lib/acosl.c

   1 /* arccos (inverse cosine) function with 'long double' argument.
   2
   3    Copyright (C) 2003-2025 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 acosl (long double x)
  38 {
  39   return acos (x);
  40 }
  41
  42 #else
  43
  44 /* Code based on glibc/sysdeps/ieee754/ldbl-128/e_asinl.c
  45    and           glibc/sysdeps/ieee754/ldbl-128/e_acosl.c.  */
  46
  47 /*
  48   Long double expansions contributed by
  49   Stephen L. Moshier <moshier@na-net.ornl.gov>
  50 */
  51
  52 /* asin(x)
  53  * Method :
  54  *      Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
  55  *      we approximate asin(x) on [0,0.5] by
  56  *              asin(x) = x + x*x^2*R(x^2)
  57  *      Between .5 and .625 the approximation is
  58  *              asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x)
  59  *      For x in [0.625,1]
  60  *              asin(x) = pi/2-2*asin(sqrt((1-x)/2))
  61  *
  62  * Special cases:
  63  *      if x is NaN, return x itself;
  64  *      if |x|>1, return NaN with invalid signal.
  65  *
  66  */
  67
  68
  69 static const long double
  70   one = 1.0L,
  71   huge = 1.0e+4932L,
  72   pi =      3.1415926535897932384626433832795028841972L,
  73   pio2_hi = 1.5707963267948966192313216916397514420986L,
  74   pio2_lo = 4.3359050650618905123985220130216759843812E-35L,
  75   pio4_hi = 7.8539816339744830961566084581987569936977E-1L,
  76
  77         /* coefficient for R(x^2) */
  78
  79   /* asin(x) = x + x^3 pS(x^2) / qS(x^2)
  80      0 <= x <= 0.5
  81      peak relative error 1.9e-35  */
  82   pS0 = -8.358099012470680544198472400254596543711E2L,
  83   pS1 =  3.674973957689619490312782828051860366493E3L,
  84   pS2 = -6.730729094812979665807581609853656623219E3L,
  85   pS3 =  6.643843795209060298375552684423454077633E3L,
  86   pS4 = -3.817341990928606692235481812252049415993E3L,
  87   pS5 =  1.284635388402653715636722822195716476156E3L,
  88   pS6 = -2.410736125231549204856567737329112037867E2L,
  89   pS7 =  2.219191969382402856557594215833622156220E1L,
  90   pS8 = -7.249056260830627156600112195061001036533E-1L,
  91   pS9 =  1.055923570937755300061509030361395604448E-3L,
  92
  93   qS0 = -5.014859407482408326519083440151745519205E3L,
  94   qS1 =  2.430653047950480068881028451580393430537E4L,
  95   qS2 = -4.997904737193653607449250593976069726962E4L,
  96   qS3 =  5.675712336110456923807959930107347511086E4L,
  97   qS4 = -3.881523118339661268482937768522572588022E4L,
  98   qS5 =  1.634202194895541569749717032234510811216E4L,
  99   qS6 = -4.151452662440709301601820849901296953752E3L,
 100   qS7 =  5.956050864057192019085175976175695342168E2L,
 101   qS8 = -4.175375777334867025769346564600396877176E1L,
 102   /* 1.000000000000000000000000000000000000000E0 */
 103
 104   /* asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x)
 105      -0.0625 <= x <= 0.0625
 106      peak relative error 3.3e-35  */
 107   rS0 = -5.619049346208901520945464704848780243887E0L,
 108   rS1 =  4.460504162777731472539175700169871920352E1L,
 109   rS2 = -1.317669505315409261479577040530751477488E2L,
 110   rS3 =  1.626532582423661989632442410808596009227E2L,
 111   rS4 = -3.144806644195158614904369445440583873264E1L,
 112   rS5 = -9.806674443470740708765165604769099559553E1L,
 113   rS6 =  5.708468492052010816555762842394927806920E1L,
 114   rS7 =  1.396540499232262112248553357962639431922E1L,
 115   rS8 = -1.126243289311910363001762058295832610344E1L,
 116   rS9 = -4.956179821329901954211277873774472383512E-1L,
 117   rS10 =  3.313227657082367169241333738391762525780E-1L,
