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 */ 14176451Sdas#include <sys/cdefs.h> 15176451Sdas__FBSDID("$FreeBSD: releng/11.0/lib/msun/src/k_tan.c 176451 2008-02-22 02:30:36Z das $"); 162116Sjkh 172116Sjkh/* __kernel_tan( x, y, k ) 18151969Sbde * kernel tan function on ~[-pi/4, pi/4] (except on -0), 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. 21141296Sdas * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned. 222116Sjkh * 232116Sjkh * Algorithm 248870Srgrimes * 1. Since tan(-x) = -tan(x), we need only to consider positive x. 25151969Sbde * 2. Callers must return tan(-0) = -0 without calling here since our 26151969Sbde * odd polynomial is not evaluated in a way that preserves -0. 27151969Sbde * Callers may do the optimization tan(x) ~ x for tiny x. 28141296Sdas * 3. tan(x) is approximated by a odd polynomial of degree 27 on 292116Sjkh * [0,0.67434] 302116Sjkh * 3 27 312116Sjkh * tan(x) ~ x + T1*x + ... + T13*x 322116Sjkh * where 338870Srgrimes * 342116Sjkh * |tan(x) 2 4 26 | -59.2 352116Sjkh * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 368870Srgrimes * | x | 378870Srgrimes * 382116Sjkh * Note: tan(x+y) = tan(x) + tan'(x)*y 392116Sjkh * ~ tan(x) + (1+x*x)*y 408870Srgrimes * Therefore, for better accuracy in computing tan(x+y), let 412116Sjkh * 3 2 2 2 2 422116Sjkh * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) 432116Sjkh * then 442116Sjkh * 3 2 452116Sjkh * tan(x+y) = x + (T1*x + (x *(r+y)+y)) 462116Sjkh * 472116Sjkh * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then 482116Sjkh * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) 492116Sjkh * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) 502116Sjkh */ 512116Sjkh 522116Sjkh#include "math.h" 532116Sjkh#include "math_private.h" 54141296Sdasstatic const double xxx[] = { 55141296Sdas 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ 56141296Sdas 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ 57141296Sdas 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ 58141296Sdas 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ 59141296Sdas 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ 60141296Sdas 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ 61141296Sdas 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ 62141296Sdas 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ 63141296Sdas 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ 64141296Sdas 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ 65141296Sdas 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ 66141296Sdas -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ 67141296Sdas 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ 68141296Sdas/* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ 69141296Sdas/* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ 70141296Sdas/* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ 712116Sjkh}; 72141296Sdas#define one xxx[13] 73141296Sdas#define pio4 xxx[14] 74141296Sdas#define pio4lo xxx[15] 75141296Sdas#define T xxx 76141296Sdas/* INDENT ON */ 772116Sjkh 7897413Salfreddouble 79141296Sdas__kernel_tan(double x, double y, int iy) { 80141296Sdas double z, r, v, w, s; 81141296Sdas int32_t ix, hx; 82141296Sdas 832116Sjkh GET_HIGH_WORD(hx,x); 84141296Sdas ix = hx & 0x7fffffff; /* high word of |x| */ 85141296Sdas if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ 86141296Sdas if (hx < 0) { 87141296Sdas x = -x; 88141296Sdas y = -y; 89141296Sdas } 90141296Sdas z = pio4 - x; 91141296Sdas w = pio4lo - y; 92141296Sdas x = z + w; 93141296Sdas y = 0.0; 942116Sjkh } 95141296Sdas z = x * x; 96141296Sdas w = z * z; 97141296Sdas /* 98141296Sdas * Break x^5*(T[1]+x^2*T[2]+...) into 99141296Sdas * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 100141296Sdas * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 101141296Sdas */ 102141296Sdas r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + 103141296Sdas w * T[11])))); 104141296Sdas v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + 105141296Sdas w * T[12]))))); 106141296Sdas s = z * x; 107141296Sdas r = y + z * (s * (r + v) + y); 108141296Sdas r += T[0] * s; 109141296Sdas w = x + r; 110141296Sdas if (ix >= 0x3FE59428) { 111141296Sdas v = (double) iy; 112141296Sdas return (double) (1 - ((hx >> 30) & 2)) * 113141296Sdas (v - 2.0 * (x - (w * w / (w + v) - r))); 1142116Sjkh } 115141296Sdas if (iy == 1) 116141296Sdas return w; 117141296Sdas else { 118141296Sdas /* 119141296Sdas * if allow error up to 2 ulp, simply return 120141296Sdas * -1.0 / (x+r) here 121141296Sdas */ 122141296Sdas /* compute -1.0 / (x+r) accurately */ 123141296Sdas double a, t; 124141296Sdas z = w; 125141296Sdas SET_LOW_WORD(z,0); 126141296Sdas v = r - (z - x); /* z+v = r+x */ 127141296Sdas t = a = -1.0 / w; /* a = -1.0/w */ 128141296Sdas SET_LOW_WORD(t,0); 129141296Sdas s = 1.0 + t * z; 130141296Sdas return t + a * (s + t * v); 1312116Sjkh } 1322116Sjkh} 133