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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.
- * ====================================================
- */
- /*
- __float128 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 */
- /* asinq(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 = 1.0Q,
- 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 = 0;
- __float128 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 */
- {
- if (huge + x > one)
- 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;
- }
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