/*-*- 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 © 2005-2020 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" 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/e_asin.c */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* asin(x) * Method : * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... * we approximate asin(x) on [0,0.5] by * asin(x) = x + x*x^2*R(x^2) * where * R(x^2) is a rational approximation of (asin(x)-x)/x^3 * and its remez error is bounded by * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75) * * For x in [0.5,1] * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; * then for x>0.98 * asin(x) = pi/2 - 2*(s+s*z*R(z)) * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) * For x<=0.98, let pio4_hi = pio2_hi/2, then * f = hi part of s; * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) * and * asin(x) = pi/2 - 2*(s+s*z*R(z)) * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) * * Special cases: * if x is NaN, return x itself; * if |x|>1, return NaN with invalid signal. * */ static const double pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ /* coefficients for R(x^2) */ pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ static double R(double z) { double_t p, q; p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5))))); q = 1.0+z*(qS1+z*(qS2+z*(qS3+z*qS4))); return p/q; } /** * Returns arc sine of 𝑥. * * @return value in range [-M_PI/2,M_PI/2] * @return NAN if 𝑥 ∈ {NAN,+INFINITY,-INFINITY} * @return NAN if 𝑥 ∉ [-1,1] */ double asin(double x) { int ng; uint32_t ix,lx; double z,r,s,f,c; union { double f; int64_t s; uint64_t i; } u = {x}; ng = u.s < 0; ix = (u.i & 0x7fffffff00000000) >> 32; /* |x| >= 1 or nan */ if (ix >= 0x3ff00000) { lx = u.i; if (!(ix-0x3ff00000 | lx)) /* asin(1) = +-pi/2 with inexact */ return x*pio2_hi + 0x1p-120f; return 0/(x-x); } /* |x| < 0.5 */ if (ix < 0x3fe00000) { /* if 0x1p-1022 <= |x| < 0x1p-26, avoid raising underflow */ if (ix < 0x3e500000 && ix >= 0x00100000) return x; return x + x*R(x*x); } /* 1 > |x| >= 0.5 */ z = (1 - fabs(x))*0.5; s = sqrt(z); r = R(z); if (ix >= 0x3fef3333) { /* if |x| > 0.975 */ x = pio2_hi-(2*(s+s*r)-pio2_lo); } else { /* f+c = sqrt(z) */ u.f = s; u.i &= 0xffffffff00000000; f = u.f; c = (z-f*f)/(s+f); x = 0.5*pio2_hi - (2*s*r - (pio2_lo-2*c) - (0.5*pio2_hi-2*f)); } return ng ? -x : x; }