k_tan.c revision 108533
1141296Sdas/* @(#)k_tan.c 5.1 93/09/24 */ 2141296Sdas/* 32116Sjkh * ==================================================== 42116Sjkh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 52116Sjkh * 62116Sjkh * Developed at SunPro, a Sun Microsystems, Inc. business. 7141296Sdas * Permission to use, copy, modify, and distribute this 82116Sjkh * software is freely granted, provided that this notice 9141296Sdas * is preserved. 102116Sjkh * ==================================================== 112116Sjkh */ 12141296Sdas 13176476Sbde#ifndef lint 142116Sjkhstatic char rcsid[] = "$FreeBSD: head/lib/msun/src/k_tan.c 108533 2003-01-01 18:49:04Z schweikh $"; 152116Sjkh#endif 16176385Sbde 17176385Sbde/* __kernel_tan( x, y, k ) 182116Sjkh * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 192116Sjkh * Input x is assumed to be bounded by ~pi/4 in magnitude. 20141296Sdas * Input y is the tail of x. 21141296Sdas * Input k indicates whether tan (if k=1) or 222116Sjkh * -1/tan (if k= -1) is returned. 232116Sjkh * 242116Sjkh * Algorithm 25176465Sbde * 1. Since tan(-x) = -tan(x), we need only to consider positive x. 26176465Sbde * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. 272116Sjkh * 3. tan(x) is approximated by an odd polynomial of degree 27 on 282116Sjkh * [0,0.67434] 292116Sjkh * 3 27 302116Sjkh * tan(x) ~ x + T1*x + ... + T13*x 312116Sjkh * where 322116Sjkh * 332116Sjkh * |tan(x) 2 4 26 | -59.2 342116Sjkh * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 352116Sjkh * | x | 362116Sjkh * 372116Sjkh * Note: tan(x+y) = tan(x) + tan'(x)*y 382116Sjkh * ~ tan(x) + (1+x*x)*y 392116Sjkh * Therefore, for better accuracy in computing tan(x+y), let 408870Srgrimes * 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" 52176385Sbde#include "math_private.h" 53176385Sbdestatic const double 54176385Sbdeone = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 55176385Sbdepio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ 56176385Sbdepio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ 572116SjkhT[] = { 582116Sjkh 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ 59176558Sbde 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ 602116Sjkh 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ 612116Sjkh 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ 622116Sjkh 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ 632116Sjkh 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ 642116Sjkh 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ 65176409Sbde 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ 662116Sjkh 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ 672116Sjkh 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ 68176409Sbde 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ 69176409Sbde -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ 70176409Sbde 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ 71176409Sbde}; 72176409Sbde 73176409Sbdedouble 74176409Sbde__kernel_tan(double x, double y, int iy) 757659Sbde{ 767659Sbde double z,r,v,w,s; 77176409Sbde int32_t ix,hx; 78176409Sbde GET_HIGH_WORD(hx,x); 79176409Sbde ix = hx&0x7fffffff; /* high word of |x| */ 807659Sbde if(ix<0x3e300000) /* x < 2**-28 */ 817659Sbde {if((int)x==0) { /* generate inexact */ 82176409Sbde u_int32_t low; 837659Sbde GET_LOW_WORD(low,x); 84176409Sbde if(((ix|low)|(iy+1))==0) return one/fabs(x); 85176409Sbde else return (iy==1)? x: -one/x; 86176409Sbde } 87176409Sbde } 88176409Sbde if(ix>=0x3FE59428) { /* |x|>=0.6744 */ 89176409Sbde if(hx<0) {x = -x; y = -y;} 90176409Sbde z = pio4-x; 91176409Sbde w = pio4lo-y; 92176409Sbde x = z+w; y = 0.0; 93176409Sbde } 94176409Sbde z = x*x; 95176409Sbde w = z*z; 967659Sbde /* Break x^5*(T[1]+x^2*T[2]+...) into 977659Sbde * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 98176409Sbde * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 99176409Sbde */ 100176409Sbde r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); 101176409Sbde v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); 102176409Sbde s = z*x; 103176409Sbde r = y + z*(s*(r+v)+y); 104176409Sbde r += T[0]*s; 105176409Sbde w = x+r; 106176409Sbde if(ix>=0x3FE59428) { 107176409Sbde v = (double)iy; 108176409Sbde return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); 109176409Sbde } 110176409Sbde if(iy==1) return w; 111176409Sbde else { /* if allow error up to 2 ulp, 112176409Sbde simply return -1.0/(x+r) here */ 113176409Sbde /* compute -1.0/(x+r) accurately */ 114176409Sbde double a,t; 115176409Sbde z = w; 116176409Sbde SET_LOW_WORD(z,0); 117176409Sbde v = r-(z - x); /* z+v = r+x */ 118176409Sbde t = a = -1.0/w; /* a = -1.0/w */ 119176409Sbde SET_LOW_WORD(t,0); 120176409Sbde s = 1.0+t*z; 121176409Sbde return t+a*(s+t*v); 122176409Sbde } 123176409Sbde} 124176409Sbde