k_tan.c revision 129980
12116Sjkh/* @(#)k_tan.c 5.1 93/09/24 */ 22116Sjkh/* 32116Sjkh * ==================================================== 4129980Sdas * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved. 52116Sjkh * 62116Sjkh * Permission to use, copy, modify, and distribute this 78870Srgrimes * software is freely granted, provided that this notice 82116Sjkh * is preserved. 92116Sjkh * ==================================================== 102116Sjkh */ 112116Sjkh 122116Sjkh#ifndef lint 1350476Speterstatic char rcsid[] = "$FreeBSD: head/lib/msun/src/k_tan.c 129980 2004-06-02 04:39:29Z das $"; 142116Sjkh#endif 152116Sjkh 162116Sjkh/* __kernel_tan( x, y, k ) 172116Sjkh * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 182116Sjkh * Input x is assumed to be bounded by ~pi/4 in magnitude. 192116Sjkh * Input y is the tail of x. 208870Srgrimes * Input k indicates whether tan (if k=1) or 212116Sjkh * -1/tan (if k= -1) is returned. 222116Sjkh * 232116Sjkh * Algorithm 248870Srgrimes * 1. Since tan(-x) = -tan(x), we need only to consider positive x. 252116Sjkh * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. 26108533Sschweikh * 3. tan(x) is approximated by an odd polynomial of degree 27 on 272116Sjkh * [0,0.67434] 282116Sjkh * 3 27 292116Sjkh * tan(x) ~ x + T1*x + ... + T13*x 302116Sjkh * where 318870Srgrimes * 322116Sjkh * |tan(x) 2 4 26 | -59.2 332116Sjkh * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 348870Srgrimes * | x | 358870Srgrimes * 362116Sjkh * Note: tan(x+y) = tan(x) + tan'(x)*y 372116Sjkh * ~ tan(x) + (1+x*x)*y 388870Srgrimes * Therefore, for better accuracy in computing tan(x+y), let 392116Sjkh * 3 2 2 2 2 402116Sjkh * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) 412116Sjkh * then 422116Sjkh * 3 2 432116Sjkh * tan(x+y) = x + (T1*x + (x *(r+y)+y)) 442116Sjkh * 452116Sjkh * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then 462116Sjkh * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) 472116Sjkh * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) 482116Sjkh */ 492116Sjkh 502116Sjkh#include "math.h" 512116Sjkh#include "math_private.h" 528870Srgrimesstatic const double 532116Sjkhone = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 542116Sjkhpio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ 552116Sjkhpio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ 562116SjkhT[] = { 572116Sjkh 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ 582116Sjkh 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ 592116Sjkh 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ 602116Sjkh 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ 612116Sjkh 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ 622116Sjkh 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ 632116Sjkh 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ 642116Sjkh 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ 652116Sjkh 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ 662116Sjkh 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ 672116Sjkh 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ 682116Sjkh -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ 692116Sjkh 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ 702116Sjkh}; 712116Sjkh 7297413Salfreddouble 7397413Salfred__kernel_tan(double x, double y, int iy) 742116Sjkh{ 752116Sjkh double z,r,v,w,s; 762116Sjkh int32_t ix,hx; 772116Sjkh GET_HIGH_WORD(hx,x); 782116Sjkh ix = hx&0x7fffffff; /* high word of |x| */ 79129980Sdas if(ix<0x3e300000) { /* x < 2**-28 */ 80129980Sdas if ((int) x == 0) { /* generate inexact */ 81129980Sdas u_int32_t low; 82129980Sdas GET_LOW_WORD(low,x); 83129980Sdas if (((ix | low) | (iy + 1)) == 0) 84129980Sdas return one / fabs(x); 85129980Sdas else { 86129980Sdas if (iy == 1) 87129980Sdas return x; 88129980Sdas else { /* compute -1 / (x+y) carefully */ 89129980Sdas double a, t; 90129980Sdas 91129980Sdas z = w = x + y; 92129980Sdas SET_LOW_WORD(z, 0); 93129980Sdas v = y - (z - x); 94129980Sdas t = a = -one / w; 95129980Sdas SET_LOW_WORD(t, 0); 96129980Sdas s = one + t * z; 97129980Sdas return t + a * (s + t * v); 98129980Sdas } 99129980Sdas } 100129980Sdas } 101129980Sdas } 1022116Sjkh if(ix>=0x3FE59428) { /* |x|>=0.6744 */ 1032116Sjkh if(hx<0) {x = -x; y = -y;} 1042116Sjkh z = pio4-x; 1052116Sjkh w = pio4lo-y; 1062116Sjkh x = z+w; y = 0.0; 1072116Sjkh } 1082116Sjkh z = x*x; 1092116Sjkh w = z*z; 1102116Sjkh /* Break x^5*(T[1]+x^2*T[2]+...) into 1112116Sjkh * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 1122116Sjkh * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 1132116Sjkh */ 1142116Sjkh r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); 1152116Sjkh v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); 1162116Sjkh s = z*x; 1172116Sjkh r = y + z*(s*(r+v)+y); 1182116Sjkh r += T[0]*s; 1192116Sjkh w = x+r; 1202116Sjkh if(ix>=0x3FE59428) { 1212116Sjkh v = (double)iy; 1222116Sjkh return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); 1232116Sjkh } 1242116Sjkh if(iy==1) return w; 1258870Srgrimes else { /* if allow error up to 2 ulp, 1262116Sjkh simply return -1.0/(x+r) here */ 1272116Sjkh /* compute -1.0/(x+r) accurately */ 1282116Sjkh double a,t; 1292116Sjkh z = w; 1302116Sjkh SET_LOW_WORD(z,0); 1312116Sjkh v = r-(z - x); /* z+v = r+x */ 1322116Sjkh t = a = -1.0/w; /* a = -1.0/w */ 1332116Sjkh SET_LOW_WORD(t,0); 1342116Sjkh s = 1.0+t*z; 1352116Sjkh return t+a*(s+t*v); 1362116Sjkh } 1372116Sjkh} 138