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/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/*
Long double expansions are
Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
and are incorporated herein by permission of the author. The author
reserves the right to distribute this material elsewhere under different
copying permissions. These modifications are distributed here under
the following terms:
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with this library; if not, write to the Free Software
Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */
/* __ieee754_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)
*
* 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.
*
*/
#include "math.h"
#include "math_private.h"
#ifdef __STDC__
static const long double
#else
static long double
#endif
one = 1.0L,
huge = 1.0e+4932L,
pio2_hi = 1.5707963267948966192021943710788178805159986950457096099853515625L,
pio2_lo = 2.9127320560933561582586004641843300502121E-20L,
pio4_hi = 7.8539816339744830960109718553940894025800E-1L,
/* coefficient for R(x^2) */
/* asin(x) = x + x^3 pS(x^2) / qS(x^2)
0 <= x <= 0.5
peak relative error 1.9e-21 */
pS0 = -1.008714657938491626019651170502036851607E1L,
pS1 = 2.331460313214179572063441834101394865259E1L,
pS2 = -1.863169762159016144159202387315381830227E1L,
pS3 = 5.930399351579141771077475766877674661747E0L,
pS4 = -6.121291917696920296944056882932695185001E-1L,
pS5 = 3.776934006243367487161248678019350338383E-3L,
qS0 = -6.052287947630949712886794360635592886517E1L,
qS1 = 1.671229145571899593737596543114258558503E2L,
qS2 = -1.707840117062586426144397688315411324388E2L,
qS3 = 7.870295154902110425886636075950077640623E1L,
qS4 = -1.568433562487314651121702982333303458814E1L;
/* 1.000000000000000000000000000000000000000E0 */
#ifdef __STDC__
long double
__ieee754_asinl (long double x)
#else
double
__ieee754_asinl (x)
long double x;
#endif
{
long double t, w, p, q, c, r, s;
int32_t ix;
u_int32_t se, i0, i1, k;
GET_LDOUBLE_WORDS (se, i0, i1, x);
ix = se & 0x7fff;
ix = (ix << 16) | (i0 >> 16);
if (ix >= 0x3fff8000)
{ /* |x|>= 1 */
if (ix == 0x3fff8000 && ((i0 - 0x80000000) | i1) == 0)
/* asin(1)=+-pi/2 with inexact */
return x * pio2_hi + x * pio2_lo;
return (x - x) / (x - x); /* asin(|x|>1) is NaN */
}
else if (ix < 0x3ffe8000)
{ /* |x|<0.5 */
if (ix < 0x3fde8000)
{ /* if |x| < 2**-33 */
if (huge + x > one)
return x; /* return x with inexact if x!=0 */
}
else
{
t = x * x;
p =
t * (pS0 +
t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
q = qS0 + t * (qS1 + t * (qS2 + t * (qS3 + t * (qS4 + t))));
w = p / q;
return x + x * w;
}
}
/* 1> |x|>= 0.5 */
w = one - fabsl (x);
t = w * 0.5;
p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
q = qS0 + t * (qS1 + t * (qS2 + t * (qS3 + t * (qS4 + t))));
s = __ieee754_sqrtl (t);
if (ix >= 0x3ffef999)
{ /* if |x| > 0.975 */
w = p / q;
t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
}
else
{
GET_LDOUBLE_WORDS (k, i0, i1, s);
i1 = 0;
SET_LDOUBLE_WORDS (w,k,i0,i1);
c = (t - w * w) / (s + w);
r = p / q;
p = 2.0 * s * r - (pio2_lo - 2.0 * c);
q = pio4_hi - 2.0 * w;
t = pio4_hi - (p - q);
}
if ((se & 0x8000) == 0)
return t;
else
return -t;
}
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