1b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project#pragma ident "@(#)k_tan.c 1.5 04/04/22 SMI" 2b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 3b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project/* 4b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * ==================================================== 5b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved. 6b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 7b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Permission to use, copy, modify, and distribute this 8b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * software is freely granted, provided that this notice 9b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * is preserved. 10b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * ==================================================== 11b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project */ 12b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 13b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project/* INDENT OFF */ 14b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project/* __kernel_tan( x, y, k ) 15b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 16b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Input x is assumed to be bounded by ~pi/4 in magnitude. 17b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Input y is the tail of x. 18b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Input k indicates whether ieee_tan (if k = 1) or -1/tan (if k = -1) is returned. 19b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 20b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Algorithm 21b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 1. Since ieee_tan(-x) = -ieee_tan(x), we need only to consider positive x. 22b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. 23b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 3. ieee_tan(x) is approximated by a odd polynomial of degree 27 on 24b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * [0,0.67434] 25b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 3 27 26b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * tan(x) ~ x + T1*x + ... + T13*x 27b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * where 28b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 29b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * |ieee_tan(x) 2 4 26 | -59.2 30b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 31b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * | x | 32b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 33b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Note: ieee_tan(x+y) = ieee_tan(x) + tan'(x)*y 34b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * ~ ieee_tan(x) + (1+x*x)*y 35b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Therefore, for better accuracy in computing ieee_tan(x+y), let 36b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 3 2 2 2 2 37b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) 38b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * then 39b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 3 2 40b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * tan(x+y) = x + (T1*x + (x *(r+y)+y)) 41b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 42b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then 43b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * tan(x) = ieee_tan(pi/4-y) = (1-ieee_tan(y))/(1+ieee_tan(y)) 44b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * = 1 - 2*(ieee_tan(y) - (ieee_tan(y)^2)/(1+ieee_tan(y))) 45b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project */ 46b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 47b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project#include "fdlibm.h" 48b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 49b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Projectstatic const double xxx[] = { 50b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ 51b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ 52b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ 53b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ 54b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ 55b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ 56b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ 57b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ 58b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ 59b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ 60b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ 61b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ 62b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ 63b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project/* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ 64b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project/* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ 65b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project/* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ 66b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project}; 67b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project#define one xxx[13] 68b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project#define pio4 xxx[14] 69b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project#define pio4lo xxx[15] 70b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project#define T xxx 71b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project/* INDENT ON */ 72b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 73b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Projectdouble 74b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project__kernel_tan(double x, double y, int iy) { 75b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project double z, r, v, w, s; 76b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project int ix, hx; 77b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 78b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project hx = __HI(x); /* high word of x */ 79b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project ix = hx & 0x7fffffff; /* high word of |x| */ 80b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if (ix < 0x3e300000) { /* x < 2**-28 */ 81b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if ((int) x == 0) { /* generate inexact */ 82b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if (((ix | __LO(x)) | (iy + 1)) == 0) 83b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project return one / ieee_fabs(x); 84b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project else { 85b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if (iy == 1) 86b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project return x; 87b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project else { /* compute -1 / (x+y) carefully */ 88b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project double a, t; 89b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project 90b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project z = w = x + y; 91b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project __LO(z) = 0; 92b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project v = y - (z - x); 93b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project t = a = -one / w; 94b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project __LO(t) = 0; 95b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project s = one + t * z; 96b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project return t + a * (s + t * v); 97b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } 98b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } 99b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } 100b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } 101b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ 102b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if (hx < 0) { 103b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project x = -x; 104b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project y = -y; 105b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } 106b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project z = pio4 - x; 107b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project w = pio4lo - y; 108b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project x = z + w; 109b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project y = 0.0; 110b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } 111b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project z = x * x; 112b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project w = z * z; 113b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project /* 114b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * Break x^5*(T[1]+x^2*T[2]+...) into 115b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 116b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 117b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project */ 118b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + 119b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project w * T[11])))); 120b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + 121b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project w * T[12]))))); 122b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project s = z * x; 123b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project r = y + z * (s * (r + v) + y); 124b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project r += T[0] * s; 125b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project w = x + r; 126b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if (ix >= 0x3FE59428) { 127b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project v = (double) iy; 128b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project return (double) (1 - ((hx >> 30) & 2)) * 129b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project (v - 2.0 * (x - (w * w / (w + v) - r))); 130b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } 131b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project if (iy == 1) 132b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project return w; 133b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project else { 134b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project /* 135b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * if allow error up to 2 ulp, simply return 136b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project * -1.0 / (x+r) here 137b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project */ 138b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project /* compute -1.0 / (x+r) accurately */ 139b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project double a, t; 140b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project z = w; 141b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project __LO(z) = 0; 142b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project v = r - (z - x); /* z+v = r+x */ 143b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project t = a = -1.0 / w; /* a = -1.0/w */ 144b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project __LO(t) = 0; 145b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project s = 1.0 + t * z; 146b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project return t + a * (s + t * v); 147b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project } 148b07e1d9fd8d9e4e03698e0bd9bf77154c5390326The Android Open Source Project} 149