/*-*- mode:c;indent-tabs-mode:t;c-basic-offset:8;tab-width:8;coding:utf-8 -*-│
│ vi: set noet ft=c ts=8 sw=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"
__static_yoink("fdlibm_notice");
__static_yoink("musl_libc_notice");
__static_yoink("freebsd_libm_notice");
/* 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;
}
#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
__weak_reference(asin, asinl);
#endif