#pragma ident "@(#)k_tan.c 1.5 04/04/22 SMI" /* * ==================================================== * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved. * * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* INDENT OFF */ /* __kernel_tan( x, y, k ) * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 * Input x is assumed to be bounded by ~pi/4 in magnitude. * Input y is the tail of x. * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned. * * Algorithm * 1. Since tan(-x) = -tan(x), we need only to consider positive x. * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. * 3. tan(x) is approximated by a odd polynomial of degree 27 on * [0,0.67434] * 3 27 * tan(x) ~ x + T1*x + ... + T13*x * where * * |tan(x) 2 4 26 | -59.2 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 * | x | * * Note: tan(x+y) = tan(x) + tan'(x)*y * ~ tan(x) + (1+x*x)*y * Therefore, for better accuracy in computing tan(x+y), let * 3 2 2 2 2 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) * then * 3 2 * tan(x+y) = x + (T1*x + (x *(r+y)+y)) * * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) */ #include "fdlibm.h" static const double xxx[] = { 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ }; #define one xxx[13] #define pio4 xxx[14] #define pio4lo xxx[15] #define T xxx /* INDENT ON */ double __kernel_tan(double x, double y, int iy) { double z, r, v, w, s; int ix, hx; hx = __HI(x); /* high word of x */ ix = hx & 0x7fffffff; /* high word of |x| */ if (ix < 0x3e300000) { /* x < 2**-28 */ if ((int) x == 0) { /* generate inexact */ if (((ix | __LO(x)) | (iy + 1)) == 0) return one / fabs(x); else { if (iy == 1) return x; else { /* compute -1 / (x+y) carefully */ double a, t; z = w = x + y; __LO(z) = 0; v = y - (z - x); t = a = -one / w; __LO(t) = 0; s = one + t * z; return t + a * (s + t * v); } } } } if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ if (hx < 0) { x = -x; y = -y; } z = pio4 - x; w = pio4lo - y; x = z + w; y = 0.0; } z = x * x; w = z * z; /* * Break x^5*(T[1]+x^2*T[2]+...) into * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) */ r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11])))); v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12]))))); s = z * x; r = y + z * (s * (r + v) + y); r += T[0] * s; w = x + r; if (ix >= 0x3FE59428) { v = (double) iy; return (double) (1 - ((hx >> 30) & 2)) * (v - 2.0 * (x - (w * w / (w + v) - r))); } if (iy == 1) return w; else { /* * if allow error up to 2 ulp, simply return * -1.0 / (x+r) here */ /* compute -1.0 / (x+r) accurately */ double a, t; z = w; __LO(z) = 0; v = r - (z - x); /* z+v = r+x */ t = a = -1.0 / w; /* a = -1.0/w */ __LO(t) = 0; s = 1.0 + t * z; return t + a * (s + t * v); } }