128 zero = 0.0Q; + +__float128 +remainderq (__float128 x, __float128 p) +{ + int64_t hx,hp; + uint64_t sx,lx,lp; + __float128 p_half; + + GET_FLT128_WORDS64(hx,lx,x); + GET_FLT128_WORDS64(hp,lp,p); + sx = hx&0x8000000000000000ULL; + hp &= 0x7fffffffffffffffLL; + hx &= 0x7fffffffffffffffLL; + + /* purge off exception values */ + if((hp|lp)==0) return (x*p)/(x*p); /* p = 0 */ + if((hx>=0x7fff000000000000LL)|| /* x not finite */ + ((hp>=0x7fff000000000000LL)&& /* p is NaN */ + (((hp-0x7fff000000000000LL)|lp)!=0))) + return (x*p)/(x*p); + + if (hp<=0x7ffdffffffffffffLL) x = fmodq (x,p+p); /* now x < 2p */ + if (((hx-hp)|(lx-lp))==0) return zero*x; + x = fabsq(x); + p = fabsq(p); + if (hp<0x0002000000000000LL) { + if(x+x>p) { + x-=p; + if(x+x>=p) x -= p; + } + } else { + p_half = 0.5Q*p; + if(x>p_half) { + x-=p; + if(x>=p_half) x -= p; + } + } + GET_FLT128_MSW64(hx,x); + SET_FLT128_MSW64(x,hx^sx); + return x; +} diff --git a/libquadmath/math/remquoq.c b/libquadmath/math/remquoq.c new file mode 100644 index 000000000..f7001afc3 --- /dev/null +++ b/libquadmath/math/remquoq.c @@ -0,0 +1,107 @@ +/* Compute remainder and a congruent to the quotient. + Copyright (C) 1997, 1999, 2002 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997 and + Jakub Jelinek <jj@ultra.linux.cz>, 1999. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + + +static const __float128 zero = 0.0; + + +__float128 +remquoq (__float128 x, __float128 y, int *quo) +{ + int64_t hx,hy; + uint64_t sx,lx,ly,qs; + int cquo; + + GET_FLT128_WORDS64 (hx, lx, x); + GET_FLT128_WORDS64 (hy, ly, y); + sx = hx & 0x8000000000000000ULL; + qs = sx ^ (hy & 0x8000000000000000ULL); + hy &= 0x7fffffffffffffffLL; + hx &= 0x7fffffffffffffffLL; + + /* Purge off exception values. */ + if ((hy | ly) == 0) + return (x * y) / (x * y); /* y = 0 */ + if ((hx >= 0x7fff000000000000LL) /* x not finite */ + || ((hy >= 0x7fff000000000000LL) /* y is NaN */ + && (((hy - 0x7fff000000000000LL) | ly) != 0))) + return (x * y) / (x * y); + + if (hy <= 0x7ffbffffffffffffLL) + x = fmodq (x, 8 * y); /* now x < 8y */ + + if (((hx - hy) | (lx - ly)) == 0) + { + *quo = qs ? -1 : 1; + return zero * x; + } + + x = fabsq (x); + y = fabsq (y); + cquo = 0; + + if (x >= 4 * y) + { + x -= 4 * y; + cquo += 4; + } + if (x >= 2 * y) + { + x -= 2 * y; + cquo += 2; + } + + if (hy < 0x0002000000000000LL) + { + if (x + x > y) + { + x -= y; + ++cquo; + if (x + x >= y) + { + x -= y; + ++cquo; + } + } + } + else + { + __float128 y_half = 0.5Q * y; + if (x > y_half) + { + x -= y; + ++cquo; + if (x >= y_half) + { + x -= y; + ++cquo; + } + } + } + + *quo = qs ? -cquo : cquo; + + if (sx) + x = -x; + return x; +} diff --git a/libquadmath/math/rintq.c b/libquadmath/math/rintq.c new file mode 100644 index 000000000..fe7a59188 --- /dev/null +++ b/libquadmath/math/rintq.c @@ -0,0 +1,66 @@ +/* s_rintl.c -- long double version of s_rint.c. + * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include "quadmath-imp.h" + +static const __float128 +TWO112[2]={ + 5.19229685853482762853049632922009600E+33L, /* 0x406F000000000000, 0 */ + -5.19229685853482762853049632922009600E+33L /* 0xC06F000000000000, 0 */ +}; + +__float128 +rintq (__float128 x) +{ + int64_t i0,j0,sx; + uint64_t i,i1; + __float128 w,t; + GET_FLT128_WORDS64(i0,i1,x); + sx = (((uint64_t)i0)>>63); + j0 = ((i0>>48)&0x7fff)-0x3fff; + if(j0<48) { + if(j0<0) { + if(((i0&0x7fffffffffffffffLL)|i1)==0) return x; + i1 |= (i0&0x0000ffffffffffffLL); + i0 &= 0xffffe00000000000ULL; + i0 |= ((i1|-i1)>>16)&0x0000800000000000LL; + SET_FLT128_MSW64(x,i0); + w = TWO112[sx]+x; + t = w-TWO112[sx]; + GET_FLT128_MSW64(i0,t); + SET_FLT128_MSW64(t,(i0&0x7fffffffffffffffLL)|(sx<<63)); + return t; + } else { + i = (0x0000ffffffffffffLL)>>j0; + if(((i0&i)|i1)==0) return x; /* x is integral */ + i>>=1; + if(((i0&i)|i1)!=0) { + if(j0==47) i1 = 0x4000000000000000ULL; else + i0 = (i0&(~i))|((0x0000200000000000LL)>>j0); + } + } + } else if (j0>111) { + if(j0==0x4000) return x+x; /* inf or NaN */ + else return x; /* x is integral */ + } else { + i = -1ULL>>(j0-48); + if((i1&i)==0) return x; /* x is integral */ + i>>=1; + if((i1&i)!=0) i1 = (i1&(~i))|((0x4000000000000000LL)>>(j0-48)); + } + SET_FLT128_WORDS64(x,i0,i1); + w = TWO112[sx]+x; + return w-TWO112[sx]; +} diff --git a/libquadmath/math/roundq.c b/libquadmath/math/roundq.c new file mode 100644 index 000000000..d7f577bd9 --- /dev/null +++ b/libquadmath/math/roundq.c @@ -0,0 +1,90 @@ +/* Round long double to integer away from zero. + Copyright (C) 1997, 1999 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997 and + Jakub Jelinek <jj@ultra.linux.cz>, 1999. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + +static const __float128 huge = 1.0E4930Q; + + +__float128 +roundq (__float128 x) +{ + int32_t j0; + uint64_t i1, i0; + + GET_FLT128_WORDS64 (i0, i1, x); + j0 = ((i0 >> 48) & 0x7fff) - 0x3fff; + if (j0 < 31) + { + if (j0 < 0) + { + if (huge + x > 0.0) + { + i0 &= 0x8000000000000000ULL; + if (j0 == -1) + i0 |= 0x3fff000000000000LL; + i1 = 0; + } + } + else + { + uint64_t i = 0x0000ffffffffffffLL >> j0; + if (((i0 & i) | i1) == 0) + /* X is integral. */ + return x; + if (huge + x > 0.0) + { + /* Raise inexact if x != 0. */ + i0 += 0x0000800000000000LL >> j0; + i0 &= ~i; + i1 = 0; + } + } + } + else if (j0 > 111) + { + if (j0 == 0x4000) + /* Inf or NaN. */ + return x + x; + else + return x; + } + else + { + uint64_t i = -1ULL >> (j0 - 48); + if ((i1 & i) == 0) + /* X is integral. */ + return x; + + if (huge + x > 0.0) + { + /* Raise inexact if x != 0. */ + uint64_t j = i1 + (1LL << (111 - j0)); + if (j < i1) + i0 += 1; + i1 = j; + } + i1 &= ~i; + } + + SET_FLT128_WORDS64 (x, i0, i1); + return x; +} diff --git a/libquadmath/math/scalblnq.c b/libquadmath/math/scalblnq.c new file mode 100644 index 000000000..75997f688 --- /dev/null +++ b/libquadmath/math/scalblnq.c @@ -0,0 +1,48 @@ +/* s_scalblnl.c -- long double version of s_scalbn.c. + * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include "quadmath-imp.h" + +static const __float128 +two114 = 2.0769187434139310514121985316880384E+34Q, /* 0x4071000000000000, 0 */ +twom114 = 4.8148248609680896326399448564623183E-35Q, /* 0x3F8D000000000000, 0 */ +huge = 1.0E+4900Q, +tiny = 1.0E-4900Q; + +__float128 +scalblnq (__float128 x, long int n) +{ + int64_t k,hx,lx; + GET_FLT128_WORDS64(hx,lx,x); + k = (hx>>48)&0x7fff; /* extract exponent */ + if (k==0) { /* 0 or subnormal x */ + if ((lx|(hx&0x7fffffffffffffffULL))==0) return x; /* +-0 */ + x *= two114; + GET_FLT128_MSW64(hx,x); + k = ((hx>>48)&0x7fff) - 114; + } + if (k==0x7fff) return x+x; /* NaN or Inf */ + k = k+n; + if (n> 50000 || k > 0x7ffe) + return huge*copysignq(huge,x); /* overflow */ + if (n< -50000) return tiny*copysignq(tiny,x); /*underflow*/ + if (k > 0) /* normal result */ + {SET_FLT128_MSW64(x,(hx&0x8000ffffffffffffULL)|(k<<48)); return x;} + if (k <= -114) + return tiny*copysignq(tiny,x); /*underflow*/ + k += 114; /* subnormal result */ + SET_FLT128_MSW64(x,(hx&0x8000ffffffffffffULL)|(k<<48)); + return x*twom114; +} diff --git a/libquadmath/math/scalbnq.c b/libquadmath/math/scalbnq.c new file mode 100644 index 000000000..b7049a70e --- /dev/null +++ b/libquadmath/math/scalbnq.c @@ -0,0 +1,48 @@ +/* s_scalbnl.c -- long double version of s_scalbn.c. + * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include "quadmath-imp.h" + +static const __float128 +two114 = 2.0769187434139310514121985316880384E+34Q, /* 0x4071000000000000, 0 */ +twom114 = 4.8148248609680896326399448564623183E-35Q, /* 0x3F8D000000000000, 0 */ +huge = 1.0E+4900Q, +tiny = 1.0E-4900Q; + +__float128 +scalbnq (__float128 x, int n) +{ + int64_t k,hx,lx; + GET_FLT128_WORDS64(hx,lx,x); + k = (hx>>48)&0x7fff; /* extract exponent */ + if (k==0) { /* 0 or subnormal x */ + if ((lx|(hx&0x7fffffffffffffffULL))==0) return x; /* +-0 */ + x *= two114; + GET_FLT128_MSW64(hx,x); + k = ((hx>>48)&0x7fff) - 114; + } + if (k==0x7fff) return x+x; /* NaN or Inf */ + k = k+n; + if (n> 50000 || k > 0x7ffe) + return huge*copysignq(huge,x); /* overflow */ + if (n< -50000) return tiny*copysignq(tiny,x); /*underflow*/ + if (k > 0) /* normal result */ + {SET_FLT128_MSW64(x,(hx&0x8000ffffffffffffULL)|(k<<48)); return x;} + if (k <= -114) + return tiny*copysignq(tiny,x); /*underflow*/ + k += 114; /* subnormal result */ + SET_FLT128_MSW64(x,(hx&0x8000ffffffffffffULL)|(k<<48)); + return x*twom114; +} diff --git a/libquadmath/math/signbitq.c b/libquadmath/math/signbitq.c new file mode 100644 index 000000000..2ddab8667 --- /dev/null +++ b/libquadmath/math/signbitq.c @@ -0,0 +1,10 @@ +#include "quadmath-imp.h" + + +int +signbitq (const __float128 x) +{ + ieee854_float128 f; + f.value = x; + return f.ieee.negative; +} diff --git a/libquadmath/math/sincos_table.c b/libquadmath/math/sincos_table.c new file mode 100644 index 000000000..b7e7c7503 --- /dev/null +++ b/libquadmath/math/sincos_table.c @@ -0,0 +1,696 @@ +/* Quad-precision floating point sine and cosine tables. + Copyright (C) 1999 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Jakub Jelinek <jj@ultra.linux.cz> + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +/* For 0.1484375 + n/128.0, n=0..82 this table contains + first 113 bits of cosine, then at least 113 additional + bits and the same for sine. + 0.1484375+82.0/128.0 is the smallest number among above defined numbers + larger than pi/4. + Computed using gmp. + */ + +#include "quadmath-imp.h" + +const __float128 __sincosq_table[] = { + +/* x = 1.48437500000000000000000000000000000e-01Q 3ffc3000000000000000000000000000 */ +/* cos(x) = 0.fd2f5320e1b790209b4dda2f98f79caaa7b873aff1014b0fbc5243766d03cb006bc837c4358 */ + 9.89003367927322909016887196069562069e-01Q, /* 3ffefa5ea641c36f2041369bb45f31ef */ + 2.15663692029265697782289400027743703e-35Q, /* 3f8bcaaa7b873aff1014b0fbc5243767 */ +/* sin(x) = 0.25dc50bc95711d0d9787d108fd438cf5959ee0bfb7a1e36e8b1a112968f356657420e9cc9ea */ + 1.47892995873409608580026675734609314e-01Q, /* 3ffc2ee285e4ab88e86cbc3e8847ea1c */ + 9.74950446464233268291647449768590886e-36Q, /* 3f8a9eb2b3dc17f6f43c6dd16342252d */ + +/* x = 1.56250000000000000000000000000000000e-01 3ffc4000000000000000000000000000 */ +/* cos(x) = 0.fce1a053e621438b6d60c76e8c45bf0a9dc71aa16f922acc10e95144ec796a249813c9cb649 */ + 9.87817783816471944100503034363211317e-01Q, /* 3ffef9c340a7cc428716dac18edd188b */ + 4.74271307836705897892468107620526395e-35Q, /* 3f8cf854ee38d50b7c915660874a8a27 */ +/* sin(x) = 0.27d66258bacd96a3eb335b365c87d59438c5142bb56a489e9b8db9d36234ffdebb6bdc22d8e */ + 1.55614992773556041209920643203516258e-01Q, /* 3ffc3eb312c5d66cb51f599ad9b2e43f */ +-7.83989563419287980121718050629497270e-36Q, /* bf8a4d78e75d7a8952b6ec2c8e48c594 */ + +/* x = 1.64062500000000000000000000000000000e-01 3ffc5000000000000000000000000000 */ +/* cos(x) = 0.fc8ffa01ba6807417e05962b0d9fdf1fddb0cc4c07d22e19e08019bffa50a6c7acdb40307a3 */ + 9.86571908399497588757337407495308409e-01Q, /* 3ffef91ff40374d00e82fc0b2c561b40 */ +-2.47327949936985362476252401212720725e-35Q, /* bf8c070112799d9fc16e8f30fbff3200 */ +/* sin(x) = 0.29cfd49b8be4f665276cab01cbf0426934906c3dd105473b226e410b1450f62e53ff7c6cce1 */ + 1.63327491736612850846866172454354370e-01Q, /* 3ffc4e7ea4dc5f27b3293b65580e5f82 */ + 1.81380344301155485770367902300754350e-36Q, /* 3f88349a48361ee882a39d913720858a */ + +/* x = 1.71875000000000000000000000000000000e-01 3ffc6000000000000000000000000000 */ +/* cos(x) = 0.fc3a6170f767ac735d63d99a9d439e1db5e59d3ef153a4265d5855850ed82b536bf361b80e3 */ + 9.85265817718213816204294709759578994e-01Q, /* 3ffef874c2e1eecf58e6bac7b3353a87 */ + 2.26568029505818066141517497778527952e-35Q, /* 3f8be1db5e59d3ef153a4265d5855851 */ +/* sin(x) = 0.2bc89f9f424de5485de7ce03b2514952b9faf5648c3244d4736feb95dbb9da49f3b58a9253b */ + 1.71030022031395019281347969239834331e-01Q, /* 3ffc5e44fcfa126f2a42ef3e701d928a */ + 7.01395875187487608875416030203241317e-36Q, /* 3f8a2a573f5eac9186489a8e6dfd72bb */ + +/* x = 1.79687500000000000000000000000000000e-01 3ffc7000000000000000000000000000 */ +/* cos(x) = 0.fbe0d7f7fef11e70aa43b8abf4f6a457cea20c8f3f676b47781f9821bbe9ce04b3c7b981c0b */ + 9.83899591489663972178309351416487245e-01Q, /* 3ffef7c1afeffde23ce154877157e9ed */ + 2.73414318948066207810486330723761265e-35Q, /* 3f8c22be75106479fb3b5a3bc0fcc10e */ +/* sin(x) = 0.2dc0bb80b49a97ffb34e8dd1f8db9df7af47ed2dcf58b12c8e7827e048cae929da02c04ecac */ + 1.78722113535153659375356241864180724e-01Q, /* 3ffc6e05dc05a4d4bffd9a746e8fc6dd */ +-1.52906926517265103202547561260594148e-36Q, /* bf8804285c09691853a769b8c3ec0fdc */ + +/* x = 1.87500000000000000000000000000000000e-01 3ffc8000000000000000000000000000 */ +/* cos(x) = 0.fb835efcf670dd2ce6fe7924697eea13ea358867e9cdb3899b783f4f9f43aa5626e8b67b3bc */ + 9.82473313101255257487327683243622495e-01Q, /* 3ffef706bdf9ece1ba59cdfcf248d2fe */ +-1.64924358891557584625463868014230342e-35Q, /* bf8b5ec15ca779816324c766487c0b06 */ +/* sin(x) = 0.2fb8205f75e56a2b56a1c4792f856258769af396e0189ef72c05e4df59a6b00e4b44a6ea515 */ + 1.86403296762269884552379983103205261e-01Q, /* 3ffc7dc102fbaf2b515ab50e23c97c2b */ + 1.76460304806826780010586715975331753e-36Q, /* 3f882c3b4d79cb700c4f7b9602f26fad */ + +/* x = 1.95312500000000000000000000000000000e-01 3ffc9000000000000000000000000000 */ +/* cos(x) = 0.fb21f7f5c156696b00ac1fe28ac5fd76674a92b4df80d9c8a46c684399005deccc41386257c */ + 9.80987069605669190469329896435309665e-01Q, /* 3ffef643efeb82acd2d601583fc5158c */ +-1.90899259410096419886996331536278461e-36Q, /* bf8844cc5ab6a5903f931badc9cbde34 */ +/* sin(x) = 0.31aec65df552876f82ece9a2356713246eba6799983d7011b0b3698d6e1da919c15d57c30c1 */ + 1.94073102892909791156055200214145404e-01Q, /* 3ffc8d7632efaa943b7c17674d11ab39 */ +-9.67304741051998267208945242944928999e-36Q, /* bf8a9b7228b30cccf851fdc9e992ce52 */ + +/* x = 2.03125000000000000000000000000000000e-01 3ffca000000000000000000000000000 */ +/* cos(x) = 0.fabca467fb3cb8f1d069f01d8ea33ade5bfd68296ecd1cc9f7b7609bbcf3676e726c3301334 */ + 9.79440951715548359998530954502987493e-01Q, /* 3ffef57948cff67971e3a0d3e03b1d46 */ + 4.42878056591560757066844797290067990e-35Q, /* 3f8cd6f2dfeb414b7668e64fbdbb04de */ +/* sin(x) = 0.33a4a5a19d86246710f602c44df4fa513f4639ce938477aeeabb82e8e0a7ed583a188879fd4 */ + 2.01731063801638804725038151164000971e-01Q, /* 3ffc9d252d0cec31233887b016226fa8 */ +-4.27513434754966978435151290617384120e-36Q, /* bf896bb02e718c5b1ee21445511f45c8 */ + +/* x = 2.10937500000000000000000000000000000e-01 3ffcb000000000000000000000000000 */ +/* cos(x) = 0.fa5365e8f1d3ca27be1db5d76ae64d983d7470a4ab0f4ccf65a2b8c67a380df949953a09bc1 */ + 9.77835053797959793331971572944454549e-01Q, /* 3ffef4a6cbd1e3a7944f7c3b6baed5cd */ +-3.79207422905180416937210853779192702e-35Q, /* bf8c933e145c7adaa7859984d2ea39cc */ +/* sin(x) = 0.3599b652f40ec999df12a0a4c8561de159c98d4e54555de518b97f48886f715d8df5f4f093e */ + 2.09376712085993643711890752724881652e-01Q, /* 3ffcaccdb297a0764ccef895052642b1 */ +-1.59470287344329449965314638482515925e-36Q, /* bf880f531b3958d5d5510d73a3405bbc */ + +/* x = 2.18750000000000000000000000000000000e-01 3ffcc000000000000000000000000000 */ +/* cos(x) = 0.f9e63e1d9e8b6f6f2e296bae5b5ed9c11fd7fa2fe11e09fc7bde901abed24b6365e72f7db4e */ + 9.76169473868635276723989035435135534e-01Q, /* 3ffef3cc7c3b3d16dede5c52d75cb6be */ +-2.87727974249481583047944860626985460e-35Q, /* bf8c31f701402e80f70fb01c210b7f2a */ +/* sin(x) = 0.378df09db8c332ce0d2b53d865582e4526ea336c768f68c32b496c6d11c1cd241bb9f1da523 */ + 2.17009581095010156760578095826055396e-01Q, /* 3ffcbc6f84edc6199670695a9ec32ac1 */ + 1.07356488794216831812829549198201194e-35Q, /* 3f8ac8a4dd466d8ed1ed1865692d8da2 */ + +/* x = 2.26562500000000000000000000000000000e-01 3ffcd000000000000000000000000000 */ +/* cos(x) = 0.f9752eba9fff6b98842beadab054a932fb0f8d5b875ae63d6b2288d09b148921aeb6e52f61b */ + 9.74444313585988980349711056045434344e-01Q, /* 3ffef2ea5d753ffed7310857d5b560a9 */ + 3.09947905955053419304514538592548333e-35Q, /* 3f8c4997d87c6adc3ad731eb59144685 */ +/* sin(x) = 0.39814cb10513453cb97b21bc1ca6a337b150c21a675ab85503bc09a436a10ab1473934e20c8 */ + 2.24629204957705292350428549796424820e-01Q, /* 3ffccc0a6588289a29e5cbd90de0e535 */ + 2.42061510849297469844695751870058679e-36Q, /* 3f889bd8a8610d33ad5c2a81de04d21b */ + +/* x = 2.34375000000000000000000000000000000e-01 3ffce000000000000000000000000000 */ +/* cos(x) = 0.f90039843324f9b940416c1984b6cbed1fc733d97354d4265788a86150493ce657cae032674 */ + 9.72659678244912752670913058267565260e-01Q, /* 3ffef20073086649f3728082d833096e */ +-3.91759231819314904966076958560252735e-35Q, /* bf8ca09701c6613465595ecd43babcf5 */ +/* sin(x) = 0.3b73c2bf6b4b9f668ef9499c81f0d965087f1753fa64b086e58cb8470515c18c1412f8c2e02 */ + 2.32235118611511462413930877746235872e-01Q, /* 3ffcdb9e15fb5a5cfb3477ca4ce40f87 */ +-4.96930483364191020075024624332928910e-36Q, /* bf89a6bde03a2b0166d3de469cd1ee3f */ + +/* x = 2.42187500000000000000000000000000000e-01 3ffcf000000000000000000000000000 */ +/* cos(x) = 0.f887604e2c39dbb20e4ec5825059a789ffc95b275ad9954078ba8a28d3fcfe9cc2c1d49697b */ + 9.70815676770349462947490545785046027e-01Q, /* 3ffef10ec09c5873b7641c9d8b04a0b3 */ + 2.97458820972393859125277682021202860e-35Q, /* 3f8c3c4ffe4ad93ad6ccaa03c5d45147 */ +/* sin(x) = 0.3d654aff15cb457a0fca854698aba33039a8a40626609204472d9d40309b626eccc6dff0ffa */ + 2.39826857830661564441369251810886574e-01Q, /* 3ffceb2a57f8ae5a2bd07e542a34c55d */ + 2.39867036569896287240938444445071448e-36Q, /* 3f88981cd45203133049022396cea018 */ + +/* x = 2.50000000000000000000000000000000000e-01 3ffd0000000000000000000000000000 */ +/* cos(x) = 0.f80aa4fbef750ba783d33cb95f94f8a41426dbe79edc4a023ef9ec13c944551c0795b84fee1 */ + 9.68912421710644784144595449494189205e-01Q, /* 3ffef01549f7deea174f07a67972bf2a */ +-5.53634706113461989398873287749326500e-36Q, /* bf89d6faf649061848ed7f704184fb0e */ +/* sin(x) = 0.3f55dda9e62aed7513bd7b8e6a3d1635dd5676648d7db525898d7086af9330f03c7f285442a */ + 2.47403959254522929596848704849389203e-01Q, /* 3ffcfaaeed4f31576ba89debdc7351e9 */ +-7.36487001108599532943597115275811618e-36Q, /* bf8a39445531336e50495b4ece51ef2a */ + +/* x = 2.57812500000000000000000000000000000e-01 3ffd0800000000000000000000000000 */ +/* cos(x) = 0.f78a098069792daabc9ee42591b7c5a68cb1ab822aeb446b3311b4ba5371b8970e2c1547ad7 */ + 9.66950029230677822008341623610531503e-01Q, /* 3ffeef141300d2f25b55793dc84b2370 */ +-4.38972214432792412062088059990480514e-35Q, /* bf8cd2cb9a72a3eea8a5dca667725a2d */ +/* sin(x) = 0.414572fd94556e6473d620271388dd47c0ba050cdb5270112e3e370e8c4705ae006426fb5d5 */ + 2.54965960415878467487556574864872628e-01Q, /* 3ffd0515cbf65155b991cf58809c4e23 */ + 2.20280377918534721005071688328074154e-35Q, /* 3f8bd47c0ba050cdb5270112e3e370e9 */ + +/* x = 2.65625000000000000000000000000000000e-01 3ffd1000000000000000000000000000 */ +/* cos(x) = 0.f7058fde0788dfc805b8fe88789e4f4253e3c50afe8b22f41159620ab5940ff7df9557c0d1f */ + 9.64928619104771009581074665315748371e-01Q, /* 3ffeee0b1fbc0f11bf900b71fd10f13d */ +-3.66685832670820775002475545602761113e-35Q, /* bf8c85ed60e1d7a80ba6e85f7534efaa */ +/* sin(x) = 0.4334033bcd90d6604f5f36c1d4b84451a87150438275b77470b50e5b968fa7962b5ffb379b7 */ + 2.62512399769153281450949626395692931e-01Q, /* 3ffd0cd00cef364359813d7cdb0752e1 */ + 3.24923677072031064673177178571821843e-36Q, /* 3f89146a1c5410e09d6ddd1c2d4396e6 */ + +/* x = 2.73437500000000000000000000000000000e-01 3ffd1800000000000000000000000000 */ +/* cos(x) = 0.f67d3a26af7d07aa4bd6d42af8c0067fefb96d5b46c031eff53627f215ea3242edc3f2e13eb */ + 9.62848314709379699899701093480214365e-01Q, /* 3ffeecfa744d5efa0f5497ada855f180 */ + 4.88986966383343450799422013051821394e-36Q, /* 3f899ffbee5b56d1b00c7bfd4d89fc85 */ +/* sin(x) = 0.452186aa5377ab20bbf2524f52e3a06a969f47166ab88cf88c111ad12c55941021ef3317a1a */ + 2.70042816718585031552755063618827102e-01Q, /* 3ffd14861aa94ddeac82efc9493d4b8f */ +-2.37608892440611310321138680065803162e-35Q, /* bf8bf956960b8e99547730773eee52ed */ + +/* x = 2.81250000000000000000000000000000000e-01 3ffd2000000000000000000000000000 */ +/* cos(x) = 0.f5f10a7bb77d3dfa0c1da8b57842783280d01ce3c0f82bae3b9d623c168d2e7c29977994451 */ + 9.60709243015561903066659350581313472e-01Q, /* 3ffeebe214f76efa7bf4183b516af085 */ +-5.87011558231583960712013351601221840e-36Q, /* bf89f35fcbf8c70fc1f5147118a770fa */ +/* sin(x) = 0.470df5931ae1d946076fe0dcff47fe31bb2ede618ebc607821f8462b639e1f4298b5ae87fd3 */ + 2.77556751646336325922023446828128568e-01Q, /* 3ffd1c37d64c6b8765181dbf8373fd20 */ +-1.35848595468998128214344668770082997e-36Q, /* bf87ce44d1219e71439f87de07b9d49c */ + +/* x = 2.89062500000000000000000000000000000e-01 3ffd2800000000000000000000000000 */ +/* cos(x) = 0.f561030ddd7a78960ea9f4a32c6521554995667f5547bafee9ec48b3155cdb0f7fd00509713 */ + 9.58511534581228627301969408154919822e-01Q, /* 3ffeeac2061bbaf4f12c1d53e94658ca */ + 2.50770779371636481145735089393154404e-35Q, /* 3f8c0aaa4cab33faaa3dd7f74f624599 */ +/* sin(x) = 0.48f948446abcd6b0f7fccb100e7a1b26eccad880b0d24b59948c7cdd49514d44b933e6985c2 */ + 2.85053745940547424587763033323252561e-01Q, /* 3ffd23e52111aaf35ac3dff32c4039e8 */ + 2.04269325885902918802700123680403749e-35Q, /* 3f8bb26eccad880b0d24b59948c7cdd5 */ + +/* x = 2.96875000000000000000000000000000000e-01 3ffd3000000000000000000000000000 */ +/* cos(x) = 0.f4cd261d3e6c15bb369c8758630d2ac00b7ace2a51c0631bfeb39ed158ba924cc91e259c195 */ + 9.56255323543175296975599942263028361e-01Q, /* 3ffee99a4c3a7cd82b766d390eb0c61a */ + 3.21616572190865997051103645135837207e-35Q, /* 3f8c56005bd671528e0318dff59cf68b */ +/* sin(x) = 0.4ae37710fad27c8aa9c4cf96c03519b9ce07dc08a1471775499f05c29f86190aaebaeb9716e */ + 2.92533342023327543624702326493913423e-01Q, /* 3ffd2b8ddc43eb49f22aa7133e5b00d4 */ + 1.93539408668704450308003687950685128e-35Q, /* 3f8b9b9ce07dc08a1471775499f05c2a */ + +/* x = 3.04687500000000000000000000000000000e-01 3ffd3800000000000000000000000000 */ +/* cos(x) = 0.f43575f94d4f6b272f5fb76b14d2a64ab52df1ee8ddf7c651034e5b2889305a9ea9015d758a */ + 9.53940747608894733981324795987611623e-01Q, /* 3ffee86aebf29a9ed64e5ebf6ed629a5 */ + 2.88075689052478602008395972924657164e-35Q, /* 3f8c3255a96f8f746efbe32881a72d94 */ +/* sin(x) = 0.4ccc7a50127e1de0cb6b40c302c651f7bded4f9e7702b0471ae0288d091a37391950907202f */ + 2.99995083378683051163248282011699944e-01Q, /* 3ffd3331e94049f877832dad030c0b19 */ + 1.35174265535697850139283361475571050e-35Q, /* 3f8b1f7bded4f9e7702b0471ae0288d1 */ + +/* x = 3.12500000000000000000000000000000000e-01 3ffd4000000000000000000000000000 */ +/* cos(x) = 0.f399f500c9e9fd37ae9957263dab8877102beb569f101ee4495350868e5847d181d50d3cca2 */ + 9.51567948048172202145488217364270962e-01Q, /* 3ffee733ea0193d3fa6f5d32ae4c7b57 */ + 6.36842628598115658308749288799884606e-36Q, /* 3f8a0ee2057d6ad3e203dc892a6a10d2 */ +/* sin(x) = 0.4eb44a5da74f600207aaa090f0734e288603ffadb3eb2542a46977b105f8547128036dcf7f0 */ + 3.07438514580380850670502958201982091e-01Q, /* 3ffd3ad129769d3d80081eaa8243c1cd */ + 1.06515172423204645839241099453417152e-35Q, /* 3f8ac510c07ff5b67d64a8548d2ef621 */ + +/* x = 3.20312500000000000000000000000000000e-01 3ffd4800000000000000000000000000 */ +/* cos(x) = 0.f2faa5a1b74e82fd61fa05f9177380e8e69b7b15a945e8e5ae1124bf3d12b0617e03af4fab5 */ + 9.49137069684463027665847421762105623e-01Q, /* 3ffee5f54b436e9d05fac3f40bf22ee7 */ + 6.84433965991637152250309190468859701e-37Q, /* 3f86d1cd36f62b528bd1cb5c22497e7a */ +/* sin(x) = 0.509adf9a7b9a5a0f638a8fa3a60a199418859f18b37169a644fdb986c21ecb00133853bc35b */ + 3.14863181319745250865036315126939016e-01Q, /* 3ffd426b7e69ee69683d8e2a3e8e9828 */ + 1.92431240212432926993057705062834160e-35Q, /* 3f8b99418859f18b37169a644fdb986c */ + +/* x = 3.28125000000000000000000000000000000e-01 3ffd5000000000000000000000000000 */ +/* cos(x) = 0.f2578a595224dd2e6bfa2eb2f99cc674f5ea6f479eae2eb580186897ae3f893df1113ca06b8 */ + 9.46648260886053321846099507295532976e-01Q, /* 3ffee4af14b2a449ba5cd7f45d65f33a */ +-4.32906339663000890941529420498824645e-35Q, /* bf8ccc5850ac85c30a8e8a53ff3cbb43 */ +/* sin(x) = 0.5280326c3cf481823ba6bb08eac82c2093f2bce3c4eb4ee3dec7df41c92c8a4226098616075 */ + 3.22268630433386625687745919893188031e-01Q, /* 3ffd4a00c9b0f3d20608ee9aec23ab21 */ +-1.49505897804759263483853908335500228e-35Q, /* bf8b3df6c0d431c3b14b11c213820be3 */ + +/* x = 3.35937500000000000000000000000000000e-01 3ffd5800000000000000000000000000 */ +/* cos(x) = 0.f1b0a5b406b526d886c55feadc8d0dcc8eb9ae2ac707051771b48e05b25b000009660bdb3e3 */ + 9.44101673557004345630017691253124860e-01Q, /* 3ffee3614b680d6a4db10d8abfd5b91a */ + 1.03812535240120229609822461172145584e-35Q, /* 3f8ab991d735c558e0e0a2ee3691c0b6 */ +/* sin(x) = 0.54643b3da29de9b357155eef0f332fb3e66c83bf4dddd9491c5eb8e103ccd92d6175220ed51 */ + 3.29654409930860171914317725126463176e-01Q, /* 3ffd5190ecf68a77a6cd5c557bbc3ccd */ +-1.22606996784743214973082192294232854e-35Q, /* bf8b04c19937c40b22226b6e3a1471f0 */ + +/* x = 3.43750000000000000000000000000000000e-01 3ffd6000000000000000000000000000 */ +/* cos(x) = 0.f105fa4d66b607a67d44e042725204435142ac8ad54dfb0907a4f6b56b06d98ee60f19e557a */ + 9.41497463127881068644511236053670815e-01Q, /* 3ffee20bf49acd6c0f4cfa89c084e4a4 */ + 3.20709366603165602071590241054884900e-36Q, /* 3f8910d450ab22b5537ec241e93dad5b */ +/* sin(x) = 0.5646f27e8bd65cbe3a5d61ff06572290ee826d9674a00246b05ae26753cdfc90d9ce81a7d02 */ + 3.37020069022253076261281754173810024e-01Q, /* 3ffd591bc9fa2f5972f8e97587fc195d */ +-2.21435756148839473677777545049890664e-35Q, /* bf8bd6f117d92698b5ffdb94fa51d98b */ + +/* x = 3.51562500000000000000000000000000000e-01 3ffd6800000000000000000000000000 */ +/* cos(x) = 0.f0578ad01ede707fa39c09dc6b984afef74f3dc8d0efb0f4c5a6b13771145b3e0446fe33887 */ + 9.38835788546265488632578305984712554e-01Q, /* 3ffee0af15a03dbce0ff473813b8d731 */ +-3.98758068773974031348585072752245458e-35Q, /* bf8ca808458611b978827859d2ca7644 */ +/* sin(x) = 0.582850a41e1dd46c7f602ea244cdbbbfcdfa8f3189be794dda427ce090b5f85164f1f80ac13 */ + 3.44365158145698408207172046472223747e-01Q, /* 3ffd60a14290787751b1fd80ba891337 */ +-3.19791885005480924937758467594051927e-36Q, /* bf89100c815c339d9061ac896f60c7dc */ + +/* x = 3.59375000000000000000000000000000000e-01 3ffd7000000000000000000000000000 */ +/* cos(x) = 0.efa559f5ec3aec3a4eb03319278a2d41fcf9189462261125fe6147b078f1daa0b06750a1654 */ + 9.36116812267055290294237411019508588e-01Q, /* 3ffedf4ab3ebd875d8749d6066324f14 */ + 3.40481591236710658435409862439032162e-35Q, /* 3f8c6a0fe7c8c4a31130892ff30a3d84 */ +/* sin(x) = 0.5a084e28e35fda2776dfdbbb5531d74ced2b5d17c0b1afc4647529d50c295e36d8ceec126c1 */ + 3.51689228994814059222584896955547016e-01Q, /* 3ffd682138a38d7f689ddb7f6eed54c7 */ + 1.75293433418270210567525412802083294e-35Q, /* 3f8b74ced2b5d17c0b1afc4647529d51 */ + +/* x = 3.67187500000000000000000000000000000e-01 3ffd7800000000000000000000000000 */ +/* cos(x) = 0.eeef6a879146af0bf9b95ea2ea0ac0d3e2e4d7e15d93f48cbd41bf8e4fded40bef69e19eafa */ + 9.33340700242548435655299229469995527e-01Q, /* 3ffeddded50f228d5e17f372bd45d416 */ +-4.75255707251679831124800898831382223e-35Q, /* bf8cf960e8d940f513605b9a15f2038e */ +/* sin(x) = 0.5be6e38ce8095542bc14ee9da0d36483e6734bcab2e07624188af5653f114eeb46738fa899d */ + 3.58991834546065053677710299152868941e-01Q, /* 3ffd6f9b8e33a025550af053ba76834e */ +-2.06772389262723368139416970257112089e-35Q, /* bf8bb7c198cb4354d1f89dbe7750a9ac */ + +/* x = 3.75000000000000000000000000000000000e-01 3ffd8000000000000000000000000000 */ +/* cos(x) = 0.ee35bf5ccac89052cd91ddb734d3a47e262e3b609db604e217053803be0091e76daf28a89b7 */ + 9.30507621912314291149476792229555481e-01Q, /* 3ffedc6b7eb9959120a59b23bb6e69a7 */ + 2.74541088551732982573335285685416092e-35Q, /* 3f8c23f13171db04edb02710b829c01e */ +/* sin(x) = 0.5dc40955d9084f48a94675a2498de5d851320ff5528a6afb3f2e24de240fce6cbed1ba0ccd6 */ + 3.66272529086047561372909351716264177e-01Q, /* 3ffd7710255764213d22a519d6892638 */ +-1.96768433534936592675897818253108989e-35Q, /* bf8ba27aecdf00aad759504c0d1db21e */ + +/* x = 3.82812500000000000000000000000000000e-01 3ffd8800000000000000000000000000 */ +/* cos(x) = 0.ed785b5c44741b4493c56bcb9d338a151c6f6b85d8f8aca658b28572c162b199680eb9304da */ + 9.27617750192851909628030798799961350e-01Q, /* 3ffedaf0b6b888e83689278ad7973a67 */ + 7.58520371916345756281201167126854712e-36Q, /* 3f8a42a38ded70bb1f1594cb1650ae58 */ +/* sin(x) = 0.5f9fb80f21b53649c432540a50e22c53057ff42ae0fdf1307760dc0093f99c8efeb2fbd7073 */ + 3.73530868238692946416839752660848112e-01Q, /* 3ffd7e7ee03c86d4d92710c950294389 */ +-1.48023494778986556048879113411517128e-35Q, /* bf8b3acfa800bd51f020ecf889f23ff7 */ + +/* x = 3.90625000000000000000000000000000000e-01 3ffd9000000000000000000000000000 */ +/* cos(x) = 0.ecb7417b8d4ee3fec37aba4073aa48f1f14666006fb431d9671303c8100d10190ec8179c41d */ + 9.24671261467036098502113014560138771e-01Q, /* 3ffed96e82f71a9dc7fd86f57480e755 */ +-4.14187124860031825108649347251175815e-35Q, /* bf8cb87075cccffc825e7134c767e1bf */ +/* sin(x) = 0.6179e84a09a5258a40e9b5face03e525f8b5753cd0105d93fe6298010c3458e84d75fe420e9 */ + 3.80766408992390192057200703388896675e-01Q, /* 3ffd85e7a1282694962903a6d7eb3810 */ +-2.02009541175208636336924533372496107e-35Q, /* bf8bada074a8ac32fefa26c019d67fef */ + +/* x = 3.98437500000000000000000000000000000e-01 3ffd9800000000000000000000000000 */ +/* cos(x) = 0.ebf274bf0bda4f62447e56a093626798d3013b5942b1abfd155aacc9dc5c6d0806a20d6b9c1 */ + 9.21668335573351918175411368202712714e-01Q, /* 3ffed7e4e97e17b49ec488fcad4126c5 */ +-1.83587995433957622948710263541479322e-35Q, /* bf8b8672cfec4a6bd4e5402eaa553362 */ +/* sin(x) = 0.6352929dd264bd44a02ea766325d8aa8bd9695fc8def3caefba5b94c9a3c873f7b2d3776ead */ + 3.87978709727025046051079690813741960e-01Q, /* 3ffd8d4a4a774992f51280ba9d98c976 */ + 8.01904783870935075844443278617586301e-36Q, /* 3f8a5517b2d2bf91bde795df74b72993 */ + +/* x = 4.06250000000000000000000000000000000e-01 3ffda000000000000000000000000000 */ +/* cos(x) = 0.eb29f839f201fd13b93796827916a78f15c85230a4e8ea4b21558265a14367e1abb4c30695a */ + 9.18609155794918267837824977718549863e-01Q, /* 3ffed653f073e403fa27726f2d04f22d */ + 2.97608282778274433460057745798409849e-35Q, /* 3f8c3c78ae429185274752590aac132d */ +/* sin(x) = 0.6529afa7d51b129631ec197c0a840a11d7dc5368b0a47956feb285caa8371c4637ef17ef01b */ + 3.95167330240934236244832640419653657e-01Q, /* 3ffd94a6be9f546c4a58c7b065f02a10 */ + 7.57560031388312550940040194042627704e-36Q, /* 3f8a423afb8a6d16148f2adfd650b955 */ + +/* x = 4.14062500000000000000000000000000000e-01 3ffda800000000000000000000000000 */ +/* cos(x) = 0.ea5dcf0e30cf03e6976ef0b1ec26515fba47383855c3b4055a99b5e86824b2cd1a691fdca7b */ + 9.15493908848301228563917732180221882e-01Q, /* 3ffed4bb9e1c619e07cd2edde163d84d */ +-3.50775517955306954815090901168305659e-35Q, /* bf8c75022dc63e3d51e25fd52b3250bd */ +/* sin(x) = 0.66ff380ba0144109e39a320b0a3fa5fd65ea0585bcbf9b1a769a9b0334576c658139e1a1cbe */ + 4.02331831777773111217105598880982387e-01Q, /* 3ffd9bfce02e805104278e68c82c28ff */ +-1.95678722882848174723569916504871563e-35Q, /* bf8ba029a15fa7a434064e5896564fcd */ + +/* x = 4.21875000000000000000000000000000000e-01 3ffdb000000000000000000000000000 */ +/* cos(x) = 0.e98dfc6c6be031e60dd3089cbdd18a75b1f6b2c1e97f79225202f03dbea45b07a5ec4efc062 */ + 9.12322784872117846492029542047341734e-01Q, /* 3ffed31bf8d8d7c063cc1ba611397ba3 */ + 7.86903886556373674267948132178845568e-36Q, /* 3f8a4eb63ed6583d2fef244a405e07b8 */ +/* sin(x) = 0.68d32473143327973bc712bcc4ccddc47630d755850c0655243b205934dc49ffed8eb76adcb */ + 4.09471777053295066122694027011452236e-01Q, /* 3ffda34c91cc50cc9e5cef1c4af31333 */ + 2.23945241468457597921655785729821354e-35Q, /* 3f8bdc47630d755850c0655243b20593 */ + +/* x = 4.29687500000000000000000000000000000e-01 3ffdb800000000000000000000000000 */ +/* cos(x) = 0.e8ba8393eca7821aa563d83491b6101189b3b101c3677f73d7bad7c10f9ee02b7ab4009739a */ + 9.09095977415431051650381735684476417e-01Q, /* 3ffed1750727d94f04354ac7b069236c */ + 1.20886014028444155733776025085677953e-35Q, /* 3f8b01189b3b101c3677f73d7bad7c11 */ +/* sin(x) = 0.6aa56d8e8249db4eb60a761fe3f9e559be456b9e13349ca99b0bfb787f22b95db3b70179615 */ + 4.16586730282041119259112448831069657e-01Q, /* 3ffdaa95b63a09276d3ad829d87f8fe8 */ +-2.00488106831998813675438269796963612e-35Q, /* bf8baa641ba9461eccb635664f404878 */ + +/* x = 4.37500000000000000000000000000000000e-01 3ffdc000000000000000000000000000 */ +/* cos(x) = 0.e7e367d2956cfb16b6aa11e5419cd0057f5c132a6455bf064297e6a76fe2b72bb630d6d50ff */ + 9.05813683425936420744516660652700258e-01Q, /* 3ffecfc6cfa52ad9f62d6d5423ca833a */ +-3.60950307605941169775676563004467163e-35Q, /* bf8c7fd4051f66acdd5207cdeb40cac5 */ +/* sin(x) = 0.6c760c14c8585a51dbd34660ae6c52ac7036a0b40887a0b63724f8b4414348c3063a637f457 */ + 4.23676257203938010361683988031102480e-01Q, /* 3ffdb1d83053216169476f4d1982b9b1 */ + 1.40484456388654470329473096579312595e-35Q, /* 3f8b2ac7036a0b40887a0b63724f8b44 */ + +/* x = 4.45312500000000000000000000000000000e-01 3ffdc800000000000000000000000000 */ +/* cos(x) = 0.e708ac84d4172a3e2737662213429e14021074d7e702e77d72a8f1101a7e70410df8273e9aa */ + 