lib/asinl.c

   1 /* arcsin (inverse sine) function with 'long double' argument.
   2
   3    Copyright (C) 2003-2024 Free Software Foundation, Inc.
   4
   5    This file is free software: you can redistribute it and/or modify
   6    it under the terms of the GNU Lesser General Public License as
   7    published by the Free Software Foundation, either version 3 of the
   8    License, or (at your option) any later version.
   9
  10    This file is distributed in the hope that it will be useful,
  11    but WITHOUT ANY WARRANTY; without even the implied warranty of
  12    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  13    GNU Lesser General Public License for more details.
  14
  15    You should have received a copy of the GNU Lesser General Public License
  16    along with this program.  If not, see <https://www.gnu.org/licenses/>.  */
  17
  18 /*
  19  * ====================================================
  20  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  21  *
  22  * Developed at SunPro, a Sun Microsystems, Inc. business.
  23  * Permission to use, copy, modify, and distribute this
  24  * software is freely granted, provided that this notice
  25  * is preserved.
  26  * ====================================================
  27  */
  28
  29 #include <config.h>
  30
  31 /* Specification.  */
  32 #include <math.h>
  33
  34 #if HAVE_SAME_LONG_DOUBLE_AS_DOUBLE
  35
  36 long double
  37 asinl (long double x)
  38 {
  39   return asin (x);
  40 }
  41
  42 #else
  43
  44 /* Code based on glibc/sysdeps/ieee754/ldbl-128/e_asinl.c.  */
  45
  46 /*
  47   Long double expansions contributed by
  48   Stephen L. Moshier <moshier@na-net.ornl.gov>
  49 */
  50
  51 /* asin(x)
  52  * Method :
  53  *      Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
  54  *      we approximate asin(x) on [0,0.5] by
  55  *              asin(x) = x + x*x^2*R(x^2)
  56  *      Between .5 and .625 the approximation is
  57  *              asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x)
  58  *      For x in [0.625,1]
  59  *              asin(x) = pi/2-2*asin(sqrt((1-x)/2))
  60  *
  61  * Special cases:
  62  *      if x is NaN, return x itself;
  63  *      if |x|>1, return NaN with invalid signal.
  64  *
  65  */
  66
  67
  68 static const long double
  69   one = 1.0L,
  70   huge = 1.0e+4932L,
  71   pio2_hi = 1.5707963267948966192313216916397514420986L,
  72   pio2_lo = 4.3359050650618905123985220130216759843812E-35L,
  73   pio4_hi = 7.8539816339744830961566084581987569936977E-1L,
  74
  75         /* coefficient for R(x^2) */
  76
  77   /* asin(x) = x + x^3 pS(x^2) / qS(x^2)
  78      0 <= x <= 0.5
  79      peak relative error 1.9e-35  */
  80   pS0 = -8.358099012470680544198472400254596543711E2L,
  81   pS1 =  3.674973957689619490312782828051860366493E3L,
  82   pS2 = -6.730729094812979665807581609853656623219E3L,
  83   pS3 =  6.643843795209060298375552684423454077633E3L,
  84   pS4 = -3.817341990928606692235481812252049415993E3L,
  85   pS5 =  1.284635388402653715636722822195716476156E3L,
  86   pS6 = -2.410736125231549204856567737329112037867E2L,
  87   pS7 =  2.219191969382402856557594215833622156220E1L,
  88   pS8 = -7.249056260830627156600112195061001036533E-1L,
  89   pS9 =  1.055923570937755300061509030361395604448E-3L,
  90
  91   qS0 = -5.014859407482408326519083440151745519205E3L,
  92   qS1 =  2.430653047950480068881028451580393430537E4L,
  93   qS2 = -4.997904737193653607449250593976069726962E4L,
  94   qS3 =  5.675712336110456923807959930107347511086E4L,
  95   qS4 = -3.881523118339661268482937768522572588022E4L,
  96   qS5 =  1.634202194895541569749717032234510811216E4L,
  97   qS6 = -4.151452662440709301601820849901296953752E3L,
  98   qS7 =  5.956050864057192019085175976175695342168E2L,
  99   qS8 = -4.175375777334867025769346564600396877176E1L,
 100   /* 1.000000000000000000000000000000000000000E0 */
 101
 102   /* asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x)
 103      -0.0625 <= x <= 0.0625
 104      peak relative error 3.3e-35  */
 105   rS0 = -5.619049346208901520945464704848780243887E0L,
 106   rS1 =  4.460504162777731472539175700169871920352E1L,
 107   rS2 = -1.317669505315409261479577040530751477488E2L,
 108   rS3 =  1.626532582423661989632442410808596009227E2L,
 109   rS4 = -3.144806644195158614904369445440583873264E1L,
 110   rS5 = -9.806674443470740708765165604769099559553E1L,
 111   rS6 =  5.708468492052010816555762842394927806920E1L,
 112   rS7 =  1.396540499232262112248553357962639431922E1L,
