k_tan.c revision 151969
1141296Sdas/* @(#)k_tan.c 1.5 04/04/22 SMI */ 2141296Sdas 32116Sjkh/* 42116Sjkh * ==================================================== 5129980Sdas * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved. 62116Sjkh * 72116Sjkh * Permission to use, copy, modify, and distribute this 88870Srgrimes * software is freely granted, provided that this notice 92116Sjkh * is preserved. 102116Sjkh * ==================================================== 112116Sjkh */ 122116Sjkh 13141296Sdas/* INDENT OFF */ 142116Sjkh#ifndef lint 1550476Speterstatic char rcsid[] = "$FreeBSD: head/lib/msun/src/k_tan.c 151969 2005-11-02 14:01:45Z bde $"; 162116Sjkh#endif 172116Sjkh 182116Sjkh/* __kernel_tan( x, y, k ) 19151969Sbde * kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854 202116Sjkh * Input x is assumed to be bounded by ~pi/4 in magnitude. 212116Sjkh * Input y is the tail of x. 22141296Sdas * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned. 232116Sjkh * 242116Sjkh * Algorithm 258870Srgrimes * 1. Since tan(-x) = -tan(x), we need only to consider positive x. 26151969Sbde * 2. Callers must return tan(-0) = -0 without calling here since our 27151969Sbde * odd polynomial is not evaluated in a way that preserves -0. 28151969Sbde * Callers may do the optimization tan(x) ~ x for tiny x. 29141296Sdas * 3. tan(x) is approximated by a odd polynomial of degree 27 on 302116Sjkh * [0,0.67434] 312116Sjkh * 3 27 322116Sjkh * tan(x) ~ x + T1*x + ... + T13*x 332116Sjkh * where 348870Srgrimes * 352116Sjkh * |tan(x) 2 4 26 | -59.2 362116Sjkh * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 378870Srgrimes * | x | 388870Srgrimes * 392116Sjkh * Note: tan(x+y) = tan(x) + tan'(x)*y 402116Sjkh * ~ tan(x) + (1+x*x)*y 418870Srgrimes * Therefore, for better accuracy in computing tan(x+y), let 422116Sjkh * 3 2 2 2 2 432116Sjkh * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) 442116Sjkh * then 452116Sjkh * 3 2 462116Sjkh * tan(x+y) = x + (T1*x + (x *(r+y)+y)) 472116Sjkh * 482116Sjkh * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then 492116Sjkh * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) 502116Sjkh * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) 512116Sjkh */ 522116Sjkh 532116Sjkh#include "math.h" 542116Sjkh#include "math_private.h" 55141296Sdasstatic const double xxx[] = { 56141296Sdas 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ 57141296Sdas 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ 58141296Sdas 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ 59141296Sdas 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ 60141296Sdas 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ 61141296Sdas 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ 62141296Sdas 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ 63141296Sdas 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ 64141296Sdas 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ 65141296Sdas 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ 66141296Sdas 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ 67141296Sdas -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ 68141296Sdas 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ 69141296Sdas/* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ 70141296Sdas/* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ 71141296Sdas/* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ 722116Sjkh}; 73141296Sdas#define one xxx[13] 74141296Sdas#define pio4 xxx[14] 75141296Sdas#define pio4lo xxx[15] 76141296Sdas#define T xxx 77141296Sdas/* INDENT ON */ 782116Sjkh 7997413Salfreddouble 80141296Sdas__kernel_tan(double x, double y, int iy) { 81141296Sdas double z, r, v, w, s; 82141296Sdas int32_t ix, hx; 83141296Sdas 842116Sjkh GET_HIGH_WORD(hx,x); 85141296Sdas ix = hx & 0x7fffffff; /* high word of |x| */ 86141296Sdas if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ 87141296Sdas if (hx < 0) { 88141296Sdas x = -x; 89141296Sdas y = -y; 90141296Sdas } 91141296Sdas z = pio4 - x; 92141296Sdas w = pio4lo - y; 93141296Sdas x = z + w; 94141296Sdas y = 0.0; 952116Sjkh } 96141296Sdas z = x * x; 97141296Sdas w = z * z; 98141296Sdas /* 99141296Sdas * Break x^5*(T[1]+x^2*T[2]+...) into 100141296Sdas * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 101141296Sdas * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 102141296Sdas */ 103141296Sdas r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + 104141296Sdas w * T[11])))); 105141296Sdas v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + 106141296Sdas w * T[12]))))); 107141296Sdas s = z * x; 108141296Sdas r = y + z * (s * (r + v) + y); 109141296Sdas r += T[0] * s; 110141296Sdas w = x + r; 111141296Sdas if (ix >= 0x3FE59428) { 112141296Sdas v = (double) iy; 113141296Sdas return (double) (1 - ((hx >> 30) & 2)) * 114141296Sdas (v - 2.0 * (x - (w * w / (w + v) - r))); 1152116Sjkh } 116141296Sdas if (iy == 1) 117141296Sdas return w; 118141296Sdas else { 119141296Sdas /* 120141296Sdas * if allow error up to 2 ulp, simply return 121141296Sdas * -1.0 / (x+r) here 122141296Sdas */ 123141296Sdas /* compute -1.0 / (x+r) accurately */ 124141296Sdas double a, t; 125141296Sdas z = w; 126141296Sdas SET_LOW_WORD(z,0); 127141296Sdas v = r - (z - x); /* z+v = r+x */ 128141296Sdas t = a = -1.0 / w; /* a = -1.0/w */ 129141296Sdas SET_LOW_WORD(t,0); 130141296Sdas s = 1.0 + t * z; 131141296Sdas return t + a * (s + t * v); 1322116Sjkh } 1332116Sjkh} 134