9.02476103237941504925183272675895999e-01Q, /* 3ffece115909a82e547c4e6ecc442685 */ + 2.26282899501344419018306295680210602e-35Q, /* 3f8be14021074d7e702e77d72a8f1102 */ +/* sin(x) = 0.6e44f8c36eb10a1c752d093c00f4d47ba446ac4c215d26b0316442f168459e677d06e7249e3 */ + 4.30739925110803197216321517850849190e-01Q, /* 3ffdb913e30dbac42871d4b424f003d3 */ + 1.54096780001629398850891218396761548e-35Q, /* 3f8b47ba446ac4c215d26b0316442f17 */ + +/* x = 4.53125000000000000000000000000000000e-01 3ffdd000000000000000000000000000 */ +/* cos(x) = 0.e62a551594b970a770b15d41d4c0e483e47aca550111df6966f9e7ac3a94ae49e6a71eb031e */ + 8.99083440560138456216544929209379307e-01Q, /* 3ffecc54aa2b2972e14ee162ba83a982 */ +-2.06772615490904370666670275154751976e-35Q, /* bf8bb7c1b8535aafeee209699061853c */ +/* sin(x) = 0.70122c5ec5028c8cff33abf4fd340ccc382e038379b09cf04f9a52692b10b72586060cbb001 */ + 4.37777302872755132861618974702796680e-01Q, /* 3ffdc048b17b140a3233fcceafd3f4d0 */ + 9.62794364503442612477117426033922467e-36Q, /* 3f8a998705c0706f36139e09f34a4d25 */ + +/* x = 4.60937500000000000000000000000000000e-01 3ffdd800000000000000000000000000 */ +/* cos(x) = 0.e54864fe33e8575cabf5bd0e5cf1b1a8bc7c0d5f61702450fa6b6539735820dd2603ae355d5 */ + 8.95635902463170698900570000446256350e-01Q, /* 3ffeca90c9fc67d0aeb957eb7a1cb9e3 */ + 3.73593741659866883088620495542311808e-35Q, /* 3f8c8d45e3e06afb0b812287d35b29cc */ +/* sin(x) = 0.71dd9fb1ff4677853acb970a9f6729c6e3aac247b1c57cea66c77413f1f98e8b9e98e49d851 */ + 4.44787960964527211433056012529525211e-01Q, /* 3ffdc7767ec7fd19de14eb2e5c2a7d9d */ +-1.67187936511493678007508371613954899e-35Q, /* bf8b6391c553db84e3a831599388bec1 */ + +/* x = 4.68750000000000000000000000000000000e-01 3ffde000000000000000000000000000 */ +/* cos(x) = 0.e462dfc670d421ab3d1a15901228f146a0547011202bf5ab01f914431859aef577966bc4fa4 */ + 8.92133699366994404723900253723788575e-01Q, /* 3ffec8c5bf8ce1a843567a342b202452 */ +-1.10771937602567314732693079264692504e-35Q, /* bf8ad72bf571fddbfa814a9fc0dd779d */ +/* sin(x) = 0.73a74b8f52947b681baf6928eb3fb021769bf4779bad0e3aa9b1cdb75ec60aad9fc63ff19d5 */ + 4.51771471491683776581688750134062870e-01Q, /* 3ffdce9d2e3d4a51eda06ebda4a3acff */ +-1.19387223016472295893794387275284505e-35Q, /* bf8afbd12c81710c8a5e38aac9c64914 */ + +/* x = 4.76562500000000000000000000000000000e-01 3ffde800000000000000000000000000 */ +/* cos(x) = 0.e379c9045f29d517c4808aa497c2057b2b3d109e76c0dc302d4d0698b36e3f0bdbf33d8e952 */ + 8.88577045028035543317609023116020980e-01Q, /* 3ffec6f39208be53aa2f890115492f84 */ + 4.12354278954664731443813655177022170e-36Q, /* 3f895ecacf44279db0370c0b5341a62d */ +/* sin(x) = 0.756f28d011d98528a44a75fc29c779bd734ecdfb582fdb74b68a4c4c4be54cfd0b2d3ad292f */ + 4.58727408216736592377295028972874773e-01Q, /* 3ffdd5bca340476614a29129d7f0a71e */ +-4.70946994194182908929251719575431779e-36Q, /* bf8990a32c4c8129f40922d25d6ceced */ + +/* x = 4.84375000000000000000000000000000000e-01 3ffdf000000000000000000000000000 */ +/* cos(x) = 0.e28d245c58baef72225e232abc003c4366acd9eb4fc2808c2ab7fe7676cf512ac7f945ae5fb */ + 8.84966156526143291697296536966647926e-01Q, /* 3ffec51a48b8b175dee444bc46557800 */ + 4.53370570288325630442037826313462165e-35Q, /* 3f8ce21b3566cf5a7e14046155bff3b4 */ +/* sin(x) = 0.77353054ca72690d4c6e171fd99e6b39fa8e1ede5f052fd2964534c75340970a3a9cd3c5c32 */ + 4.65655346585160182681199512507546779e-01Q, /* 3ffddcd4c15329c9a43531b85c7f667a */ +-1.56282598978971872478619772155305961e-35Q, /* bf8b4c60571e121a0fad02d69bacb38b */ + +/* x = 4.92187500000000000000000000000000000e-01 3ffdf800000000000000000000000000 */ +/* cos(x) = 0.e19cf580eeec046aa1422fa74807ecefb2a1911c94e7b5f20a00f70022d940193691e5bd790 */ + 8.81301254251340599140161908298100173e-01Q, /* 3ffec339eb01ddd808d542845f4e9010 */ +-1.43419192312116687783945619009629445e-35Q, /* bf8b3104d5e6ee36b184a0df5ff08ffe */ +/* sin(x) = 0.78f95b0560a9a3bd6df7bd981dc38c61224d08bc20631ea932e605e53b579e9e0767dfcbbcb */ + 4.72554863751304451146551317808516942e-01Q, /* 3ffde3e56c1582a68ef5b7def660770e */ + 9.31324774957768018850224267625371204e-36Q, /* 3f8a8c2449a117840c63d5265cc0bca7 */ + +/* x = 5.00000000000000000000000000000000000e-01 3ffe0000000000000000000000000000 */ +/* cos(x) = 0.e0a94032dbea7cedbddd9da2fafad98556566b3a89f43eabd72350af3e8b19e801204d8fe2e */ + 8.77582561890372716116281582603829681e-01Q, /* 3ffec1528065b7d4f9db7bbb3b45f5f6 */ +-2.89484960181363924855192538540698851e-35Q, /* bf8c33d54d4ca62bb05e0aa146e57a86 */ +/* sin(x) = 0.7abba1d12c17bfa1d92f0d93f60ded9992f45b4fcaf13cd58b303693d2a0db47db35ae8a3a9 */ + 4.79425538604203000273287935215571402e-01Q, /* 3ffdeaee8744b05efe8764bc364fd838 */ +-1.38426977616718318950175848639381926e-35Q, /* bf8b2666d0ba4b0350ec32a74cfc96c3 */ + +/* x = 5.07812500000000000000000000000000000e-01 3ffe0400000000000000000000000000 */ +/* cos(x) = 0.dfb20840f3a9b36f7ae2c515342890b5ec583b8366cc2b55029e95094d31112383f2553498b */ + 8.73810306413054508282556837071377159e-01Q, /* 3ffebf641081e75366def5c58a2a6851 */ + 1.25716864497849302237218128599994785e-35Q, /* 3f8b0b5ec583b8366cc2b55029e95095 */ +/* sin(x) = 0.7c7bfdaf13e5ed17212f8a7525bfb113aba6c0741b5362bb8d59282a850b63716bca0c910f0 */ + 4.86266951793275574311011306895834993e-01Q, /* 3ffdf1eff6bc4f97b45c84be29d496ff */ +-1.12269393250914752644352376448094271e-35Q, /* bf8add8a8b27f17c9593a88e54dafaaf */ + +/* x = 5.15625000000000000000000000000000000e-01 3ffe0800000000000000000000000000 */ +/* cos(x) = 0.deb7518814a7a931bbcc88c109cd41c50bf8bb48f20ae8c36628d1d3d57574f7dc58f27d91c */ + 8.69984718058417388828915599901466243e-01Q, /* 3ffebd6ea310294f526377991182139b */ +-4.68168638300575626782741319792183837e-35Q, /* bf8cf1d7a03a25b86fa8b9e4ceb97161 */ +/* sin(x) = 0.7e3a679daaf25c676542bcb4028d0964172961c921823a4ef0c3a9070d886dbd073f6283699 */ + 4.93078685753923057265136552753487121e-01Q, /* 3ffdf8e99e76abc9719d950af2d00a34 */ + 7.06498693112535056352301101088624950e-36Q, /* 3f8a2c82e52c3924304749de187520e2 */ + +/* x = 5.23437500000000000000000000000000000e-01 3ffe0c00000000000000000000000000 */ +/* cos(x) = 0.ddb91ff318799172bd2452d0a3889f5169c64a0094bcf0b8aa7dcf0d7640a2eba68955a80be */ + 8.66106030320656714696616831654267220e-01Q, /* 3ffebb723fe630f322e57a48a5a14711 */ + 2.35610597588322493119667003904687628e-35Q, /* 3f8bf5169c64a0094bcf0b8aa7dcf0d7 */ +/* sin(x) = 0.7ff6d8a34bd5e8fa54c97482db5159df1f24e8038419c0b448b9eea8939b5d4dfcf40900257 */ + 4.99860324733013463819556536946425724e-01Q, /* 3ffdffdb628d2f57a3e95325d20b6d45 */ + 1.94636052312235297538564591686645139e-35Q, /* 3f8b9df1f24e8038419c0b448b9eea89 */ + +/* x = 5.31250000000000000000000000000000000e-01 3ffe1000000000000000000000000000 */ +/* cos(x) = 0.dcb7777ac420705168f31e3eb780ce9c939ecada62843b54522f5407eb7f21e556059fcd734 */ + 8.62174479934880504367162510253324274e-01Q, /* 3ffeb96eeef58840e0a2d1e63c7d6f02 */ +-3.71556818317533582234562471835771823e-35Q, /* bf8c8b1b6309a92cebde255d6e855fc1 */ +/* sin(x) = 0.81b149ce34caa5a4e650f8d09fd4d6aa74206c32ca951a93074c83b2d294d25dbb0f7fdfad2 */ + 5.06611454814257367642296000893867192e-01Q, /* 3ffe0362939c69954b49cca1f1a13faa */ +-3.10963699824274155702706043065967062e-35Q, /* bf8c4aac5efc9e69ab572b67c59be269 */ + +/* x = 5.39062500000000000000000000000000000e-01 3ffe1400000000000000000000000000 */ +/* cos(x) = 0.dbb25c25b8260c14f6e7bc98ec991b70c65335198b0ab628bad20cc7b229d4dd62183cfa055 */ + 8.58190306862660347046629564970494649e-01Q, /* 3ffeb764b84b704c1829edcf7931d932 */ + 2.06439574601190798155563653000684861e-35Q, /* 3f8bb70c65335198b0ab628bad20cc7b */ +/* sin(x) = 0.8369b434a372da7eb5c8a71fe36ce1e0b2b493f6f5cb2e38bcaec2a556b3678c401940d1c3c */ + 5.13331663943471218288801270215706878e-01Q, /* 3ffe06d3686946e5b4fd6b914e3fc6da */ +-2.26614796466671970772244932848067224e-35Q, /* bf8be1f4d4b6c090a34d1c743513d5ab */ + +/* x = 5.46875000000000000000000000000000000e-01 3ffe1800000000000000000000000000 */ +/* cos(x) = 0.daa9d20860827063fde51c09e855e9932e1b17143e7244fd267a899d41ae1f3bc6a0ec42e27 */ + 8.54153754277385385143451785105103176e-01Q, /* 3ffeb553a410c104e0c7fbca3813d0ac */ +-1.68707534013095152873222061722573172e-35Q, /* bf8b66cd1e4e8ebc18dbb02d9857662c */ +/* sin(x) = 0.852010f4f0800521378bd8dd614753d080c2e9e0775ffc609947b9132f5357404f464f06a58 */ + 5.20020541953727004760213699874674730e-01Q, /* 3ffe0a4021e9e1000a426f17b1bac28f */ +-3.32415021330884924833711842866896734e-35Q, /* bf8c617bf9e8b0fc45001cfb35c23767 */ + +/* x = 5.54687500000000000000000000000000000e-01 3ffe1c00000000000000000000000000 */ +/* cos(x) = 0.d99ddd44e44a43d4d4a3a3ed95204106fd54d78e8c7684545c0da0b7c2c72be7a89b7c182ad */ + 8.50065068549420263957072899177793617e-01Q, /* 3ffeb33bba89c89487a9a94747db2a41 */ +-4.73753917078785974356016104842568442e-35Q, /* bf8cf7c81559438b9c4bdd5d1f92fa42 */ +/* sin(x) = 0.86d45935ab396cb4e421e822dee54f3562dfcefeaa782184c23401d231f5ad981a1cc195b18 */ + 5.26677680590386730710789410624833901e-01Q, /* 3ffe0da8b26b5672d969c843d045bdcb */ +-3.67066148195515214077582496518566735e-35Q, /* bf8c8654e901880aac3ef3d9ee5ff16e */ + +/* x = 5.62500000000000000000000000000000000e-01 3ffe2000000000000000000000000000 */ +/* cos(x) = 0.d88e820b1526311dd561efbc0c1a9a5375eb26f65d246c5744b13ca26a7e0fd42556da843c8 */ + 8.45924499231067954459723078597493262e-01Q, /* 3ffeb11d04162a4c623baac3df781835 */ + 1.98054947141989878179164342925274053e-35Q, /* 3f8ba5375eb26f65d246c5744b13ca27 */ +/* sin(x) = 0.88868625b4e1dbb2313310133022527200c143a5cb16637cb7daf8ade82459ff2e98511f40f */ + 5.33302673536020173329131103308161529e-01Q, /* 3ffe110d0c4b69c3b764626620266045 */ +-3.42715291319551615996993795226755157e-35Q, /* bf8c6c6ff9f5e2d1a74ce41a41283a91 */ + +/* x = 5.70312500000000000000000000000000000e-01 3ffe2400000000000000000000000000 */ +/* cos(x) = 0.d77bc4985e93a607c9d868b906bbc6bbe3a04258814acb0358468b826fc91bd4d814827f65e */ + 8.41732299041338366963111794309701085e-01Q, /* 3ffeaef78930bd274c0f93b0d1720d78 */ +-4.30821936750410026005408345400225948e-35Q, /* bf8cca20e2fded3bf5a9a7e53dcba3ed */ +/* sin(x) = 0.8a3690fc5bfc11bf9535e2739a8512f448a41251514bbed7fc18d530f9b4650fcbb2861b0aa */ + 5.39895116435204405041660709903993340e-01Q, /* 3ffe146d21f8b7f8237f2a6bc4e7350a */ + 1.42595803521626714477253741404712093e-35Q, /* 3f8b2f448a41251514bbed7fc18d5310 */ + +/* x = 5.78125000000000000000000000000000000e-01 3ffe2800000000000000000000000000 */ +/* cos(x) = 0.d665a937b4ef2b1f6d51bad6d988a4419c1d7051faf31a9efa151d7631117efac03713f950a */ + 8.37488723850523685315353348917240617e-01Q, /* 3ffeaccb526f69de563edaa375adb311 */ + 2.72761997872084533045777718677326179e-35Q, /* 3f8c220ce0eb828fd798d4f7d0a8ebb2 */ +/* sin(x) = 0.8be472f9776d809af2b88171243d63d66dfceeeb739cc894e023fbc165a0e3f26ff729c5d57 */ + 5.46454606919203564403349553749411001e-01Q, /* 3ffe17c8e5f2eedb0135e57102e2487b */ +-2.11870230730160315420936523771864858e-35Q, /* bf8bc29920311148c63376b1fdc043ea */ + +/* x = 5.85937500000000000000000000000000000e-01 3ffe2c00000000000000000000000000 */ +/* cos(x) = 