 113   rS8 = -1.126243289311910363001762058295832610344E1L,
 114   rS9 = -4.956179821329901954211277873774472383512E-1L,
 115   rS10 =  3.313227657082367169241333738391762525780E-1L,
 116
 117   sS0 = -4.645814742084009935700221277307007679325E0L,
 118   sS1 =  3.879074822457694323970438316317961918430E1L,
 119   sS2 = -1.221986588013474694623973554726201001066E2L,
 120   sS3 =  1.658821150347718105012079876756201905822E2L,
 121   sS4 = -4.804379630977558197953176474426239748977E1L,
 122   sS5 = -1.004296417397316948114344573811562952793E2L,
 123   sS6 =  7.530281592861320234941101403870010111138E1L,
 124   sS7 =  1.270735595411673647119592092304357226607E1L,
 125   sS8 = -1.815144839646376500705105967064792930282E1L,
 126   sS9 = -7.821597334910963922204235247786840828217E-2L,
 127   /*  1.000000000000000000000000000000000000000E0 */
 128
 129  asinr5625 =  5.9740641664535021430381036628424864397707E-1L;
 130
 131
 132 long double
 133 asinl (long double x)
 134 {
 135   long double y, t, p, q;
 136   int sign;
 137
 138   sign = 1;
 139   y = x;
 140   if (x < 0.0L)
 141     {
 142       sign = -1;
 143       y = -x;
 144     }
 145
 146   if (y >= 1.0L)        /* |x|>= 1 */
 147     {
 148       if (y == 1.0L)
 149         /* asin(1)=+-pi/2 with inexact */
 150         return x * pio2_hi + x * pio2_lo;
 151
 152       return (x - x) / (x - x); /* asin(|x|>1) is NaN */
 153     }
 154   else if (y < 0.5L) /* |x| < 0.5 */
 155     {
 156       if (y < 0.000000000000000006938893903907228377647697925567626953125L) /* |x| < 2**-57 */
 157         if (huge + y > one)
 158           return y;             /* return x with inexact if x!=0 */
 159
 160       t = x * x;
 161       p = (((((((((pS9 * t
 162                    + pS8) * t
 163                   + pS7) * t
 164                  + pS6) * t
 165                 + pS5) * t
 166                + pS4) * t
 167               + pS3) * t
 168              + pS2) * t
 169             + pS1) * t
 170            + pS0) * t;
 171
 172       q = (((((((( t
 173                   + qS8) * t
 174                  + qS7) * t
 175                 + qS6) * t
 176                + qS5) * t
 177               + qS4) * t
 178              + qS3) * t
 179             + qS2) * t
 180            + qS1) * t
 181         + qS0;
 182
 183       return x + x * (p / q);
 184     }
 185
 186   else if (y < 0.625) /* 0.625 */
 187     {
 188       t = y - 0.5625;
 189       p = ((((((((((rS10 * t
 190                     + rS9) * t
 191                    + rS8) * t
 192                   + rS7) * t
 193                  + rS6) * t
 194                 + rS5) * t
 195                + rS4) * t
 196               + rS3) * t
 197              + rS2) * t
 198             + rS1) * t
 199            + rS0) * t;
 200
 201       q = ((((((((( t
 202                     + sS9) * t
 203                   + sS8) * t
 204                  + sS7) * t
 205                 + sS6) * t
 206                + sS5) * t
 207               + sS4) * t
 208              + sS3) * t
 209             + sS2) * t
 210            + sS1) * t
 211         + sS0;
 212       t = asinr5625 + p / q;
 213     }
 214   else
 215     t = pio2_hi + pio2_lo - 2 * asinl (sqrtl ((1 - y) / 2));
 216
 217   return t * sign;
 218 }
 219
 220 #endif
 221
 222 #if 0
 223 int
 224 main (void)
 225 {
 226   printf ("%.18Lg %.18Lg\n",
 227           asinl (1.0L),
 228           1.5707963267948966192313216916397514420984L);
 229   printf ("%.18Lg %.18Lg\n",
 230           asinl (0.7071067811865475244008443621048490392848L),
 231           0.7853981633974483096156608458198757210492L);
 232   printf ("%.18Lg %.18Lg\n",
 233           asinl (0.5L),
 234           0.5235987755982988730771072305465838140328L);
 235   printf ("%.18Lg %.18Lg\n",
 236           asinl (0.3090169943749474241022934171828190588600L),
 237           0.3141592653589793238462643383279502884196L);
 238   printf ("%.18Lg %.18Lg\n",
 239           asinl (-1.0L),
 240           -1.5707963267948966192313216916397514420984L);
 241   printf ("%.18Lg %.18Lg\n",
 242           asinl (-0.7071067811865475244008443621048490392848L),
 243           -0.7853981633974483096156608458198757210492L);
 244   printf ("%.18Lg %.18Lg\n",
 245           asinl (-0.5L),
 246           -0.5235987755982988730771072305465838140328L);
 247   printf ("%.18Lg %.18Lg\n",
 248           asinl (-0.3090169943749474241022934171828190588600L),
 249           -0.3141592653589793238462643383279502884196L);
 250 }
 251 #endif