k_tan.c revision 151969
1177633Sdfr/* @(#)k_tan.c 1.5 04/04/22 SMI */ 2177633Sdfr 3177633Sdfr/* 4177633Sdfr * ==================================================== 5177633Sdfr * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved. 6177633Sdfr * 7177633Sdfr * Permission to use, copy, modify, and distribute this 8177633Sdfr * software is freely granted, provided that this notice 9177633Sdfr * is preserved. 10177633Sdfr * ==================================================== 11177633Sdfr */ 12177633Sdfr 13177633Sdfr/* INDENT OFF */ 14177633Sdfr#ifndef lint 15177633Sdfrstatic char rcsid[] = "$FreeBSD: head/lib/msun/src/k_tan.c 151969 2005-11-02 14:01:45Z bde $"; 16177633Sdfr#endif 17177633Sdfr 18177633Sdfr/* __kernel_tan( x, y, k ) 19177633Sdfr * kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854 20177633Sdfr * Input x is assumed to be bounded by ~pi/4 in magnitude. 21177633Sdfr * Input y is the tail of x. 22177633Sdfr * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned. 23177633Sdfr * 24177633Sdfr * Algorithm 25177633Sdfr * 1. Since tan(-x) = -tan(x), we need only to consider positive x. 26177633Sdfr * 2. Callers must return tan(-0) = -0 without calling here since our 27177633Sdfr * odd polynomial is not evaluated in a way that preserves -0. 28177633Sdfr * Callers may do the optimization tan(x) ~ x for tiny x. 29177633Sdfr * 3. tan(x) is approximated by a odd polynomial of degree 27 on 30177633Sdfr * [0,0.67434] 31177633Sdfr * 3 27 32177633Sdfr * tan(x) ~ x + T1*x + ... + T13*x 33177633Sdfr * where 34177633Sdfr * 35177633Sdfr * |tan(x) 2 4 26 | -59.2 36177633Sdfr * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 37177633Sdfr * | x | 38177633Sdfr * 39177633Sdfr * Note: tan(x+y) = tan(x) + tan'(x)*y 40177633Sdfr * ~ tan(x) + (1+x*x)*y 41177633Sdfr * Therefore, for better accuracy in computing tan(x+y), let 42177633Sdfr * 3 2 2 2 2 43177633Sdfr * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) 44177633Sdfr * then 45177633Sdfr * 3 2 46177633Sdfr * tan(x+y) = x + (T1*x + (x *(r+y)+y)) 47177633Sdfr * 48177633Sdfr * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then 49177633Sdfr * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) 50177633Sdfr * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) 51193650Srwatson */ 52177633Sdfr 53177633Sdfr#include "math.h" 54177633Sdfr#include "math_private.h" 55177633Sdfrstatic const double xxx[] = { 56184588Sdfr 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ 57184588Sdfr 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ 58184588Sdfr 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ 59184588Sdfr 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ 60184588Sdfr 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ 61184588Sdfr 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ 62184588Sdfr 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ 63177633Sdfr 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ 64177633Sdfr 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ 65177633Sdfr 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ 66177633Sdfr 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ 67177633Sdfr -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ 68177633Sdfr 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ 69177633Sdfr/* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ 70177633Sdfr/* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ 71177633Sdfr/* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ 72177633Sdfr}; 73177633Sdfr#define one xxx[13] 74177633Sdfr#define pio4 xxx[14] 75177633Sdfr#define pio4lo xxx[15] 76177633Sdfr#define T xxx 77177633Sdfr/* INDENT ON */ 78177633Sdfr 79177633Sdfrdouble 80177633Sdfr__kernel_tan(double x, double y, int iy) { 81177633Sdfr double z, r, v, w, s; 82177633Sdfr int32_t ix, hx; 83177633Sdfr 84177633Sdfr GET_HIGH_WORD(hx,x); 85177633Sdfr ix = hx & 0x7fffffff; /* high word of |x| */ 86177633Sdfr if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ 87177633Sdfr if (hx < 0) { 88184588Sdfr x = -x; 89184588Sdfr y = -y; 90177633Sdfr } 91177633Sdfr z = pio4 - x; 92177633Sdfr w = pio4lo - y; 93177633Sdfr x = z + w; 94177633Sdfr y = 0.0; 95177633Sdfr } 96177633Sdfr z = x * x; 97177633Sdfr w = z * z; 98177633Sdfr /* 99177633Sdfr * Break x^5*(T[1]+x^2*T[2]+...) into 100177633Sdfr * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 101184588Sdfr * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 102184588Sdfr */ 103184588Sdfr r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + 104184588Sdfr w * T[11])))); 105184588Sdfr v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + 106177633Sdfr w * T[12]))))); 107177633Sdfr s = z * x; 108177633Sdfr r = y + z * (s * (r + v) + y); 109177633Sdfr r += T[0] * s; 110177633Sdfr w = x + r; 111177633Sdfr if (ix >= 0x3FE59428) { 112177633Sdfr v = (double) iy; 113184588Sdfr return (double) (1 - ((hx >> 30) & 2)) * 114184588Sdfr (v - 2.0 * (x - (w * w / (w + v) - r))); 115184588Sdfr } 116184588Sdfr if (iy == 1) 117184588Sdfr return w; 118184588Sdfr else { 119184588Sdfr /* 120184588Sdfr * if allow error up to 2 ulp, simply return 121184588Sdfr * -1.0 / (x+r) here 122184588Sdfr */ 123184588Sdfr /* compute -1.0 / (x+r) accurately */ 124184588Sdfr double a, t; 125184588Sdfr z = w; 126184588Sdfr SET_LOW_WORD(z,0); 127184588Sdfr v = r - (z - x); /* z+v = r+x */ 128184588Sdfr t = a = -1.0 / w; /* a = -1.0/w */ 129184588Sdfr SET_LOW_WORD(t,0); 130184588Sdfr s = 1.0 + t * z; 131184588Sdfr return t + a * (s + t * v); 132184588Sdfr } 133184588Sdfr} 134184588Sdfr