0.d54c3441844897fc8f853f0655f1ba695eba9fbfd7439dbb1171d862d9d9146ca5136f825ac */ + 8.33194032664581363070224042208032321e-01Q, /* 3ffeaa98688308912ff91f0a7e0cabe3 */ + 4.39440050052045486567668031751259899e-35Q, /* 3f8cd34af5d4fdfeba1cedd88b8ec317 */ +/* sin(x) = 0.8d902565817ee7839bce3cd128060119492cd36d42d82ada30d7f8bde91324808377ddbf5d4 */ + 5.52980744630527369849695082681623667e-01Q, /* 3ffe1b204acb02fdcf07379c79a2500c */ + 8.26624790417342895897164123189984127e-37Q, /* 3f8719492cd36d42d82ada30d7f8bde9 */ + +/* x = 5.93750000000000000000000000000000000e-01 3ffe3000000000000000000000000000 */ +/* cos(x) = 0.d42f6a1b9f0168cdf031c2f63c8d9304d86f8d34cb1d5fccb68ca0f2241427fc18d1fd5bbdf */ + 8.28848487609325734810171790119116638e-01Q, /* 3ffea85ed4373e02d19be06385ec791b */ + 1.43082508100496581719048175506239770e-35Q, /* 3f8b304d86f8d34cb1d5fccb68ca0f22 */ +/* sin(x) = 0.8f39a191b2ba6122a3fa4f41d5a3ffd421417d46f19a22230a14f7fcc8fce5c75b4b28b29d1 */ + 5.59473131247366877384844006003116688e-01Q, /* 3ffe1e7343236574c24547f49e83ab48 */ +-1.28922620524163922306886952100992796e-37Q, /* bf845ef5f415c8732eeee7af584019b8 */ + +/* x = 6.01562500000000000000000000000000000e-01 3ffe3400000000000000000000000000 */ +/* cos(x) = 0.d30f4f392c357ab0661c5fa8a7d9b26627846fef214b1d19a22379ff9eddba087cf410eb097 */ + 8.24452353914429207485643598212356053e-01Q, /* 3ffea61e9e72586af560cc38bf514fb3 */ + 3.79160239225080026987031418939026741e-35Q, /* 3f8c93313c237f790a58e8cd111bcffd */ +/* sin(x) = 0.90e0e0d81ca678796cc92c8ea8c2815bc72ca78abe571bfa8576aacc571e096a33237e0e830 */ + 5.65931370507905990773159095689276114e-01Q, /* 3ffe21c1c1b0394cf0f2d992591d5185 */ + 1.02202775968053982310991962521535027e-36Q, /* 3f875bc72ca78abe571bfa8576aacc57 */ + +/* x = 6.09375000000000000000000000000000000e-01 3ffe3800000000000000000000000000 */ +/* cos(x) = 0.d1ebe81a95ee752e48a26bcd32d6e922d7eb44b8ad2232f6930795e84b56317269b9dd1dfa6 */ + 8.20005899897234008255550633876556043e-01Q, /* 3ffea3d7d0352bdcea5c9144d79a65ae */ +-1.72008811955230823416724332297991247e-35Q, /* bf8b6dd2814bb4752ddcd096cf86a17b */ +/* sin(x) = 0.9285dc9bc45dd9ea3d02457bcce59c4175aab6ff7929a8d287195525fdace200dba032874fb */ + 5.72355068234507240384953706824503608e-01Q, /* 3ffe250bb93788bbb3d47a048af799cb */ + 2.12572273479933123944580199464514529e-35Q, /* 3f8bc4175aab6ff7929a8d2871955260 */ + +/* x = 6.17187500000000000000000000000000000e-01 3ffe3c00000000000000000000000000 */ +/* cos(x) = 0.d0c5394d772228195e25736c03574707de0af1ca344b13bd3914bfe27518e9e426f5deff1e1 */ + 8.15509396946375476876345384201386217e-01Q, /* 3ffea18a729aee445032bc4ae6d806af */ +-4.28589138410712954051679139949341961e-35Q, /* bf8cc7c10fa871ae5da76216375a00ec */ +/* sin(x) = 0.94288e48bd0335fc41c4cbd2920497a8f5d1d8185c99fa0081f90c27e2a53ffdd208a0dbe69 */ + 5.78743832357770354521111378581385347e-01Q, /* 3ffe28511c917a066bf8838997a52409 */ + 1.77998063432551282609698670002456093e-35Q, /* 3f8b7a8f5d1d8185c99fa0081f90c27e */ + +/* x = 6.25000000000000000000000000000000000e-01 3ffe4000000000000000000000000000 */ +/* cos(x) = 0.cf9b476c897c25c5bfe750dd3f308eaf7bcc1ed00179a256870f4200445043dcdb1974b5878 */ + 8.10963119505217902189534803941080724e-01Q, /* 3ffe9f368ed912f84b8b7fcea1ba7e61 */ + 1.10481292856794436426051402418804358e-35Q, /* 3f8ad5ef7983da002f344ad0e1e84009 */ +/* sin(x) = 0.95c8ef544210ec0b91c49bd2aa09e8515fa61a156ebb10f5f8c232a6445b61ebf3c2ec268f9 */ + 5.85097272940462154805399314150080459e-01Q, /* 3ffe2b91dea88421d817238937a55414 */ +-1.78164576278056195136525335403380464e-35Q, /* bf8b7aea059e5ea9144ef0a073dcd59c */ + +/* x = 6.32812500000000000000000000000000000e-01 3ffe4400000000000000000000000000 */ +/* cos(x) = 0.ce6e171f92f2e27f32225327ec440ddaefae248413efc0e58ceee1ae369aabe73f88c87ed1a */ + 8.06367345055103913698795406077297399e-01Q, /* 3ffe9cdc2e3f25e5c4fe6444a64fd888 */ + 1.04235088143133625463876245029180850e-35Q, /* 3f8abb5df5c490827df81cb19ddc35c7 */ +/* sin(x) = 0.9766f93cd18413a6aafc1cfc6fc28abb6817bf94ce349901ae3f48c3215d3eb60acc5f78903 */ + 5.91415002201316315087000225758031236e-01Q, /* 3ffe2ecdf279a308274d55f839f8df85 */ + 8.07390238063560077355762466502569603e-36Q, /* 3f8a576d02f7f299c6932035c7e91864 */ + +/* x = 6.40625000000000000000000000000000000e-01 3ffe4800000000000000000000000000 */ +/* cos(x) = 0.cd3dad1b5328a2e459f993f4f5108819faccbc4eeba9604e81c7adad51cc8a2561631a06826 */ + 8.01722354098418450607492605652964208e-01Q, /* 3ffe9a7b5a36a65145c8b3f327e9ea21 */ + 6.09487851305233089325627939458963741e-36Q, /* 3f8a033f599789dd752c09d038f5b5aa */ +/* sin(x) = 0.9902a58a45e27bed68412b426b675ed503f54d14c8172e0d373f42cadf04daf67319a7f94be */ + 5.97696634538701531238647618967334337e-01Q, /* 3ffe32054b148bc4f7dad0825684d6cf */ +-2.49527608940873714527427941350461554e-35Q, /* bf8c0957e0559759bf468f964605e9a9 */ + +/* x = 6.48437500000000000000000000000000000e-01 3ffe4c00000000000000000000000000 */ +/* cos(x) = 0.cc0a0e21709883a3ff00911e11a07ee3bd7ea2b04e081be99be0264791170761ae64b8b744a */ + 7.97028430141468342004642741431945296e-01Q, /* 3ffe98141c42e1310747fe01223c2341 */ +-8.35364432831812599727083251866305534e-37Q, /* bf871c42815d4fb1f7e416641fd9b86f */ +/* sin(x) = 0.9a9bedcdf01b38d993f3d7820781de292033ead73b89e28f39313dbe3a6e463f845b5fa8490 */ + 6.03941786554156657267270287527367726e-01Q, /* 3ffe3537db9be03671b327e7af040f04 */ +-2.54578992328947177770363936132309779e-35Q, /* bf8c0eb6fe60a94623b0eb863676120e */ + +/* x = 6.56250000000000000000000000000000000e-01 3ffe5000000000000000000000000000 */ +/* cos(x) = 0.cad33f00658fe5e8204bbc0f3a66a0e6a773f87987a780b243d7be83b3db1448ca0e0e62787 */ + 7.92285859677178543141501323781709399e-01Q, /* 3ffe95a67e00cb1fcbd04097781e74cd */ + 2.47519558228473167879248891673807645e-35Q, /* 3f8c07353b9fc3cc3d3c05921ebdf41e */ +/* sin(x) = 0.9c32cba2b14156ef05256c4f857991ca6a547cd7ceb1ac8a8e62a282bd7b9183648a462bd04 */ + 6.10150077075791371273742393566183220e-01Q, /* 3ffe386597456282adde0a4ad89f0af3 */ + 1.33842237929938963780969418369150532e-35Q, /* 3f8b1ca6a547cd7ceb1ac8a8e62a282c */ + +/* x = 6.64062500000000000000000000000000000e-01 3ffe5400000000000000000000000000 */ +/* cos(x) = 0.c99944936cf48c8911ff93fe64b3ddb7981e414bdaf6aae1203577de44878c62bc3bc9cf7b9 */ + 7.87494932167606083931328295965533034e-01Q, /* 3ffe93328926d9e9191223ff27fcc968 */ +-2.57915385618070637156514241185180920e-35Q, /* bf8c12433f0df5a1284aa8f6fe54410e */ +/* sin(x) = 0.9dc738ad14204e689ac582d0f85826590feece34886cfefe2e08cf2bb8488d55424dc9d3525 */ + 6.16321127181550943005700433761731837e-01Q, /* 3ffe3b8e715a28409cd1358b05a1f0b0 */ + 2.88497530050197716298085892460478666e-35Q, /* 3f8c32c87f7671a44367f7f17046795e */ + +/* x = 6.71875000000000000000000000000000000e-01 3ffe5800000000000000000000000000 */ +/* cos(x) = 0.c85c23c26ed7b6f014ef546c47929682122876bfbf157de0aff3c4247d820c746e32cd4174f */ + 7.82655940026272796930787447428139026e-01Q, /* 3ffe90b84784ddaf6de029dea8d88f25 */ + 1.69332045679237919427807771288506254e-35Q, /* 3f8b682122876bfbf157de0aff3c4248 */ +/* sin(x) = 0.9f592e9b66a9cf906a3c7aa3c10199849040c45ec3f0a747597311038101780c5f266059dbf */ + 6.22454560222343683041926705090443330e-01Q, /* 3ffe3eb25d36cd539f20d478f5478203 */ + 1.91974786921147072717621236192269859e-35Q, /* 3f8b9849040c45ec3f0a747597311038 */ + +/* x = 6.79687500000000000000000000000000000e-01 3ffe5c00000000000000000000000000 */ +/* cos(x) = 0.c71be181ecd6875ce2da5615a03cca207d9adcb9dfb0a1d6c40a4f0056437f1a59ccddd06ee */ + 7.77769178600317903122203513685412863e-01Q, /* 3ffe8e37c303d9ad0eb9c5b4ac2b407a */ +-4.05296033424632846931240580239929672e-35Q, /* bf8caefc13291a31027af149dfad87fd */ +/* sin(x) = 0.a0e8a725d33c828c11fa50fd9e9a15ffecfad43f3e534358076b9b0f6865694842b1e8c67dc */ + 6.28550001845029662028004327939032867e-01Q, /* 3ffe41d14e4ba679051823f4a1fb3d34 */ + 1.65507421184028099672784511397428852e-35Q, /* 3f8b5ffecfad43f3e534358076b9b0f7 */ + +/* x = 6.87500000000000000000000000000000000e-01 3ffe6000000000000000000000000000 */ +/* cos(x) = 0.c5d882d2ee48030c7c07d28e981e34804f82ed4cf93655d2365389b716de6ad44676a1cc5da */ + 7.72834946152471544810851845913425178e-01Q, /* 3ffe8bb105a5dc900618f80fa51d303c */ + 3.94975229341211664237241534741146939e-35Q, /* 3f8ca4027c176a67c9b2ae91b29c4db9 */ +/* sin(x) = 0.a2759c0e79c35582527c32b55f5405c182c66160cb1d9eb7bb0b7cdf4ad66f317bda4332914 */ + 6.34607080015269296850309914203671436e-01Q, /* 3ffe44eb381cf386ab04a4f8656abea8 */ + 4.33025916939968369326060156455927002e-36Q, /* 3f897060b1985832c767adeec2df37d3 */ + +/* x = 6.95312500000000000000000000000000000e-01 3ffe6400000000000000000000000000 */ +/* cos(x) = 0.c4920cc2ec38fb891b38827db08884fc66371ac4c2052ca8885b981bbcfd3bb7b093ee31515 */ + 7.67853543842850365879920759114193964e-01Q, /* 3ffe89241985d871f712367104fb6111 */ + 3.75100035267325597157244776081706979e-36Q, /* 3f893f198dc6b130814b2a2216e606ef */ +/* sin(x) = 0.a400072188acf49cd6b173825e038346f105e1301afe642bcc364cea455e21e506e3e927ed8 */ + 6.40625425040230409188409779413961021e-01Q, /* 3ffe48000e431159e939ad62e704bc07 */ + 2.46542747294664049615806500747173281e-36Q, /* 3f88a37882f0980d7f3215e61b267523 */ + +/* x = 7.03125000000000000000000000000000000e-01 3ffe6800000000000000000000000000 */ +/* cos(x) = 0.c348846bbd3631338ffe2bfe9dd1381a35b4e9c0c51b4c13fe376bad1bf5caacc4542be0aa9 */ + 7.62825275710576250507098753625429792e-01Q, /* 3ffe869108d77a6c62671ffc57fd3ba2 */ + 4.22067411888601505004748939382325080e-35Q, /* 3f8cc0d1ada74e0628da609ff1bb5d69 */ +/* sin(x) = 0.a587e23555bb08086d02b9c662cdd29316c3e9bd08d93793634a21b1810cce73bdb97a99b9e */ + 6.46604669591152370524042159882800763e-01Q, /* 3ffe4b0fc46aab761010da05738cc59c */ +-3.41742981816219412415674365946079826e-35Q, /* bf8c6b6749e0b217b9364364e5aef274 */ + +/* x = 7.10937500000000000000000000000000000e-01 3ffe6c00000000000000000000000000 */ +/* cos(x) = 0.c1fbeef380e4ffdd5a613ec8722f643ffe814ec2343e53adb549627224fdc9f2a7b77d3d69f */ + 7.57750448655219342240234832230493361e-01Q, /* 3ffe83f7dde701c9ffbab4c27d90e45f */ +-2.08767968311222650582659938787920125e-35Q, /* bf8bbc0017eb13dcbc1ac524ab69d8de */ +/* sin(x) = 0.a70d272a76a8d4b6da0ec90712bb748b96dabf88c3079246f3db7eea6e58ead4ed0e2843303 */ + 6.52544448725765956407573982284767763e-01Q, /* 3ffe4e1a4e54ed51a96db41d920e2577 */ +-8.61758060284379660697102362141557170e-36Q, /* bf8a6e8d24a80ee79f0db721849022b2 */ + +/* x = 7.18750000000000000000000000000000000e-01 3ffe7000000000000000000000000000 */ +/* cos(x) = 0.c0ac518c8b6ae710ba37a3eeb90cb15aebcb8bed4356fb507a48a6e97de9aa6d9660116b436 */ + 7.52629372418066476054541324847143116e-01Q, /* 3ffe8158a31916d5ce21746f47dd7219 */ + 3.71306958657663189665450864311104571e-35Q, /* 3f8c8ad75e5c5f6a1ab7da83d245374c */ +/* sin(x) = 0.a88fcfebd9a8dd47e2f3c76ef9e2439920f7e7fbe735f8bcc985491ec6f12a2d4214f8cfa99 */ + 6.58444399910567541589583954884041989e-01Q, /* 3ffe511f9fd7b351ba8fc5e78eddf3c5 */ +-4.54412944084300330523721391865787219e-35Q, /* bf8ce336f840c020c6503a19b3d5b70a */ + +/* x = 7.26562500000000000000000000000000000e-01 