/* origin: FreeBSD /usr/src/lib/msun/src/k_tan.c */ /* * ==================================================== * 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. * ==================================================== */ /* __tan( x, y, k ) * kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854 * Input x is assumed to be bounded by ~pi/4 in magnitude. * Input y is the tail of x. * Input odd indicates whether tan (if odd = 0) or -1/tan (if odd = 1) is returned. * * Algorithm * 1. Since tan(-x) = -tan(x), we need only to consider positive x. * 2. Callers must return tan(-0) = -0 without calling here since our * odd polynomial is not evaluated in a way that preserves -0. * Callers may do the optimization tan(x) ~ x for tiny x. * 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 "libm.h" static const double T[] = { 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 */ }, pio4 = 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ pio4lo = 3.06161699786838301793e-17; /* 3C81A626, 33145C07 */ double __tan(double x, double y, int odd) { double_t z, r, v, w, s, a; double w0, a0; uint32_t hx; int big, sign; GET_HIGH_WORD(hx,x); big = (hx&0x7fffffff) >= 0x3FE59428; /* |x| >= 0.6744 */ if (big) { sign = hx>>31; if (sign) { x = -x; y = -y; } x = (pio4 - x) + (pio4lo - y); 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) + s*T[0]; w = x + r; if (big) { s = 1 - 2*odd; v = s - 2.0 * (x + (r - w*w/(w + s))); return sign ? -v : v; } if (!odd) return w; /* -1.0/(x+r) has up to 2ulp error, so compute it accurately */ w0 = w; SET_LOW_WORD(w0, 0); v = r - (w0 - x); /* w0+v = r+x */ a0 = a = -1.0 / w; SET_LOW_WORD(a0, 0); return a0 + a*(1.0 + a0*w0 + a0*v); }