11dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* @(#)k_tan.c 1.5 04/04/22 SMI */ 21dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 31dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* 41dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * ==================================================== 51dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved. 61dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 71dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Permission to use, copy, modify, and distribute this 81dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * software is freely granted, provided that this notice 91dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * is preserved. 101dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * ==================================================== 111dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 121dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 131dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* INDENT OFF */ 14a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes#include <sys/cdefs.h> 15a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes__FBSDID("$FreeBSD$"); 161dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 171dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* __kernel_tan( x, y, k ) 181dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854 191dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Input x is assumed to be bounded by ~pi/4 in magnitude. 201dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Input y is the tail of x. 211dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned. 221dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 231dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Algorithm 241dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 1. Since tan(-x) = -tan(x), we need only to consider positive x. 251dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 2. Callers must return tan(-0) = -0 without calling here since our 261dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * odd polynomial is not evaluated in a way that preserves -0. 271dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Callers may do the optimization tan(x) ~ x for tiny x. 281dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 3. tan(x) is approximated by a odd polynomial of degree 27 on 291dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * [0,0.67434] 301dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 3 27 311dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * tan(x) ~ x + T1*x + ... + T13*x 321dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * where 331dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 341dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * |tan(x) 2 4 26 | -59.2 351dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 361dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * | x | 371dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 381dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Note: tan(x+y) = tan(x) + tan'(x)*y 391dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * ~ tan(x) + (1+x*x)*y 401dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Therefore, for better accuracy in computing tan(x+y), let 411dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 3 2 2 2 2 421dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) 431dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * then 441dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 3 2 451dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * tan(x+y) = x + (T1*x + (x *(r+y)+y)) 461dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 471dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then 481dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) 491dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) 501dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 511dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 521dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#include "math.h" 531dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#include "math_private.h" 541dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectstatic const double xxx[] = { 551dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ 561dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ 571dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ 581dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ 591dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ 601dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ 611dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ 621dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ 631dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ 641dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ 651dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ 661dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ 671dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ 681dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ 691dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ 701dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ 711dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project}; 721dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#define one xxx[13] 731dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#define pio4 xxx[14] 741dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#define pio4lo xxx[15] 751dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#define T xxx 761dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* INDENT ON */ 771dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 781dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectdouble 791dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project__kernel_tan(double x, double y, int iy) { 801dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project double z, r, v, w, s; 811dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project int32_t ix, hx; 821dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 831dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project GET_HIGH_WORD(hx,x); 841dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project ix = hx & 0x7fffffff; /* high word of |x| */ 851dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ 861dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if (hx < 0) { 871dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project x = -x; 881dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project y = -y; 891dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 901dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project z = pio4 - x; 911dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project w = pio4lo - y; 921dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project x = z + w; 931dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project y = 0.0; 941dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 951dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project z = x * x; 961dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project w = z * z; 971dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project /* 981dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Break x^5*(T[1]+x^2*T[2]+...) into 991dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 1001dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 1011dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 1021dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + 1031dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project w * T[11])))); 1041dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + 1051dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project w * T[12]))))); 1061dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project s = z * x; 1071dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project r = y + z * (s * (r + v) + y); 1081dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project r += T[0] * s; 1091dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project w = x + r; 1101dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if (ix >= 0x3FE59428) { 1111dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project v = (double) iy; 1121dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return (double) (1 - ((hx >> 30) & 2)) * 1131dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project (v - 2.0 * (x - (w * w / (w + v) - r))); 1141dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 1151dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if (iy == 1) 1161dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return w; 1171dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project else { 1181dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project /* 1191dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * if allow error up to 2 ulp, simply return 1201dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * -1.0 / (x+r) here 1211dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 1221dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project /* compute -1.0 / (x+r) accurately */ 1231dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project double a, t; 1241dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project z = w; 1251dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project SET_LOW_WORD(z,0); 1261dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project v = r - (z - x); /* z+v = r+x */ 1271dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project t = a = -1.0 / w; /* a = -1.0/w */ 1281dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project SET_LOW_WORD(t,0); 1291dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project s = 1.0 + t * z; 1301dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return t + a * (s + t * v); 1311dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 1321dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project} 133