3ffe7400000000000000000000000000 */ +/* cos(x) = 0.bf59b17550a4406875969296567cf3e3b4e483061877c02811c6cae85fad5a6c3da58f49292 */ + 7.47462359563216166669700384714767552e-01Q, /* 3ffe7eb362eaa14880d0eb2d252cacfa */ +-9.11094340926220027288083639048016945e-36Q, /* bf8a8389636f9f3cf107fafdc726a2f4 */ +/* sin(x) = 0.aa0fd66eddb921232c28520d3911b8a03193b47f187f1471ac216fbcd5bb81029294d3a73f1 */ + 6.64304163042946276515506587432846246e-01Q, /* 3ffe541facddbb7242465850a41a7223 */ + 4.26004843895378210155889028714676019e-35Q, /* 3f8cc5018c9da3f8c3f8a38d610b7de7 */ + +/* x = 7.34375000000000000000000000000000000e-01 3ffe7800000000000000000000000000 */ +/* cos(x) = 0.be0413f84f2a771c614946a88cbf4da1d75a5560243de8f2283fefa0ea4a48468a52d51d8b3 */ + 7.42249725458501306991347253449610537e-01Q, /* 3ffe7c0827f09e54ee38c2928d51197f */ +-3.78925270049800913539923473871287550e-35Q, /* bf8c92f1452d54fede10b86ebe0082f9 */ +/* sin(x) = 0.ab8d34b36acd987210ed343ec65d7e3adc2e7109fce43d55c8d57dfdf55b9e01d2cc1f1b9ec */ + 6.70123380473162894654531583500648495e-01Q, /* 3ffe571a6966d59b30e421da687d8cbb */ +-1.33165852952743729897634069393684656e-36Q, /* bf87c523d18ef6031bc2aa372a82020b */ + +/* x = 7.42187500000000000000000000000000000e-01 3ffe7c00000000000000000000000000 */ +/* cos(x) = 0.bcab7e6bfb2a14a9b122c574a376bec98ab14808c64a4e731b34047e217611013ac99c0f25d */ + 7.36991788256240741057089385586450844e-01Q, /* 3ffe7956fcd7f654295362458ae946ed */ + 4.72358938637974850573747497460125519e-35Q, /* 3f8cf64c558a404632527398d9a023f1 */ +/* sin(x) = 0.ad07e4c409d08c4fa3a9057bb0ac24b8636e74e76f51e09bd6b2319707cbd9f5e254643897a */ + 6.75901697026178809189642203142423973e-01Q, /* 3ffe5a0fc98813a1189f47520af76158 */ + 2.76252586616364878801928456702948857e-35Q, /* 3f8c25c31b73a73b7a8f04deb5918cb8 */ + +/* x = 7.50000000000000000000000000000000000e-01 3ffe8000000000000000000000000000 */ +/* cos(x) = 0.bb4ff632a908f73ec151839cb9d993b4e0bfb8f20e7e44e6e4aee845e35575c3106dbe6fd06 */ + 7.31688868873820886311838753000084529e-01Q, /* 3ffe769fec655211ee7d82a3073973b3 */ + 1.48255637548931697184991710293198620e-35Q, /* 3f8b3b4e0bfb8f20e7e44e6e4aee845e */ +/* sin(x) = 0.ae7fe0b5fc786b2d966e1d6af140a488476747c2646425fc7533f532cd044cb10a971a49a6a */ + 6.81638760023334166733241952779893908e-01Q, /* 3ffe5cffc16bf8f0d65b2cdc3ad5e281 */ + 2.74838775935027549024224114338667371e-35Q, /* 3f8c24423b3a3e1323212fe3a99fa996 */ + +/* x = 7.57812500000000000000000000000000000e-01 3ffe8400000000000000000000000000 */ +/* cos(x) = 0.b9f180ba77dd0751628e135a9508299012230f14becacdd14c3f8862d122de5b56d55b53360 */ + 7.26341290974108590410147630237598973e-01Q, /* 3ffe73e30174efba0ea2c51c26b52a10 */ + 3.12683579338351123545814364980658990e-35Q, /* 3f8c4c80911878a5f6566e8a61fc4317 */ +/* sin(x) = 0.aff522a954f2ba16d9defdc416e33f5e9a5dfd5a6c228e0abc4d521327ff6e2517a7b3851dd */ + 6.87334219303873534951703613035647220e-01Q, /* 3ffe5fea4552a9e5742db3bdfb882dc6 */ + 4.76739454455410744997012795035529128e-35Q, /* 3f8cfaf4d2efead361147055e26a9099 */ + +/* x = 7.65625000000000000000000000000000000e-01 3ffe8800000000000000000000000000 */ +/* cos(x) = 0.b890237d3bb3c284b614a0539016bfa1053730bbdf940fa895e185f8e58884d3dda15e63371 */ + 7.20949380945696418043812784148447688e-01Q, /* 3ffe712046fa776785096c2940a7202d */ + 4.78691285733673379499536326050811832e-35Q, /* 3f8cfd0829b985defca07d44af0c2fc7 */ +/* sin(x) = 0.b167a4c90d63c4244cf5493b7cc23bd3c3c1225e078baa0c53d6d400b926281f537a1a260e6 */ + 6.92987727246317910281815490823048210e-01Q, /* 3ffe62cf49921ac7884899ea9276f984 */ + 4.50089871077663557180849219529189918e-35Q, /* 3f8cde9e1e0912f03c5d50629eb6a006 */ + +/* x = 7.73437500000000000000000000000000000e-01 3ffe8c00000000000000000000000000 */ +/* cos(x) = 0.b72be40067aaf2c050dbdb7a14c3d7d4f203f6b3f0224a4afe55d6ec8e92b508fd5c5984b3b */ + 7.15513467882981573520620561289896903e-01Q, /* 3ffe6e57c800cf55e580a1b7b6f42988 */ +-3.02191815581445336509438104625489192e-35Q, /* bf8c41586fe04a607eedada80d51489c */ +/* sin(x) = 0.b2d7614b1f3aaa24df2d6e20a77e1ca3e6d838c03e29c1bcb026e6733324815fadc9eb89674 */ + 6.98598938789681741301929277107891591e-01Q, /* 3ffe65aec2963e755449be5adc414efc */ + 2.15465226809256290914423429408722521e-35Q, /* 3f8bca3e6d838c03e29c1bcb026e6733 */ + +/* x = 7.81250000000000000000000000000000000e-01 3ffe9000000000000000000000000000 */ +/* cos(x) = 0.b5c4c7d4f7dae915ac786ccf4b1a498d3e73b6e5e74fe7519d9c53ee6d6b90e881bddfc33e1 */ + 7.10033883566079674974121643959490219e-01Q, /* 3ffe6b898fa9efb5d22b58f0d99e9635 */ +-4.09623224763692443220896752907902465e-35Q, /* bf8cb3960c6248d0c580c573131d608d */ +/* sin(x) = 0.b44452709a59752905913765434a59d111f0433eb2b133f7d103207e2aeb4aae111ddc385b3 */ + 7.04167511454533672780059509973942844e-01Q, /* 3ffe6888a4e134b2ea520b226eca8695 */ +-2.87259372740393348676633610275598640e-35Q, /* bf8c3177707de60a6a76604177e6fc0f */ + +/* x = 7.89062500000000000000000000000000000e-01 3ffe9400000000000000000000000000 */ +/* cos(x) = 0.b45ad4975b1294cadca4cf40ec8f22a68cd14b175835239a37e63acb85e8e9505215df18140 */ + 7.04510962440574606164129481545916976e-01Q, /* 3ffe68b5a92eb6252995b9499e81d91e */ + 2.60682037357042658395360726992048803e-35Q, /* 3f8c1534668a58bac1a91cd1bf31d65c */ +/* sin(x) = 0.b5ae7285bc10cf515753847e8f8b7a30e0a580d929d770103509880680f7b8b0e8ad23b65d8 */ + 7.09693105363899724959669028139035515e-01Q, /* 3ffe6b5ce50b78219ea2aea708fd1f17 */ +-4.37026016974122945368562319136420097e-36Q, /* bf8973c7d69fc9b58a23fbf2bd9dfe60 */ +}; diff --git a/libquadmath/math/sincosq.c b/libquadmath/math/sincosq.c new file mode 100644 index 000000000..b7c221486 --- /dev/null +++ b/libquadmath/math/sincosq.c @@ -0,0 +1,68 @@ +/* Compute sine and cosine of argument. + Copyright (C) 1997, 1999 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997 and + Jakub Jelinek <jj@ultra.linux.cz>. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + +void +sincosq (__float128 x, __float128 *sinx, __float128 *cosx) +{ + int64_t ix; + + /* High word of x. */ + GET_FLT128_MSW64 (ix, x); + + /* |x| ~< pi/4 */ + ix &= 0x7fffffffffffffffLL; + if (ix <= 0x3ffe921fb54442d1LL) + __quadmath_kernel_sincosq (x, 0.0Q, sinx, cosx, 0); + else if (ix >= 0x7fff000000000000LL) + { + /* sin(Inf or NaN) is NaN */ + *sinx = *cosx = x - x; + } + else + { + /* Argument reduction needed. */ + __float128 y[2]; + int n; + + n = __quadmath_rem_pio2q (x, y); + switch (n & 3) + { + case 0: + __quadmath_kernel_sincosq (y[0], y[1], sinx, cosx, 1); + break; + case 1: + __quadmath_kernel_sincosq (y[0], y[1], cosx, sinx, 1); + *cosx = -*cosx; + break; + case 2: + __quadmath_kernel_sincosq (y[0], y[1], sinx, cosx, 1); + *sinx = -*sinx; + *cosx = -*cosx; + break; + default: + __quadmath_kernel_sincosq (y[0], y[1], cosx, sinx, 1); + *sinx = -*sinx; + break; + } + } +} diff --git a/libquadmath/math/sincosq_kernel.c b/libquadmath/math/sincosq_kernel.c new file mode 100644 index 000000000..578d1828f --- /dev/null +++ b/libquadmath/math/sincosq_kernel.c @@ -0,0 +1,163 @@ +/* Quad-precision floating point sine and cosine on <-pi/4,pi/4>. + Copyright (C) 1999 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Jakub Jelinek <jj@ultra.linux.cz> + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + +static const __float128 c[] = { +#define ONE c[0] + 1.00000000000000000000000000000000000E+00Q, /* 3fff0000000000000000000000000000 */ + +/* cos x ~ ONE + x^2 ( SCOS1 + SCOS2 * x^2 + ... + SCOS4 * x^6 + SCOS5 * x^8 ) + x in <0,1/256> */ +#define SCOS1 c[1] +#define SCOS2 c[2] +#define SCOS3 c[3] +#define SCOS4 c[4] +#define SCOS5 c[5] +-5.00000000000000000000000000000000000E-01Q, /* bffe0000000000000000000000000000 */ + 4.16666666666666666666666666556146073E-02Q, /* 3ffa5555555555555555555555395023 */ +-1.38888888888888888888309442601939728E-03Q, /* bff56c16c16c16c16c16a566e42c0375 */ + 2.48015873015862382987049502531095061E-05Q, /* 3fefa01a01a019ee02dcf7da2d6d5444 */ +-2.75573112601362126593516899592158083E-07Q, /* bfe927e4f5dce637cb0b54908754bde0 */ + +/* cos x ~ ONE + x^2 ( COS1 + COS2 * x^2 + ... + COS7 * x^12 + COS8 * x^14 ) + x in <0,0.1484375> */ +#define COS1 c[6] +#define COS2 c[7] +#define COS3 c[8] +#define COS4 c[9] +#define COS5 c[10] +#define COS6 c[11] +#define COS7 c[12] +#define COS8 c[13] +-4.99999999999999999999999999999999759E-01Q, /* bffdfffffffffffffffffffffffffffb */ + 4.16666666666666666666666666651287795E-02Q, /* 3ffa5555555555555555555555516f30 */ +-1.38888888888888888888888742314300284E-03Q, /* bff56c16c16c16c16c16c16a463dfd0d */ + 2.48015873015873015867694002851118210E-05Q, /* 3fefa01a01a01a01a0195cebe6f3d3a5 */ +-2.75573192239858811636614709689300351E-07Q, /* bfe927e4fb7789f5aa8142a22044b51f */ + 2.08767569877762248667431926878073669E-09Q, /* 3fe21eed8eff881d1e9262d7adff4373 */ +-1.14707451049343817400420280514614892E-11Q, /* bfda9397496922a9601ed3d4ca48944b */ + 4.77810092804389587579843296923533297E-14Q, /* 3fd2ae5f8197cbcdcaf7c3fb4523414c */ + +/* sin x ~ ONE * x + x^3 ( SSIN1 + SSIN2 * x^2 + ... + SSIN4 * x^6 + SSIN5 * x^8 ) + x in <0,1/256> */ +#define SSIN1 c[14] +#define SSIN2 c[15] +#define SSIN3 c[16] +#define SSIN4 c[17] +#define SSIN5 c[18] +-1.66666666666666666666666666666666659E-01Q, /* bffc5555555555555555555555555555 */ + 8.33333333333333333333333333146298442E-03Q, /* 3ff81111111111111111111110fe195d */ +-1.98412698412698412697726277416810661E-04Q, /* bff2a01a01a01a01a019e7121e080d88 */ + 2.75573192239848624174178393552189149E-06Q, /* 3fec71de3a556c640c6aaa51aa02ab41 */ +-2.50521016467996193495359189395805639E-08Q, /* bfe5ae644ee90c47dc71839de75b2787 */ + +/* sin x ~ ONE * x + x^3 ( SIN1 + SIN2 * x^2 + ... + SIN7 * x^12 + SIN8 * x^14 ) + x in <0,0.1484375> */ +#define SIN1 c[19] +#define SIN2 c[20] +#define SIN3 c[21] +#define SIN4 c[22] +#define SIN5 c[23] +#define SIN6 c[24] +#define SIN7 c[25] +#define SIN8 c[26] +-1.66666666666666666666666666666666538e-01Q, /* bffc5555555555555555555555555550 */ + 8.33333333333333333333333333307532934e-03Q, /* 3ff811111111111111111111110e7340 */ +-1.98412698412698412698412534478712057e-04Q, /* bff2a01a01a01a01a01a019e7a626296 */ + 2.75573192239858906520896496653095890e-06Q, /* 3fec71de3a556c7338fa38527474b8f5 */ +-2.50521083854417116999224301266655662e-08Q, /* bfe5ae64567f544e16c7de65c2ea551f */ + 1.60590438367608957516841576404938118e-10Q, /* 3fde6124613a811480538a9a41957115 */ +-7.64716343504264506714019494041582610e-13Q, /* bfd6ae7f3d5aef30c7bc660b060ef365 */ + 2.81068754939739570236322404393398135e-15Q, /* 3fce9510115aabf87aceb2022a9a9180 */ +}; + +#define SINCOSQ_COS_HI 0 +#define SINCOSQ_COS_LO 1 +#define SINCOSQ_SIN_HI 2 +#define SINCOSQ_SIN_LO 3 +extern const __float128 __sincosq_table[]; + +void +__quadmath_kernel_sincosq(__float128 x, __float128 y, __float128 *sinx, + __float128 *cosx, int iy) +{ + __float128 h, l, z, sin_l, cos_l_m1; + int64_t ix; + uint32_t tix, hix, index; + GET_FLT128_MSW64 (ix, x); + tix = ((uint64_t)ix) >> 32; + tix &= ~0x80000000; /* tix = |x|'s high 32 bits */ + if (tix < 0x3ffc3000) /* |x| < 0.1484375 */ + { + /* Argument is small enough to approximate it by a Chebyshev + polynomial of degree 16(17). */ + if (tix < 0x3fc60000) /* |x| < 2^-57 */ + if (!