111 lines
3.9 KiB
C
111 lines
3.9 KiB
C
/* origin: FreeBSD /usr/src/lib/msun/src/k_tan.c */
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/*
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* ====================================================
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* Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
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*
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/* __tan( x, y, k )
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* kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
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* Input x is assumed to be bounded by ~pi/4 in magnitude.
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* Input y is the tail of x.
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* Input odd indicates whether tan (if odd = 0) or -1/tan (if odd = 1) is returned.
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*
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* Algorithm
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* 1. Since tan(-x) = -tan(x), we need only to consider positive x.
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* 2. Callers must return tan(-0) = -0 without calling here since our
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* odd polynomial is not evaluated in a way that preserves -0.
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* Callers may do the optimization tan(x) ~ x for tiny x.
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* 3. tan(x) is approximated by a odd polynomial of degree 27 on
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* [0,0.67434]
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* 3 27
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* tan(x) ~ x + T1*x + ... + T13*x
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* where
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*
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* |tan(x) 2 4 26 | -59.2
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* |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
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* | x |
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*
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* Note: tan(x+y) = tan(x) + tan'(x)*y
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* ~ tan(x) + (1+x*x)*y
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* Therefore, for better accuracy in computing tan(x+y), let
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* 3 2 2 2 2
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* r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
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* then
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* 3 2
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* tan(x+y) = x + (T1*x + (x *(r+y)+y))
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*
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* 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
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* tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
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* = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
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*/
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#include "libm.h"
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static const double T[] = {
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3.33333333333334091986e-01, /* 3FD55555, 55555563 */
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1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
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5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
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2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
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8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
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3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
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1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
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5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
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2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
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7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
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7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
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-1.85586374855275456654e-05, /* BEF375CB, DB605373 */
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2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
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},
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pio4 = 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
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pio4lo = 3.06161699786838301793e-17; /* 3C81A626, 33145C07 */
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double __tan(double x, double y, int odd)
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{
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double_t z, r, v, w, s, a;
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double w0, a0;
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uint32_t hx;
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int big, sign;
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GET_HIGH_WORD(hx,x);
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big = (hx&0x7fffffff) >= 0x3FE59428; /* |x| >= 0.6744 */
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if (big) {
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sign = hx>>31;
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if (sign) {
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x = -x;
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y = -y;
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}
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x = (pio4 - x) + (pio4lo - y);
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y = 0.0;
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}
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z = x * x;
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w = z * z;
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/*
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* Break x^5*(T[1]+x^2*T[2]+...) into
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* x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
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* x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
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*/
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r = T[1] + w*(T[3] + w*(T[5] + w*(T[7] + w*(T[9] + w*T[11]))));
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v = z*(T[2] + w*(T[4] + w*(T[6] + w*(T[8] + w*(T[10] + w*T[12])))));
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s = z * x;
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r = y + z*(s*(r + v) + y) + s*T[0];
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w = x + r;
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if (big) {
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s = 1 - 2*odd;
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v = s - 2.0 * (x + (r - w*w/(w + s)));
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return sign ? -v : v;
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}
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if (!odd)
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return w;
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/* -1.0/(x+r) has up to 2ulp error, so compute it accurately */
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w0 = w;
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SET_LOW_WORD(w0, 0);
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v = r - (w0 - x); /* w0+v = r+x */
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a0 = a = -1.0 / w;
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SET_LOW_WORD(a0, 0);
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return a0 + a*(1.0 + a0*w0 + a0*v);
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}
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