262 lines
7.6 KiB
C
262 lines
7.6 KiB
C
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/*
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Long double expansions are
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Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
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and are incorporated herein by permission of the author. The author
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reserves the right to distribute this material elsewhere under different
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copying permissions. These modifications are distributed here under the
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following terms:
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This library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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This library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with this library; if not, see
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<https://www.gnu.org/licenses/>. */
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/* __ieee754_asin(x)
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* Method :
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* Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
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* we approximate asin(x) on [0,0.5] by
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* asin(x) = x + x*x^2*R(x^2)
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* Between .5 and .625 the approximation is
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* asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x)
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* For x in [0.625,1]
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* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
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* Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
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* then for x>0.98
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* asin(x) = pi/2 - 2*(s+s*z*R(z))
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* = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
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* For x<=0.98, let pio4_hi = pio2_hi/2, then
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* f = hi part of s;
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* c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
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* and
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* asin(x) = pi/2 - 2*(s+s*z*R(z))
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* = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
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* = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
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*
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* Special cases:
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* if x is NaN, return x itself;
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* if |x|>1, return NaN with invalid signal.
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*
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*/
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#include <float.h>
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#include <math.h>
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#include <math-barriers.h>
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#include <math_private.h>
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#include <math-underflow.h>
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#include <libm-alias-finite.h>
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static const _Float128
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one = 1,
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huge = L(1.0e+4932),
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pio2_hi = L(1.5707963267948966192313216916397514420986),
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pio2_lo = L(4.3359050650618905123985220130216759843812E-35),
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pio4_hi = L(7.8539816339744830961566084581987569936977E-1),
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/* coefficient for R(x^2) */
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/* asin(x) = x + x^3 pS(x^2) / qS(x^2)
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0 <= x <= 0.5
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peak relative error 1.9e-35 */
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pS0 = L(-8.358099012470680544198472400254596543711E2),
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pS1 = L(3.674973957689619490312782828051860366493E3),
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pS2 = L(-6.730729094812979665807581609853656623219E3),
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pS3 = L(6.643843795209060298375552684423454077633E3),
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pS4 = L(-3.817341990928606692235481812252049415993E3),
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pS5 = L(1.284635388402653715636722822195716476156E3),
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pS6 = L(-2.410736125231549204856567737329112037867E2),
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pS7 = L(2.219191969382402856557594215833622156220E1),
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pS8 = L(-7.249056260830627156600112195061001036533E-1),
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pS9 = L(1.055923570937755300061509030361395604448E-3),
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qS0 = L(-5.014859407482408326519083440151745519205E3),
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qS1 = L(2.430653047950480068881028451580393430537E4),
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qS2 = L(-4.997904737193653607449250593976069726962E4),
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qS3 = L(5.675712336110456923807959930107347511086E4),
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qS4 = L(-3.881523118339661268482937768522572588022E4),
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qS5 = L(1.634202194895541569749717032234510811216E4),
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qS6 = L(-4.151452662440709301601820849901296953752E3),
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qS7 = L(5.956050864057192019085175976175695342168E2),
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qS8 = L(-4.175375777334867025769346564600396877176E1),
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/* 1.000000000000000000000000000000000000000E0 */
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/* asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x)
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-0.0625 <= x <= 0.0625
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peak relative error 3.3e-35 */
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rS0 = L(-5.619049346208901520945464704848780243887E0),
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rS1 = L(4.460504162777731472539175700169871920352E1),
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rS2 = L(-1.317669505315409261479577040530751477488E2),
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rS3 = L(1.626532582423661989632442410808596009227E2),
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rS4 = L(-3.144806644195158614904369445440583873264E1),
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rS5 = L(-9.806674443470740708765165604769099559553E1),
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rS6 = L(5.708468492052010816555762842394927806920E1),
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rS7 = L(1.396540499232262112248553357962639431922E1),
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rS8 = L(-1.126243289311910363001762058295832610344E1),
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rS9 = L(-4.956179821329901954211277873774472383512E-1),
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rS10 = L(3.313227657082367169241333738391762525780E-1),
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sS0 = L(-4.645814742084009935700221277307007679325E0),
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sS1 = L(3.879074822457694323970438316317961918430E1),
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sS2 = L(-1.221986588013474694623973554726201001066E2),
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sS3 = L(1.658821150347718105012079876756201905822E2),
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sS4 = L(-4.804379630977558197953176474426239748977E1),
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sS5 = L(-1.004296417397316948114344573811562952793E2),
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sS6 = L(7.530281592861320234941101403870010111138E1),
