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asin.c
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/* Copyright JS Foundation and other contributors, http://js.foundation
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*
* This file is based on work under the following copyright and permission
* notice:
*
* 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.
*
* @(#)e_asin.c 1.3 95/01/18
*/
#include "jerry-math-internal.h"
/* 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.
*/
#define one 1.00000000000000000000e+00 /* 0x3FF00000, 0x00000000 */
#define huge 1.000e+300
#define pio2_hi 1.57079632679489655800e+00 /* 0x3FF921FB, 0x54442D18 */
#define pio2_lo 6.12323399573676603587e-17 /* 0x3C91A626, 0x33145C07 */
#define pio4_hi 7.85398163397448278999e-01 /* 0x3FE921FB, 0x54442D18 */
/* coefficient for R(x^2) */
#define pS0 1.66666666666666657415e-01 /* 0x3FC55555, 0x55555555 */
#define pS1 -3.25565818622400915405e-01 /* 0xBFD4D612, 0x03EB6F7D */
#define pS2 2.01212532134862925881e-01 /* 0x3FC9C155, 0x0E884455 */
#define pS3 -4.00555345006794114027e-02 /* 0xBFA48228, 0xB5688F3B */
#define pS4 7.91534994289814532176e-04 /* 0x3F49EFE0, 0x7501B288 */
#define pS5 3.47933107596021167570e-05 /* 0x3F023DE1, 0x0DFDF709 */
#define qS1 -2.40339491173441421878e+00 /* 0xC0033A27, 0x1C8A2D4B */
#define qS2 2.02094576023350569471e+00 /* 0x40002AE5, 0x9C598AC8 */
#define qS3 -6.88283971605453293030e-01 /* 0xBFE6066C, 0x1B8D0159 */
#define qS4 7.70381505559019352791e-02 /* 0x3FB3B8C5, 0xB12E9282 */
double
asin (double x)
{
double t, p, q, c, r, s;
double_accessor w;
int hx, ix;
hx = __HI (x);
ix = hx & 0x7fffffff;
if (ix >= 0x3ff00000) /* |x| >= 1 */
{
if (((ix - 0x3ff00000) | __LO (x)) == 0) /* asin(1) = +-pi/2 with inexact */
{
return x * pio2_hi + x * pio2_lo;
}
return NAN; /* asin(|x|>1) is NaN */
}
else if (ix < 0x3fe00000) /* |x| < 0.5 */
{
if (ix < 0x3e400000) /* if |x| < 2**-27 */
{
if (huge + x > one) /* return x with inexact if x != 0 */
{
return x;
}
}
t = x * x;
p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
w.dbl = p / q;
return x + x * w.dbl;
}
/* 1 > |x| >= 0.5 */
w.dbl = one - fabs (x);
t = w.dbl * 0.5;
p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
s = sqrt (t);
if (ix >= 0x3FEF3333) /* if |x| > 0.975 */
{
w.dbl = p / q;
t = pio2_hi - (2.0 * (s + s * w.dbl) - pio2_lo);
}
else
{
w.dbl = s;
w.as_int.lo = 0;
c = (t - w.dbl * w.dbl) / (s + w.dbl);
r = p / q;
p = 2.0 * s * r - (pio2_lo - 2.0 * c);
q = pio4_hi - 2.0 * w.dbl;
t = pio4_hi - (p - q);
}
if (hx > 0)
{
return t;
}
else
{
return -t;
}
} /* asin */
#undef one
#undef huge
#undef pio2_hi
#undef pio2_lo
#undef pio4_hi
#undef pS0
#undef pS1
#undef pS2
#undef pS3
#undef pS4
#undef pS5
#undef qS1
#undef qS2
#undef qS3
#undef qS4