k_tan.c revision 141296
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 141296 2005-02-04 18:26:06Z das $"; 162116Sjkh#endif 172116Sjkh 182116Sjkh/* __kernel_tan( x, y, k ) 192116Sjkh * kernel tan function on [-pi/4, pi/4], 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. 262116Sjkh * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. 27141296Sdas * 3. tan(x) is approximated by a odd polynomial of degree 27 on 282116Sjkh * [0,0.67434] 292116Sjkh * 3 27 302116Sjkh * tan(x) ~ x + T1*x + ... + T13*x 312116Sjkh * where 328870Srgrimes * 332116Sjkh * |tan(x) 2 4 26 | -59.2 342116Sjkh * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 358870Srgrimes * | x | 368870Srgrimes * 372116Sjkh * Note: tan(x+y) = tan(x) + tan'(x)*y 382116Sjkh * ~ tan(x) + (1+x*x)*y 398870Srgrimes * Therefore, for better accuracy in computing tan(x+y), let 402116Sjkh * 3 2 2 2 2 412116Sjkh * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) 422116Sjkh * then 432116Sjkh * 3 2 442116Sjkh * tan(x+y) = x + (T1*x + (x *(r+y)+y)) 452116Sjkh * 462116Sjkh * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then 472116Sjkh * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) 482116Sjkh * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) 492116Sjkh */ 502116Sjkh 512116Sjkh#include "math.h" 522116Sjkh#include "math_private.h" 53141296Sdasstatic const double xxx[] = { 54141296Sdas 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ 55141296Sdas 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ 56141296Sdas 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ 57141296Sdas 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ 58141296Sdas 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ 59141296Sdas 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ 60141296Sdas 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ 61141296Sdas 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ 62141296Sdas 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ 63141296Sdas 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ 64141296Sdas 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ 65141296Sdas -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ 66141296Sdas 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ 67141296Sdas/* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ 68141296Sdas/* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ 69141296Sdas/* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ 702116Sjkh}; 71141296Sdas#define one xxx[13] 72141296Sdas#define pio4 xxx[14] 73141296Sdas#define pio4lo xxx[15] 74141296Sdas#define T xxx 75141296Sdas/* INDENT ON */ 762116Sjkh 7797413Salfreddouble 78141296Sdas__kernel_tan(double x, double y, int iy) { 79141296Sdas double z, r, v, w, s; 80141296Sdas int32_t ix, hx; 81141296Sdas 822116Sjkh GET_HIGH_WORD(hx,x); 83141296Sdas ix = hx & 0x7fffffff; /* high word of |x| */ 84141296Sdas if (ix < 0x3e300000) { /* x < 2**-28 */ 85129980Sdas if ((int) x == 0) { /* generate inexact */ 86129980Sdas u_int32_t low; 87129980Sdas GET_LOW_WORD(low,x); 88129980Sdas if (((ix | low) | (iy + 1)) == 0) 89129980Sdas return one / fabs(x); 90129980Sdas else { 91129980Sdas if (iy == 1) 92129980Sdas return x; 93129980Sdas else { /* compute -1 / (x+y) carefully */ 94129980Sdas double a, t; 95129980Sdas 96129980Sdas z = w = x + y; 97129980Sdas SET_LOW_WORD(z, 0); 98129980Sdas v = y - (z - x); 99129980Sdas t = a = -one / w; 100129980Sdas SET_LOW_WORD(t, 0); 101129980Sdas s = one + t * z; 102129980Sdas return t + a * (s + t * v); 103129980Sdas } 104129980Sdas } 105129980Sdas } 106129980Sdas } 107141296Sdas if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ 108141296Sdas if (hx < 0) { 109141296Sdas x = -x; 110141296Sdas y = -y; 111141296Sdas } 112141296Sdas z = pio4 - x; 113141296Sdas w = pio4lo - y; 114141296Sdas x = z + w; 115141296Sdas y = 0.0; 1162116Sjkh } 117141296Sdas z = x * x; 118141296Sdas w = z * z; 119141296Sdas /* 120141296Sdas * Break x^5*(T[1]+x^2*T[2]+...) into 121141296Sdas * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 122141296Sdas * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 123141296Sdas */ 124141296Sdas r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + 125141296Sdas w * T[11])))); 126141296Sdas v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + 127141296Sdas w * T[12]))))); 128141296Sdas s = z * x; 129141296Sdas r = y + z * (s * (r + v) + y); 130141296Sdas r += T[0] * s; 131141296Sdas w = x + r; 132141296Sdas if (ix >= 0x3FE59428) { 133141296Sdas v = (double) iy; 134141296Sdas return (double) (1 - ((hx >> 30) & 2)) * 135141296Sdas (v - 2.0 * (x - (w * w / (w + v) - r))); 1362116Sjkh } 137141296Sdas if (iy == 1) 138141296Sdas return w; 139141296Sdas else { 140141296Sdas /* 141141296Sdas * if allow error up to 2 ulp, simply return 142141296Sdas * -1.0 / (x+r) here 143141296Sdas */ 144141296Sdas /* compute -1.0 / (x+r) accurately */ 145141296Sdas double a, t; 146141296Sdas z = w; 147141296Sdas SET_LOW_WORD(z,0); 148141296Sdas v = r - (z - x); /* z+v = r+x */ 149141296Sdas t = a = -1.0 / w; /* a = -1.0/w */ 150141296Sdas SET_LOW_WORD(t,0); 151141296Sdas s = 1.0 + t * z; 152141296Sdas return t + a * (s + t * v); 1532116Sjkh } 1542116Sjkh} 155