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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.
* ====================================================
*/
#include "mathl.h"
/*
Long double expansions contributed by
Stephen L. Moshier <moshier@na-net.ornl.gov>
*/
/* 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)
* Between .5 and .625 the approximation is
* asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x)
* For x in [0.625,1]
* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN with invalid signal.
*
*/
#include <math.h>
static const long double
one = 1.0L,
huge = 1.0e+4932L,
pio2_hi = 1.5707963267948966192313216916397514420986L,
pio2_lo = 4.3359050650618905123985220130216759843812E-35L,
pio4_hi = 7.8539816339744830961566084581987569936977E-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-35 */
pS0 = -8.358099012470680544198472400254596543711E2L,
pS1 = 3.674973957689619490312782828051860366493E3L,
pS2 = -6.730729094812979665807581609853656623219E3L,
pS3 = 6.643843795209060298375552684423454077633E3L,
pS4 = -3.817341990928606692235481812252049415993E3L,
pS5 = 1.284635388402653715636722822195716476156E3L,
pS6 = -2.410736125231549204856567737329112037867E2L,
pS7 = 2.219191969382402856557594215833622156220E1L,
pS8 = -7.249056260830627156600112195061001036533E-1L,
pS9 = 1.055923570937755300061509030361395604448E-3L,
qS0 = -5.014859407482408326519083440151745519205E3L,
qS1 = 2.430653047950480068881028451580393430537E4L,
qS2 = -4.997904737193653607449250593976069726962E4L,
qS3 = 5.675712336110456923807959930107347511086E4L,
qS4 = -3.881523118339661268482937768522572588022E4L,
qS5 = 1.634202194895541569749717032234510811216E4L,
qS6 = -4.151452662440709301601820849901296953752E3L,
qS7 = 5.956050864057192019085175976175695342168E2L,
qS8 = -4.175375777334867025769346564600396877176E1L,
/* 1.000000000000000000000000000000000000000E0 */
/* asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x)
-0.0625 <= x <= 0.0625
peak relative error 3.3e-35 */
rS0 = -5.619049346208901520945464704848780243887E0L,
rS1 = 4.460504162777731472539175700169871920352E1L,
rS2 = -1.317669505315409261479577040530751477488E2L,
rS3 = 1.626532582423661989632442410808596009227E2L,
rS4 = -3.144806644195158614904369445440583873264E1L,
rS5 = -9.806674443470740708765165604769099559553E1L,
rS6 = 5.708468492052010816555762842394927806920E1L,
rS7 = 1.396540499232262112248553357962639431922E1L,
rS8 = -1.126243289311910363001762058295832610344E1L,
rS9 = -4.956179821329901954211277873774472383512E-1L,
rS10 = 3.313227657082367169241333738391762525780E-1L,
sS0 = -4.645814742084009935700221277307007679325E0L,
sS1 = 3.879074822457694323970438316317961918430E1L,
sS2 = -1.221986588013474694623973554726201001066E2L,
sS3 = 1.658821150347718105012079876756201905822E2L,
sS4 = -4.804379630977558197953176474426239748977E1L,
sS5 = -1.004296417397316948114344573811562952793E2L,
sS6 = 7.530281592861320234941101403870010111138E1L,
sS7 = 1.270735595411673647119592092304357226607E1L,
sS8 = -1.815144839646376500705105967064792930282E1L,
sS9 = -7.821597334910963922204235247786840828217E-2L,
/* 1.000000000000000000000000000000000000000E0 */
asinr5625 = 5.9740641664535021430381036628424864397707E-1L;
long double
asinl (long double x)
{
long double y, t, p, q;
int sign;
sign = 1;
y = x;
if (x < 0.0L)
{
sign = -1;
y = -x;
}
if (y >= 1.0L) /* |x|>= 1 */
{
if (y == 1.0L)
/* 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 (y < 0.5L) /* |x| < 0.5 */
{
if (y < 0.000000000000000006938893903907228377647697925567626953125L) /* |x| < 2**-57 */
if (huge + y > one)
return y; /* return x with inexact if x!=0 */
t = x * x;
p = (((((((((pS9 * t
+ pS8) * t
+ pS7) * t
+ pS6) * t
+ pS5) * t
+ pS4) * t
+ pS3) * t
+ pS2) * t
+ pS1) * t
+ pS0) * t;
q = (((((((( t
+ qS8) * t
+ qS7) * t
+ qS6) * t
+ qS5) * t
+ qS4) * t
+ qS3) * t
+ qS2) * t
+ qS1) * t
+ qS0;
return x + x * (p / q);
}
else if (y < 0.625) /* 0.625 */
{
t = y - 0.5625;
p = ((((((((((rS10 * t
+ rS9) * t
+ rS8) * t
+ rS7) * t
+ rS6) * t
+ rS5) * t
+ rS4) * t
+ rS3) * t
+ rS2) * t
+ rS1) * t
+ rS0) * t;
q = ((((((((( t
+ sS9) * t
+ sS8) * t
+ sS7) * t
+ sS6) * t
+ sS5) * t
+ sS4) * t
+ sS3) * t
+ sS2) * t
+ sS1) * t
+ sS0;
t = asinr5625 + p / q;
}
else
t = pio2_hi + pio2_lo - 2 * asinl(sqrtl((1-y)/2));
return t * sign;
}
#if 0
main()
{
printf ("%.18Lg %.18Lg\n",
asinl(1.0L),
1.5707963267948966192313216916397514420984L);
printf ("%.18Lg %.18Lg\n",
asinl(0.7071067811865475244008443621048490392848L),
0.7853981633974483096156608458198757210492L);
printf ("%.18Lg %.18Lg\n",
asinl(0.5L),
0.5235987755982988730771072305465838140328L);
printf ("%.18Lg %.18Lg\n",
asinl(0.3090169943749474241022934171828190588600L),
0.3141592653589793238462643383279502884196L);
printf ("%.18Lg %.18Lg\n",
asinl(-1.0L),
-1.5707963267948966192313216916397514420984L);
printf ("%.18Lg %.18Lg\n",
asinl(-0.7071067811865475244008443621048490392848L),
-0.7853981633974483096156608458198757210492L);
printf ("%.18Lg %.18Lg\n",
asinl(-0.5L),
-0.5235987755982988730771072305465838140328L);
printf ("%.18Lg %.18Lg\n",
asinl(-0.3090169943749474241022934171828190588600L),
-0.3141592653589793238462643383279502884196L);
}
#endif