| /* 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); |
| } |