((int)x)) /* generate inexact */ + { + *sinx = x; + *cosx = ONE; + return; + } + z = x * x; + *sinx = x + (x * (z*(SIN1+z*(SIN2+z*(SIN3+z*(SIN4+ + z*(SIN5+z*(SIN6+z*(SIN7+z*SIN8))))))))); + *cosx = ONE + (z*(COS1+z*(COS2+z*(COS3+z*(COS4+ + z*(COS5+z*(COS6+z*(COS7+z*COS8)))))))); + } + else + { + /* So that we don't have to use too large polynomial, we find + l and h such that x = l + h, where fabsl(l) <= 1.0/256 with 83 + possible values for h. We look up cosl(h) and sinl(h) in + pre-computed tables, compute cosl(l) and sinl(l) using a + Chebyshev polynomial of degree 10(11) and compute + sinl(h+l) = sinl(h)cosl(l) + cosl(h)sinl(l) and + cosl(h+l) = cosl(h)cosl(l) - sinl(h)sinl(l). */ + index = 0x3ffe - (tix >> 16); + hix = (tix + (0x200 << index)) & (0xfffffc00 << index); + x = fabsq (x); + switch (index) + { + case 0: index = ((45 << 10) + hix - 0x3ffe0000) >> 8; break; + case 1: index = ((13 << 11) + hix - 0x3ffd0000) >> 9; break; + default: + case 2: index = (hix - 0x3ffc3000) >> 10; break; + } + + SET_FLT128_WORDS64(h, ((uint64_t)hix) << 32, 0); + if (iy) + l = y - (h - x); + else + l = x - h; + z = l * l; + sin_l = l*(ONE+z*(SSIN1+z*(SSIN2+z*(SSIN3+z*(SSIN4+z*SSIN5))))); + cos_l_m1 = z*(SCOS1+z*(SCOS2+z*(SCOS3+z*(SCOS4+z*SCOS5)))); + z = __sincosq_table [index + SINCOSQ_SIN_HI] + + (__sincosq_table [index + SINCOSQ_SIN_LO] + + (__sincosq_table [index + SINCOSQ_SIN_HI] * cos_l_m1) + + (__sincosq_table [index + SINCOSQ_COS_HI] * sin_l)); + *sinx = (ix < 0) ? -z : z; + *cosx = __sincosq_table [index + SINCOSQ_COS_HI] + + (__sincosq_table [index + SINCOSQ_COS_LO] + - (__sincosq_table [index + SINCOSQ_SIN_HI] * sin_l + - __sincosq_table [index + SINCOSQ_COS_HI] * cos_l_m1)); + } +} diff --git a/libquadmath/math/sinhq.c b/libquadmath/math/sinhq.c new file mode 100644 index 000000000..5492180a2 --- /dev/null +++ b/libquadmath/math/sinhq.c @@ -0,0 +1,111 @@ +/* e_sinhl.c -- __float128 version of e_sinh.c. + * Conversion to __float128 by Ulrich Drepper, + * Cygnus Support, drepper@cygnus.com. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* Changes for 128-bit __float128 are + Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> + and are incorporated herein by permission of the author. The author + reserves the right to distribute this material elsewhere under different + copying permissions. These modifications are distributed here under + the following terms: + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + +/* __ieee754_sinhl(x) + * Method : + * mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2 + * 1. Replace x by |x| (sinhl(-x) = -sinhl(x)). + * 2. + * E + E/(E+1) + * 0 <= x <= 25 : sinhl(x) := --------------, E=expm1l(x) + * 2 + * + * 25 <= x <= lnovft : sinhl(x) := expl(x)/2 + * lnovft <= x <= ln2ovft: sinhl(x) := expl(x/2)/2 * expl(x/2) + * ln2ovft < x : sinhl(x) := x*shuge (overflow) + * + * Special cases: + * sinhl(x) is |x| if x is +INF, -INF, or NaN. + * only sinhl(0)=0 is exact for finite x. + */ + +#include "quadmath-imp.h" + +static const __float128 one = 1.0, shuge = 1.0e4931Q, + ovf_thresh = 1.1357216553474703894801348310092223067821E4Q; + +__float128 +sinhq (__float128 x) +{ + __float128 t, w, h; + uint32_t jx, ix; + ieee854_float128 u; + + /* Words of |x|. */ + u.value = x; + jx = u.words32.w0; + ix = jx & 0x7fffffff; + + /* x is INF or NaN */ + if (ix >= 0x7fff0000) + return x + x; + + h = 0.5Q; + if (jx & 0x80000000) + h = -h; + + /* Absolute value of x. */ + u.words32.w0 = ix; + + /* |x| in [0,40], return sign(x)*0.5*(E+E/(E+1))) */ + if (ix <= 0x40044000) + { + if (ix < 0x3fc60000) /* |x| < 2^-57 */ + if (shuge + x > one) + return x; /* sinh(tiny) = tiny with inexact */ + t = expm1q (u.value); + if (ix < 0x3fff0000) + return h * (2.0Q * t - t * t / (t + one)); + return h * (t + t / (t + one)); + } + + /* |x| in [40, log(maxdouble)] return 0.5*exp(|x|) */ + if (ix <= 0x400c62e3) /* 11356.375 */ + return h * expq (u.value); + + /* |x| in [log(maxdouble), overflowthreshold] + Overflow threshold is log(2 * maxdouble). */ + if (u.value <= ovf_thresh) + { + w = expq (0.5Q * u.value); + t = h * w; + return t * w; + } + + /* |x| > overflowthreshold, sinhl(x) overflow */ + return x * shuge; +} diff --git a/libquadmath/math/sinq.c b/libquadmath/math/sinq.c new file mode 100644 index 000000000..76254a373 --- /dev/null +++ b/libquadmath/math/sinq.c @@ -0,0 +1,82 @@ +/* s_sinl.c -- long double version of s_sin.c. + * Conversion to long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* sinl(x) + * Return sine function of x. + * + * kernel function: + * __kernel_sinl ... sine function on [-pi/4,pi/4] + * __kernel_cosl ... cose function on [-pi/4,pi/4] + * __ieee754_rem_pio2l ... argument reduction routine + * + * Method. + * Let S,C and T denote the sin, cos and tan respectively on + * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 + * in [-pi/4 , +pi/4], and let n = k mod 4. + * We have + * + * n sin(x) cos(x) tan(x) + * ---------------------------------------------------------- + * 0 S C T + * 1 C -S -1/T + * 2 -S -C T + * 3 -C S -1/T + * ---------------------------------------------------------- + * + * Special cases: + * Let trig be any of sin, cos, or tan. + * trig(+-INF) is NaN, with signals; + * trig(NaN) is that NaN; + * + * Accuracy: + * TRIG(x) returns trig(x) nearly rounded + */ + +#include "quadmath-imp.h" + +__float128 +sinq (__float128 x) +{ + __float128 y[2],z=0.0Q; + int64_t n, ix; + + /* High word of x. */ + GET_FLT128_MSW64(ix,x); + + /* |x| ~< pi/4 */ + ix &= 0x7fffffffffffffffLL; + if(ix <= 0x3ffe921fb54442d1LL) + return __quadmath_kernel_sinq(x,z,0); + + /* sin(Inf or NaN) is NaN */ + else if (ix>=0x7fff000000000000LL) { + if (ix == 0x7fff000000000000LL) { + GET_FLT128_LSW64(n,x); + } + return x-x; + } + + /* argument reduction needed */ + else { + n = __quadmath_rem_pio2q(x,y); + switch(n&3) { + case 0: return __quadmath_kernel_sinq(y[0],y[1],1); + case 1: return __quadmath_kernel_cosq(y[0],y[1]); + case 2: return -__quadmath_kernel_sinq(y[0],y[1],1); + default: + return -__quadmath_kernel_cosq(y[0],y[1]); + } + } +} diff --git a/libquadmath/math/sinq_kernel.c b/libquadmath/math/sinq_kernel.c new file mode 100644 index 000000000..395fcaba9 --- /dev/null +++ b/libquadmath/math/sinq_kernel.c @@ -0,0 +1,131 @@ +/* Quad-precision floating point sine on <-pi/4,pi/4>. + Copyright (C) 1999 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Jakub Jelinek <jj@ultra.linux.cz> + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + +static const __float128 c[] = { +#define ONE c[0] + 1.00000000000000000000000000000000000E+00Q, /* 3fff0000000000000000000000000000 */ + +/* cos x ~ ONE + x^2 ( SCOS1 + SCOS2 * x^2 + ... + SCOS4 * x^6 + SCOS5 * x^8 ) + x in <0,1/256> */ +#define SCOS1 c[1] +#define SCOS2 c[2] +#define SCOS3 c[3] +#define SCOS4 c[4] +#define SCOS5 c[5] +-5.00000000000000000000000000000000000E-01Q, /* bffe0000000000000000000000000000 */ + 4.16666666666666666666666666556146073E-02Q, /* 3ffa5555555555555555555555395023 */ +-1.38888888888888888888309442601939728E-03Q, /* bff56c16c16c16c16c16a566e42c0375 */ + 2.48015873015862382987049502531095061E-05Q, /* 3fefa01a01a019ee02dcf7da2d6d5444 */ +-2.75573112601362126593516899592158083E-07Q, /* bfe927e4f5dce637cb0b54908754bde0 */ + +/* sin x ~ ONE * x + x^3 ( SIN1 + SIN2 * x^2 + ... + SIN7 * x^12 + SIN8 * x^14 ) + x in <0,0.1484375> */ +#define SIN1 c[6] +#define SIN2 c[7] +#define SIN3 c[8] +#define SIN4 c[9] +#define SIN5 c[10] +#define SIN6 c[11] +#define SIN7 c[12] +#define SIN8 c[13] +-1.66666666666666666666666666666666538e-01Q, /* bffc5555555555555555555555555550 */ + 8.33333333333333333333333333307532934e-03Q, /* 3ff811111111111111111111110e7340 */ +-1.98412698412698412698412534478712057e-04Q, /* bff2a01a01a01a01a01a019e7a626296 */ + 2.75573192239858906520896496653095890e-06Q, /* 3fec71de3a556c7338fa38527474b8f5 */ +-2.50521083854417116999224301266655662e-08Q, /* bfe5ae64567f544e16c7de65c2ea551f */ + 1.60590438367608957516841576404938118e-10Q, /* 3fde6124613a811480538a9a41957115 */ +-7.64716343504264506714019494041582610e-13Q, /* bfd6ae7f3d5aef30c7bc660b060ef365 */ + 2.81068754939739570236322404393398135e-15Q, /* 3fce9510115aabf87aceb2022a9a9180 */ + +/* sin x ~ ONE * x + x^3 ( SSIN1 + SSIN2 * x^2 + ... + SSIN4 * x^6 + SSIN5 * x^8 ) + x in <0,1/256> */ +#define SSIN1 c[14] +#define SSIN2 c[15] +#define SSIN3 c[16] +#define SSIN4 c[17] +#define SSIN5 c[18] +-1.66666666666666666666666666666666659E-01Q, /* bffc5555555555555555555555555555 */ + 8.33333333333333333333333333146298442E-03Q, /* 3ff81111111111111111111110fe195d */ +-1.98412698412698412697726277416810661E-04Q, /* bff2a01a01a01a01a019e7121e080d88 */ + 2.75573192239848624174178393552189149E-06Q, /* 3fec71de3a556c640c6aaa51aa02ab41 */ +-2.50521016467996193495359189395805639E-08Q, /* bfe5ae644ee90c47dc71839de75b2787 */ +}; + +#define SINCOSQ_COS_HI 0 +#define SINCOSQ_COS_LO 1 +#define SINCOSQ_SIN_HI 2 +#define SINCOSQ_SIN_LO 3 +extern const __float128 __sincosq_table[]; + +__float128 +__quadmath_kernel_sinq (__float128 x, __float128 y, int iy) +{ + __float128 h, l, z, sin_l, cos_l_m1; + int64_t ix; + uint32_t tix, hix, index; + GET_FLT128_MSW64 (ix, x); + tix = ((uint64_t)ix) >> 32; + tix &= ~0x80000000; /* tix = |x|'s high 32 bits */ + if (tix < 0x3ffc3000) /* |x| < 0.1484375 */ + { + /* Argument is small enough to approximate it by a Chebyshev + polynomial of degree 17. */ + if (tix < 0x3fc60000) /* |x| < 2^-57 */ + if (!((int)x)) return x; /* generate inexact */ + z = x * x; + return x + (x * (z*(SIN1+z*(SIN2+z*(SIN3+z*(SIN4+ + z*(SIN5+z*(SIN6+z*(SIN7+z*SIN8))))))))); + } + else + { + /* So that we don't have to use too large polynomial, we find + l and h such that x = l + h, where fabsl(l) <= 1.0/256 with 83 + possible values for h. We look up cosl(h) and sinl(h) in + pre-computed tables, compute cosl(l) and sinl(l) using a + Chebyshev polynomial of degree 10(11) and compute + sinl(h+l) = sinl(h)cosl(l) + cosl(h)sinl(l). */ + index = 0x3ffe - (tix >> 16); + hix = (tix + (0x200 << index)) & (0xfffffc00 << index); + x = fabsq (x); + switch (index) + { + case 0: index = ((45 << 10) + hix - 0x3ffe0000) >> 8; break; + case 1: index = ((13 << 11) + hix - 0x3ffd0000) >> 9; break; + default: + case 2: index = (hix - 0x3ffc3000) >> 10; break; + } + + SET_FLT128_WORDS64(h, ((uint64_t)hix) << 32, 0); + if (iy) + l = y - (h - x); + else + l = x - h; + z = l * l; + sin_l = l*(ONE+z*(SSIN1+z*(SSIN2+z*(SSIN3+z*(SSIN4+z*SSIN5))))); + cos_l_m1 = z*(SCOS1+z*(SCOS2+z*(SCOS3+z*(SCOS4+z*SCOS5)))); + z = __sincosq_table [index + SINCOSQ_SIN_HI] + + (__sincosq_table [index + SINCOSQ_SIN_LO] + + (__sincosq_table [index + SINCOSQ_SIN_HI] * cos_l_m1) + + (__sincosq_table [index + SINCOSQ_COS_HI] * sin_l)); + return (ix < 0) ? -z : z; + } +} diff --git a/libquadmath/math/sqrtq.c b/libquadmath/math/sqrtq.c new file mode 100644 index 000000000..6ed4605ed --- /dev/null +++ b/libquadmath/math/sqrtq.c @@ -0,0 +1,57 @@ +#include "quadmath-imp.h" +#include <math.h> +#include <float.h> + +__float128 +sqrtq (const __float128 x) +{ + __float128 y; + int exp; + + if (x == 0) + return x; + + if (isnanq (x)) + return x; + + if (x < 0) + return nanq (""); + + if (x <= DBL_MAX && x >= DBL_MIN) + { + /* Use double result as starting point. */ + y = sqrt ((double) x); + + /* Two Newton iterations. */ + y -= 0.5q * (y - x / y); + y -= 0.5q * (y - x / y); + return y; + } + +#ifdef HAVE_SQRTL + if (x <= LDBL_MAX && x >= LDBL_MIN) + { + /* Use long double result as starting point. */ + y = sqrtl ((long double) x); + + /* One Newton iteration. */ + y -= 0.5q * (y - x / y); + return y; + } +#endif + + /* If we're outside of the range of C types, we have to compute + the initial guess the hard way. */ + y = frexpq (x, &exp); + if (exp % 2) + y *= 2, exp--; + + y = sqrt (y); + y = scalbnq (y, exp / 2); + + /* Two Newton iterations. */ + y -= 0.5q * (y - x / y); + y -= 0.5q * (y - x / y); + return y; +} + diff --git a/libquadmath/math/tanhq.c b/libquadmath/math/tanhq.c new file mode 100644 index 000000000..4ef4fd021 --- /dev/null +++ b/libquadmath/math/tanhq.c @@ -0,0 +1,94 @@ +/* s_tanhl.c -- __float128 version of s_tanh.c. + * Conversion to __float128 by Ulrich Drepper, + * Cygnus Support, drepper@cygnus.com. