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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 contributed by
Stephen L. Moshier <moshier@na-net.ornl.gov>
*/
/* __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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