/*-*- mode:c;indent-tabs-mode:t;c-basic-offset:8;tab-width:8;coding:utf-8 -*-│ │vi: set et ft=c ts=8 tw=8 fenc=utf-8 :vi│ ╚──────────────────────────────────────────────────────────────────────────────╝ │ │ │ Musl Libc │ │ Copyright © 2005-2014 Rich Felker, et al. │ │ │ │ Permission is hereby granted, free of charge, to any person obtaining │ │ a copy of this software and associated documentation files (the │ │ "Software"), to deal in the Software without restriction, including │ │ without limitation the rights to use, copy, modify, merge, publish, │ │ distribute, sublicense, and/or sell copies of the Software, and to │ │ permit persons to whom the Software is furnished to do so, subject to │ │ the following conditions: │ │ │ │ The above copyright notice and this permission notice shall be │ │ included in all copies or substantial portions of the Software. │ │ │ │ THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, │ │ EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF │ │ MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. │ │ IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY │ │ CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, │ │ TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE │ │ SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. │ │ │ ╚─────────────────────────────────────────────────────────────────────────────*/ #include "libc/math.h" #include "libc/tinymath/kernel.internal.h" asm(".ident\t\"\\n\\n\ fdlibm (fdlibm license)\\n\ Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.\""); asm(".ident\t\"\\n\\n\ Musl libc (MIT License)\\n\ Copyright 2005-2014 Rich Felker, et. al.\""); asm(".include \"libc/disclaimer.inc\""); /* clang-format off */ /* 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))) */ #define ASUINT64(f) ((union{double _f; uint64_t _i;}){f})._i #define ASDOUBLE(i) ((union{uint64_t _i; double _f;}){i})._f #define GET_HIGH_WORD(hi,d) (hi) = ASUINT64(d) >> 32 #define GET_LOW_WORD(lo,d) (lo) = (uint32_t)ASUINT64(d) #define SET_LOW_WORD(d,lo) INSERT(d, ASUINT64(d)>>32, lo) #define INSERT(d,hi,lo) (d)=ASDOUBLE((uint64_t)(hi)<<32|(uint32_t)(lo)) 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); }