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sS7 = L(1.270735595411673647119592092304357226607E1),
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sS8 = L(-1.815144839646376500705105967064792930282E1),
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sS9 = L(-7.821597334910963922204235247786840828217E-2),
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/* 1.000000000000000000000000000000000000000E0 */
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asinr5625 = L(5.9740641664535021430381036628424864397707E-1);
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_Float128
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__ieee754_asinl (_Float128 x)
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{
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_Float128 t, w, p, q, c, r, s;
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int32_t ix, sign, flag;
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ieee854_long_double_shape_type u;
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flag = 0;
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u.value = x;
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sign = u.parts32.w0;
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ix = sign & 0x7fffffff;
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u.parts32.w0 = ix; /* |x| */
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if (ix >= 0x3fff0000) /* |x|>= 1 */
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{
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if (ix == 0x3fff0000
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&& (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
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/* asin(1)=+-pi/2 with inexact */
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return x * pio2_hi + x * pio2_lo;
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return (x - x) / (x - x); /* asin(|x|>1) is NaN */
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}
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else if (ix < 0x3ffe0000) /* |x| < 0.5 */
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{
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if (ix < 0x3fc60000) /* |x| < 2**-57 */
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{
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math_check_force_underflow (x);
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_Float128 force_inexact = huge + x;
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math_force_eval (force_inexact);
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return x; /* return x with inexact if x!=0 */
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}
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else
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{
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t = x * x;
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/* Mark to use pS, qS later on. */
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flag = 1;
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}
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}
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else if (ix < 0x3ffe4000) /* 0.625 */
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{
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t = u.value - 0.5625;
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p = ((((((((((rS10 * t
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+ rS9) * t
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+ rS8) * t
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+ rS7) * t
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+ rS6) * t
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+ rS5) * t
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+ rS4) * t
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+ rS3) * t
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+ rS2) * t
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+ rS1) * t
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+ rS0) * t;
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q = ((((((((( t
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+ sS9) * t
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+ sS8) * t
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+ sS7) * t
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+ sS6) * t
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+ sS5) * t
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+ sS4) * t
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+ sS3) * t
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+ sS2) * t
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+ sS1) * t
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+ sS0;
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t = asinr5625 + p / q;
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if ((sign & 0x80000000) == 0)
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return t;
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else
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return -t;
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}
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else
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{
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/* 1 > |x| >= 0.625 */
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w = one - u.value;
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t = w * 0.5;
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}
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p = (((((((((pS9 * t
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+ pS8) * t
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+ pS7) * t
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+ pS6) * t
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+ pS5) * t
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+ pS4) * t
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+ pS3) * t
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+ pS2) * t
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+ pS1) * t
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+ pS0) * t;
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q = (((((((( t
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+ qS8) * t
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+ qS7) * t
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+ qS6) * t
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+ qS5) * t
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+ qS4) * t
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+ qS3) * t
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+ qS2) * t
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+ qS1) * t
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+ qS0;
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if (flag) /* 2^-57 < |x| < 0.5 */
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{
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w = p / q;
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return x + x * w;
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}
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s = sqrtl (t);
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if (ix >= 0x3ffef333) /* |x| > 0.975 */
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{
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w = p / q;
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t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
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}
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else
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{
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u.value = s;
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u.parts32.w3 = 0;
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u.parts32.w2 = 0;
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w = u.value;
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c = (t - w * w) / (s + w);
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r = p / q;
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p = 2.0 * s * r - (pio2_lo - 2.0 * c);
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q = pio4_hi - 2.0 * w;
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t = pio4_hi - (p - q);
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}
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if ((sign & 0x80000000) == 0)
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return t;
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else
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return -t;
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}
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libm_alias_finite (__ieee754_asinl, __asinl)
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