k_tan.c revision 8870
12116Sjkh/* @(#)k_tan.c 5.1 93/09/24 */ 22116Sjkh/* 32116Sjkh * ==================================================== 42116Sjkh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 52116Sjkh * 62116Sjkh * Developed at SunPro, a Sun Microsystems, Inc. business. 72116Sjkh * Permission to use, copy, modify, and distribute this 88870Srgrimes * software is freely granted, provided that this notice 92116Sjkh * is preserved. 102116Sjkh * ==================================================== 112116Sjkh */ 122116Sjkh 132116Sjkh#ifndef lint 148870Srgrimesstatic char rcsid[] = "$Id: k_tan.c,v 1.1.1.1 1994/08/19 09:39:45 jkh Exp $"; 152116Sjkh#endif 162116Sjkh 172116Sjkh/* __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. 202116Sjkh * Input y is the tail of x. 218870Srgrimes * Input k indicates whether tan (if k=1) or 222116Sjkh * -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. 272116Sjkh * 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" 532116Sjkh#ifdef __STDC__ 548870Srgrimesstatic const double 552116Sjkh#else 568870Srgrimesstatic double 572116Sjkh#endif 582116Sjkhone = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 592116Sjkhpio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ 602116Sjkhpio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ 612116SjkhT[] = { 622116Sjkh 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ 632116Sjkh 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ 642116Sjkh 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ 652116Sjkh 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ 662116Sjkh 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ 672116Sjkh 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ 682116Sjkh 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ 692116Sjkh 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ 702116Sjkh 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ 712116Sjkh 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ 722116Sjkh 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ 732116Sjkh -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ 742116Sjkh 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ 752116Sjkh}; 762116Sjkh 772116Sjkh#ifdef __STDC__ 782116Sjkh double __kernel_tan(double x, double y, int iy) 792116Sjkh#else 802116Sjkh double __kernel_tan(x, y, iy) 812116Sjkh double x,y; int iy; 822116Sjkh#endif 832116Sjkh{ 842116Sjkh double z,r,v,w,s; 852116Sjkh int32_t ix,hx; 862116Sjkh GET_HIGH_WORD(hx,x); 872116Sjkh ix = hx&0x7fffffff; /* high word of |x| */ 882116Sjkh if(ix<0x3e300000) /* x < 2**-28 */ 892116Sjkh {if((int)x==0) { /* generate inexact */ 902116Sjkh u_int32_t low; 912116Sjkh GET_LOW_WORD(low,x); 922116Sjkh if(((ix|low)|(iy+1))==0) return one/fabs(x); 932116Sjkh else return (iy==1)? x: -one/x; 942116Sjkh } 952116Sjkh } 962116Sjkh if(ix>=0x3FE59428) { /* |x|>=0.6744 */ 972116Sjkh if(hx<0) {x = -x; y = -y;} 982116Sjkh z = pio4-x; 992116Sjkh w = pio4lo-y; 1002116Sjkh x = z+w; y = 0.0; 1012116Sjkh } 1022116Sjkh z = x*x; 1032116Sjkh w = z*z; 1042116Sjkh /* Break x^5*(T[1]+x^2*T[2]+...) into 1052116Sjkh * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 1062116Sjkh * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 1072116Sjkh */ 1082116Sjkh r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); 1092116Sjkh v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); 1102116Sjkh s = z*x; 1112116Sjkh r = y + z*(s*(r+v)+y); 1122116Sjkh r += T[0]*s; 1132116Sjkh w = x+r; 1142116Sjkh if(ix>=0x3FE59428) { 1152116Sjkh v = (double)iy; 1162116Sjkh return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); 1172116Sjkh } 1182116Sjkh if(iy==1) return w; 1198870Srgrimes else { /* if allow error up to 2 ulp, 1202116Sjkh simply return -1.0/(x+r) here */ 1212116Sjkh /* compute -1.0/(x+r) accurately */ 1222116Sjkh double a,t; 1232116Sjkh z = w; 1242116Sjkh SET_LOW_WORD(z,0); 1252116Sjkh v = r-(z - x); /* z+v = r+x */ 1262116Sjkh t = a = -1.0/w; /* a = -1.0/w */ 1272116Sjkh SET_LOW_WORD(t,0); 1282116Sjkh s = 1.0+t*z; 1292116Sjkh return t+a*(s+t*v); 1302116Sjkh } 1312116Sjkh} 132