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* Changes for 128-bit __float128 contributed by + Stephen L. Moshier <moshier@na-net.ornl.gov> */ + +/* tanhl(x) + * Return the Hyperbolic Tangent of x + * + * Method : + * x -x + * e - e + * 0. tanhl(x) is defined to be ----------- + * x -x + * e + e + * 1. reduce x to non-negative by tanhl(-x) = -tanhl(x). + * 2. 0 <= x <= 2**-57 : tanhl(x) := x*(one+x) + * -t + * 2**-57 < x <= 1 : tanhl(x) := -----; t = expm1l(-2x) + * t + 2 + * 2 + * 1 <= x <= 40.0 : tanhl(x) := 1- ----- ; t=expm1l(2x) + * t + 2 + * 40.0 < x <= INF : tanhl(x) := 1. + * + * Special cases: + * tanhl(NaN) is NaN; + * only tanhl(0)=0 is exact for finite argument. + */ + +#include "quadmath-imp.h" + +static const __float128 one = 1.0Q, two = 2.0Q, tiny = 1.0e-4900Q; + +__float128 +tanhq (__float128 x) +{ + __float128 t, z; + uint32_t jx, ix; + ieee854_float128 u; + + /* Words of |x|. */ + u.value = x; + jx = u.words32.w0; + ix = jx & 0x7fffffff; + /* x is INF or NaN */ + if (ix >= 0x7fff0000) + { + /* for NaN it's not important which branch: tanhl(NaN) = NaN */ + if (jx & 0x80000000) + return one / x - one; /* tanhl(-inf)= -1; */ + else + return one / x + one; /* tanhl(+inf)=+1 */ + } + + /* |x| < 40 */ + if (ix < 0x40044000) + { + if (u.value == 0) + return x; /* x == +- 0 */ + if (ix < 0x3fc60000) /* |x| < 2^-57 */ + return x * (one + tiny); /* tanh(small) = small */ + u.words32.w0 = ix; /* Absolute value of x. */ + if (ix >= 0x3fff0000) + { /* |x| >= 1 */ + t = expm1q (two * u.value); + z = one - two / (t + two); + } + else + { + t = expm1q (-two * u.value); + z = -t / (t + two); + } + /* |x| > 40, return +-1 */ + } + else + { + z = one - tiny; /* raised inexact flag */ + } + return (jx & 0x80000000) ? -z : z; +} diff --git a/libquadmath/math/tanq.c b/libquadmath/math/tanq.c new file mode 100644 index 000000000..e1ec6aae8 --- /dev/null +++ b/libquadmath/math/tanq.c @@ -0,0 +1,237 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* + Long double expansions are + Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> + and are incorporated herein by permission of the author. The author + reserves the right to distribute this material elsewhere under different + copying permissions. These modifications are distributed here under + the following terms: + + This library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + This library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with this library; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + +/* __quadmath_kernel_tanq( x, y, k ) + * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 + * Input x is assumed to be bounded by ~pi/4 in magnitude. + * Input y is the tail of x. + * Input k indicates whether tan (if k=1) or + * -1/tan (if k= -1) is returned. + * + * Algorithm + * 1. Since tan(-x) = -tan(x), we need only to consider positive x. + * 2. if x < 2^-57, return x with inexact if x!=0. + * 3. tan(x) is approximated by a rational form x + x^3 / 3 + x^5 R(x^2) + * on [0,0.67433]. + * + * Note: tan(x+y) = tan(x) + tan'(x)*y + * ~ tan(x) + (1+x*x)*y + * Therefore, for better accuracy in computing tan(x+y), let + * r = x^3 * R(x^2) + * then + * tan(x+y) = x + (x^3 / 3 + (x^2 *(r+y)+y)) + * + * 4. For x in [0.67433,pi/4], let y = pi/4 - x, then + * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) + * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) + */ + +#include "quadmath-imp.h" + + + +static const __float128 + one = 1.0Q, + pio4hi = 7.8539816339744830961566084581987569936977E-1Q, + pio4lo = 2.1679525325309452561992610065108379921906E-35Q, + + /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2) + 0 <= x <= 0.6743316650390625 + Peak relative error 8.0e-36 */ + TH = 3.333333333333333333333333333333333333333E-1Q, + T0 = -1.813014711743583437742363284336855889393E7Q, + T1 = 1.320767960008972224312740075083259247618E6Q, + T2 = -2.626775478255838182468651821863299023956E4Q, + T3 = 1.764573356488504935415411383687150199315E2Q, + T4 = -3.333267763822178690794678978979803526092E-1Q, + + U0 = -1.359761033807687578306772463253710042010E8Q, + U1 = 6.494370630656893175666729313065113194784E7Q, + U2 = -4.180787672237927475505536849168729386782E6Q, + U3 = 8.031643765106170040139966622980914621521E4Q, + U4 = -5.323131271912475695157127875560667378597E2Q; + /* 1.000000000000000000000000000000000000000E0 */ + + +static __float128 +__quadmath_kernel_tanq (__float128 x, __float128 y, int iy) +{ + __float128 z, r, v, w, s; + int32_t ix, sign = 1; + ieee854_float128 u, u1; + + u.value = x; + ix = u.words32.w0 & 0x7fffffff; + if (ix < 0x3fc60000) /* x < 2**-57 */ + { + if ((int) x == 0) + { /* generate inexact */ + if ((ix | u.words32.w1 | u.words32.w2 | u.words32.w3 + | (iy + 1)) == 0) + return one / fabsq (x); + else + return (iy == 1) ? x : -one / x; + } + } + if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */ + { + if ((u.words32.w0 & 0x80000000) != 0) + { + x = -x; + y = -y; + sign = -1; + } + else + sign = 1; + z = pio4hi - x; + w = pio4lo - y; + x = z + w; + y = 0.0; + } + z = x * x; + r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4))); + v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z)))); + r = r / v; + + s = z * x; + r = y + z * (s * r + y); + r += TH * s; + w = x + r; + if (ix >= 0x3ffe5942) + { + v = (__float128) iy; + w = (v - 2.0Q * (x - (w * w / (w + v) - r))); + if (sign < 0) + w = -w; + return w; + } + if (iy == 1) + return w; + else + { /* if allow error up to 2 ulp, + simply return -1.0/(x+r) here */ + /* compute -1.0/(x+r) accurately */ + u1.value = w; + u1.words32.w2 = 0; + u1.words32.w3 = 0; + v = r - (u1.value - x); /* u1+v = r+x */ + z = -1.0 / w; + u.value = z; + u.words32.w2 = 0; + u.words32.w3 = 0; + s = 1.0 + u.value * u1.value; + return u.value + z * (s + u.value * v); + } +} + + + + + + + +/* s_tanl.c -- long double version of s_tan.c. + * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz. + */ + +/* @(#)s_tan.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* tanl(x) + * Return tangent function of x. + * + * kernel function: + * __kernel_tanq ... tangent function on [-pi/4,pi/4] + * __ieee754_rem_pio2q ... argument reduction routine + * + * Method. + * Let S,C and T denote the sin, cos and tan respectively on + * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 + * in [-pi/4 , +pi/4], and let n = k mod 4. + * We have + * + * n sin(x) cos(x) tan(x) + * ---------------------------------------------------------- + * 0 S C T + * 1 C -S -1/T + * 2 -S -C T + * 3 -C S -1/T + * ---------------------------------------------------------- + * + * Special cases: + * Let trig be any of sin, cos, or tan. + * trig(+-INF) is NaN, with signals; + * trig(NaN) is that NaN; + * + * Accuracy: + * TRIG(x) returns trig(x) nearly rounded + */ + + +__float128 +tanq (__float128 x) +{ + __float128 y[2],z=0.0Q; + int64_t n, ix; + + /* High word of x. */ + GET_FLT128_MSW64(ix,x); + + /* |x| ~< pi/4 */ + ix &= 0x7fffffffffffffffLL; + if(ix <= 0x3ffe921fb54442d1LL) return __quadmath_kernel_tanq(x,z,1); + + /* tanl(Inf or NaN) is NaN */ + else if (ix>=0x7fff000000000000LL) { + if (ix == 0x7fff000000000000LL) { + GET_FLT128_LSW64(n,x); + } + return x-x; /* NaN */ + } + + /* argument reduction needed */ + else { + n = __quadmath_rem_pio2q(x,y); + /* 1 -- n even, -1 -- n odd */ + return __quadmath_kernel_tanq(y[0],y[1],1-((n&1)<<1)); + } +} diff --git a/libquadmath/math/tgammaq.c b/libquadmath/math/tgammaq.c new file mode 100644 index 000000000..3305b6484 --- /dev/null +++ b/libquadmath/math/tgammaq.c @@ -0,0 +1,50 @@ +/* Implementation of gamma function according to ISO C. + Copyright (C) 1997, 1999, 2002, 2004 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997 and + Jakub Jelinek <jj@ultra.linux.cz, 1999. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + + +__float128 +tgammaq (__float128 x) +{ + /* We don't have a real gamma implementation now. We'll use lgamma + and the exp function. But due to the required boundary + conditions we must check some values separately. */ + int64_t hx; + uint64_t lx; + + GET_FLT128_WORDS64 (hx, lx, x); + + if (((hx & 0x7fffffffffffffffLL) | lx) == 0) + /* Return value for x == 0 is Inf with divide by zero exception. */ + return 1.0 / x; + + if (hx < 0 && (uint64_t) hx < 0xffff000000000000ULL && rintq (x) == x) + /* Return value for integer x < 0 is NaN with invalid exception. */ + return (x - x) / (x - x); + + if (hx == 0xffff000000000000ULL && lx == 0) + /* x == -Inf. According to ISO this is NaN. */ + return x - x; + + /* XXX FIXME. */ + return expq (lgammaq (x)); +} diff --git a/libquadmath/math/truncq.c b/libquadmath/math/truncq.c new file mode 100644 index 000000000..608871785 --- /dev/null +++ b/libquadmath/math/truncq.c @@ -0,0 +1,54 @@ +/* Truncate argument to nearest integral value not larger than the argument. + Copyright (C) 1997, 1999 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997 and + Jakub Jelinek <jj@ultra.linux.cz>, 1999. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include "quadmath-imp.h" + + +__float128 +truncq (__float128 x) +{ + int32_t j0; + uint64_t i0, i1, sx; + + GET_FLT128_WORDS64 (i0, i1, x); + sx = i0 & 0x8000000000000000ULL; + j0 = ((i0 >> 48) & 0x7fff) - 0x3fff; + if (j0 < 48) + { + if (j0 < 0) + /* The magnitude of the number is < 1 so the result is +-0. */ + SET_FLT128_WORDS64 (x, sx, 0); + else + SET_FLT128_WORDS64 (x, i0 & ~(0x0000ffffffffffffLL >> j0), 0); + } + else if (j0 > 111) + { + if (j0 == 0x4000) + /* x is inf or NaN. */ + return x + x; + } + else + { + SET_FLT128_WORDS64 (x, i0, i1 & ~(0xffffffffffffffffULL >> (j0 - 48))); + } + + return x; +} |