/*
* ====================================================
* 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 <
[email protected]>
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, see
<
http://www.gnu.org/licenses/>. */
/* __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)
* 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))
* 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 "quadmath-imp.h"
static const __float128
one = 1,
huge = 1.0e+4932Q,
pio2_hi = 1.5707963267948966192313216916397514420986Q,
pio2_lo = 4.3359050650618905123985220130216759843812E-35Q,
pio4_hi = 7.8539816339744830961566084581987569936977E-1Q,
/* 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 = -8.358099012470680544198472400254596543711E2Q,
pS1 = 3.674973957689619490312782828051860366493E3Q,
pS2 = -6.730729094812979665807581609853656623219E3Q,
pS3 = 6.643843795209060298375552684423454077633E3Q,
pS4 = -3.817341990928606692235481812252049415993E3Q,
pS5 = 1.284635388402653715636722822195716476156E3Q,
pS6 = -2.410736125231549204856567737329112037867E2Q,
pS7 = 2.219191969382402856557594215833622156220E1Q,
pS8 = -7.249056260830627156600112195061001036533E-1Q,
pS9 = 1.055923570937755300061509030361395604448E-3Q,
qS0 = -5.014859407482408326519083440151745519205E3Q,
qS1 = 2.430653047950480068881028451580393430537E4Q,
qS2 = -4.997904737193653607449250593976069726962E4Q,
qS3 = 5.675712336110456923807959930107347511086E4Q,
qS4 = -3.881523118339661268482937768522572588022E4Q,
qS5 = 1.634202194895541569749717032234510811216E4Q,
qS6 = -4.151452662440709301601820849901296953752E3Q,
qS7 = 5.956050864057192019085175976175695342168E2Q,
qS8 = -4.175375777334867025769346564600396877176E1Q,
/* 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 = -5.619049346208901520945464704848780243887E0Q,
rS1 = 4.460504162777731472539175700169871920352E1Q,
rS2 = -1.317669505315409261479577040530751477488E2Q,
rS3 = 1.626532582423661989632442410808596009227E2Q,
rS4 = -3.144806644195158614904369445440583873264E1Q,
rS5 = -9.806674443470740708765165604769099559553E1Q,
rS6 = 5.708468492052010816555762842394927806920E1Q,
rS7 = 1.396540499232262112248553357962639431922E1Q,
rS8 = -1.126243289311910363001762058295832610344E1Q,
rS9 = -4.956179821329901954211277873774472383512E-1Q,
rS10 = 3.313227657082367169241333738391762525780E-1Q,
sS0 = -4.645814742084009935700221277307007679325E0Q,
sS1 = 3.879074822457694323970438316317961918430E1Q,
sS2 = -1.221986588013474694623973554726201001066E2Q,
sS3 = 1.658821150347718105012079876756201905822E2Q,
sS4 = -4.804379630977558197953176474426239748977E1Q,
sS5 = -1.004296417397316948114344573811562952793E2Q,
sS6 = 7.530281592861320234941101403870010111138E1Q,
sS7 = 1.270735595411673647119592092304357226607E1Q,
sS8 = -1.815144839646376500705105967064792930282E1Q,
sS9 = -7.821597334910963922204235247786840828217E-2Q,
/* 1.000000000000000000000000000000000000000E0 */
asinr5625 = 5.9740641664535021430381036628424864397707E-1Q;
__float128
asinq (__float128 x)
{
__float128 t, w, p, q, c, r, s;
int32_t ix, sign, flag;
ieee854_float128 u;
flag = 0;
u.value = x;
sign = u.words32.w0;
ix = sign & 0x7fffffff;
u.words32.w0 = ix; /* |x| */
if (ix >= 0x3fff0000) /* |x|>= 1 */
{
if (ix == 0x3fff0000
&& (u.words32.w1 | u.words32.w2 | u.words32.w3) == 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 < 0x3ffe0000) /* |x| < 0.5 */
{
if (ix < 0x3fc60000) /* |x| < 2**-57 */
{
math_check_force_underflow (x);
__float128 force_inexact = huge + x;
math_force_eval (force_inexact);
return x; /* return x with inexact if x!=0 */
}
else
{
t = x * x;
/* Mark to use pS, qS later on. */
flag = 1;
}
}
else if (ix < 0x3ffe4000) /* 0.625 */
{
t = u.value - 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;
if ((sign & 0x80000000) == 0)
return t;
else
return -t;
}
else
{
/* 1 > |x| >= 0.625 */
w = one - u.value;
t = w * 0.5;
}
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;
if (flag) /* 2^-57 < |x| < 0.5 */
{
w = p / q;
return x + x * w;
}
s = sqrtq (t);
if (ix >= 0x3ffef333) /* |x| > 0.975 */
{
w = p / q;
t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
}
else
{
u.value = s;
u.words32.w3 = 0;
u.words32.w2 = 0;
w = u.value;
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 ((sign & 0x80000000) == 0)
return t;
else
return -t;
}
