diff options
Diffstat (limited to 'libquadmath/math/acosq.c')
| -rw-r--r-- | libquadmath/math/acosq.c | 317 |
1 files changed, 317 insertions, 0 deletions
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; + } +} |