 118
 119   sS0 = -4.645814742084009935700221277307007679325E0L,
 120   sS1 =  3.879074822457694323970438316317961918430E1L,
 121   sS2 = -1.221986588013474694623973554726201001066E2L,
 122   sS3 =  1.658821150347718105012079876756201905822E2L,
 123   sS4 = -4.804379630977558197953176474426239748977E1L,
 124   sS5 = -1.004296417397316948114344573811562952793E2L,
 125   sS6 =  7.530281592861320234941101403870010111138E1L,
 126   sS7 =  1.270735595411673647119592092304357226607E1L,
 127   sS8 = -1.815144839646376500705105967064792930282E1L,
 128   sS9 = -7.821597334910963922204235247786840828217E-2L,
 129   /*  1.000000000000000000000000000000000000000E0 */
 130
 131  asinr5625 =  5.9740641664535021430381036628424864397707E-1L;
 132
 133
 134 long double
 135 acosl (long double x)
 136 {
 137   long double t, p, q;
 138
 139   if (x < 0.0L)
 140     {
 141       t = pi - acosl (-x);
 142       if (huge + x > one) /* return with inexact */
 143         return t;
 144     }
 145
 146   if (x >= 1.0L)        /* |x|>= 1 */
 147     {
 148       if (x == 1.0L)
 149         return 0.0L;   /* return zero */
 150
 151       return (x - x) / (x - x); /* asin(|x|>1) is NaN */
 152     }
 153
 154   else if (x < 0.5L) /* |x| < 0.5 */
 155     {
 156       if (x < 0.000000000000000006938893903907228377647697925567626953125L) /* |x| < 2**-57 */
 157         /* acos(0)=+-pi/2 with inexact */
 158         return x * pio2_hi + x * pio2_lo;
 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 pio2_hi - (x + x * (p / q) - pio2_lo);
 184     }
 185
 186   else if (x < 0.625) /* 0.625 */
 187     {
 188       t = x - 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
 213       return (pio2_hi - asinr5625) - (p / q - pio2_lo);
 214     }
 215   else
 216     return 2 * asinl (sqrtl ((1 - x) / 2));
 217 }
 218
 219 #endif
 220
 221 #if 0
 222 int
 223 main (void)
 224 {
 225   printf ("%.18Lg %.18Lg\n",
 226           acosl (1.0L),
 227           1.5707963267948966192313216916397514420984L -
 228           1.5707963267948966192313216916397514420984L);
 229   printf ("%.18Lg %.18Lg\n",
 230           acosl (0.7071067811865475244008443621048490392848L),
 231           1.5707963267948966192313216916397514420984L -
 232           0.7853981633974483096156608458198757210492L);
 233   printf ("%.18Lg %.18Lg\n",
 234           acosl (0.5L),
 235           1.5707963267948966192313216916397514420984L -
 236           0.5235987755982988730771072305465838140328L);
 237   printf ("%.18Lg %.18Lg\n",
 238           acosl (0.3090169943749474241022934171828190588600L),
 239           1.5707963267948966192313216916397514420984L -
 240           0.3141592653589793238462643383279502884196L);
 241   printf ("%.18Lg %.18Lg\n",
 242           acosl (-1.0L),
 243           1.5707963267948966192313216916397514420984L -
 244           -1.5707963267948966192313216916397514420984L);
 245   printf ("%.18Lg %.18Lg\n",
 246           acosl (-0.7071067811865475244008443621048490392848L),
 247           1.5707963267948966192313216916397514420984L -
 248           -0.7853981633974483096156608458198757210492L);
 249   printf ("%.18Lg %.18Lg\n",
 250           acosl (-0.5L),
 251           1.5707963267948966192313216916397514420984L -
 252           -0.5235987755982988730771072305465838140328L);
 253   printf ("%.18Lg %.18Lg\n",
 254           acosl (-0.3090169943749474241022934171828190588600L),
 255           1.5707963267948966192313216916397514420984L -
 256           -0.3141592653589793238462643383279502884196L);
 257 }
 258 #endif