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Diffstat (limited to 'libquadmath/math/tanq.c')
| -rw-r--r-- | libquadmath/math/tanq.c | 237 |
1 files changed, 237 insertions, 0 deletions
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)); + } +} |