lib/acosl.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, a Sun Microsystems, Inc. business.
   6  * Permission to use, copy, modify, and distribute this
   7  * software is freely granted, provided that this notice
   8  * is preserved.
   9  * ====================================================
  10  */
  11
  12 #include <config.h>
  13
  14 /* Specification.  */
  15 #include <math.h>
  16
  17 #if HAVE_SAME_LONG_DOUBLE_AS_DOUBLE
  18
  19 long double
  20 acosl (long double x)
  21 {
  22   return acos (x);
  23 }
  24
  25 #else
  26
  27 /* Code based on glibc/sysdeps/ieee754/ldbl-128/e_asinl.c
  28    and           glibc/sysdeps/ieee754/ldbl-128/e_acosl.c.  */
  29
  30 /*
  31   Long double expansions contributed by
  32   Stephen L. Moshier <moshier@na-net.ornl.gov>
  33 */
  34
  35 /* asin(x)
  36  * Method :
  37  *      Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
  38  *      we approximate asin(x) on [0,0.5] by
  39  *              asin(x) = x + x*x^2*R(x^2)
  40  *      Between .5 and .625 the approximation is
  41  *              asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x)
  42  *      For x in [0.625,1]
  43  *              asin(x) = pi/2-2*asin(sqrt((1-x)/2))
  44  *
  45  * Special cases:
  46  *      if x is NaN, return x itself;
  47  *      if |x|>1, return NaN with invalid signal.
  48  *
  49  */
  50
  51
  52 static const long double
  53   one = 1.0L,
  54   huge = 1.0e+4932L,
  55   pi =      3.1415926535897932384626433832795028841972L,
  56   pio2_hi = 1.5707963267948966192313216916397514420986L,
  57   pio2_lo = 4.3359050650618905123985220130216759843812E-35L,
  58   pio4_hi = 7.8539816339744830961566084581987569936977E-1L,
  59
  60         /* coefficient for R(x^2) */
  61
  62   /* asin(x) = x + x^3 pS(x^2) / qS(x^2)
  63      0 <= x <= 0.5
  64      peak relative error 1.9e-35  */
  65   pS0 = -8.358099012470680544198472400254596543711E2L,
  66   pS1 =  3.674973957689619490312782828051860366493E3L,
  67   pS2 = -6.730729094812979665807581609853656623219E3L,
  68   pS3 =  6.643843795209060298375552684423454077633E3L,
  69   pS4 = -3.817341990928606692235481812252049415993E3L,
  70   pS5 =  1.284635388402653715636722822195716476156E3L,
  71   pS6 = -2.410736125231549204856567737329112037867E2L,
  72   pS7 =  2.219191969382402856557594215833622156220E1L,
  73   pS8 = -7.249056260830627156600112195061001036533E-1L,
  74   pS9 =  1.055923570937755300061509030361395604448E-3L,
  75
  76   qS0 = -5.014859407482408326519083440151745519205E3L,
  77   qS1 =  2.430653047950480068881028451580393430537E4L,
  78   qS2 = -4.997904737193653607449250593976069726962E4L,
  79   qS3 =  5.675712336110456923807959930107347511086E4L,
  80   qS4 = -3.881523118339661268482937768522572588022E4L,
  81   qS5 =  1.634202194895541569749717032234510811216E4L,
  82   qS6 = -4.151452662440709301601820849901296953752E3L,
  83   qS7 =  5.956050864057192019085175976175695342168E2L,
  84   qS8 = -4.175375777334867025769346564600396877176E1L,
  85   /* 1.000000000000000000000000000000000000000E0 */
  86
  87   /* asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x)
  88      -0.0625 <= x <= 0.0625
  89      peak relative error 3.3e-35  */
  90   rS0 = -5.619049346208901520945464704848780243887E0L,
  91   rS1 =  4.460504162777731472539175700169871920352E1L,
  92   rS2 = -1.317669505315409261479577040530751477488E2L,
  93   rS3 =  1.626532582423661989632442410808596009227E2L,
  94   rS4 = -3.144806644195158614904369445440583873264E1L,
  95   rS5 = -9.806674443470740708765165604769099559553E1L,
  96   rS6 =  5.708468492052010816555762842394927806920E1L,
  97   rS7 =  1.396540499232262112248553357962639431922E1L,
  98   rS8 = -1.126243289311910363001762058295832610344E1L,
  99   rS9 = -4.956179821329901954211277873774472383512E-1L,
 100   rS10 =  3.313227657082367169241333738391762525780E-1L,
 101
 102   sS0 = -4.645814742084009935700221277307007679325E0L,
 103   sS1 =  3.879074822457694323970438316317961918430E1L,
 104   sS2 = -1.221986588013474694623973554726201001066E2L,
 105   sS3 =  1.658821150347718105012079876756201905822E2L,
 106   sS4 = -4.804379630977558197953176474426239748977E1L,
 107   sS5 = -1.004296417397316948114344573811562952793E2L,
 108   sS6 =  7.530281592861320234941101403870010111138E1L,
 109   sS7 =  1.270735595411673647119592092304357226607E1L,
 110   sS8 = -1.815144839646376500705105967064792930282E1L,
 111   sS9 = -7.821597334910963922204235247786840828217E-2L,
 112   /*  1.000000000000000000000000000000000000000E0 */
 113
 114  asinr5625 =  5.9740641664535021430381036628424864397707E-1L;
 115
 116
 117 long double
 118 acosl (long double x)
 119 {
 120   long double t, p, q;
 121
 122   if (x < 0.0L)
 123     {
 124       t = pi - acosl (-x);
 125       if (huge + x > one) /* return with inexact */
 126         return t;
 127     }
 128
 129   if (x >= 1.0L)        /* |x|>= 1 */
 130     {
 131       if (x == 1.0L)
 132         return 0.0L;   /* return zero */
 133
 134       return (x - x) / (x - x); /* asin(|x|>1) is NaN */
 135     }
 136
 137   else if (x < 0.5L) /* |x| < 0.5 */
 138     {
 139       if (x < 0.000000000000000006938893903907228377647697925567626953125L) /* |x| < 2**-57 */
 140         /* acos(0)=+-pi/2 with inexact */
 141         return x * pio2_hi + x * pio2_lo;
 142
 143       t = x * x;
 144       p = (((((((((pS9 * t
 145                    + pS8) * t
 146                   + pS7) * t
 147                  + pS6) * t
 148                 + pS5) * t
 149                + pS4) * t
 150               + pS3) * t
 151              + pS2) * t
 152             + pS1) * t
 153            + pS0) * t;
 154
 155       q = (((((((( t
 156                   + qS8) * t
 157                  + qS7) * t
 158                 + qS6) * t
 159                + qS5) * t
 160               + qS4) * t
 161              + qS3) * t
 162             + qS2) * t
 163            + qS1) * t
 164         + qS0;
 165
 166       return pio2_hi - (x + x * (p / q) - pio2_lo);
 167     }
 168
 169   else if (x < 0.625) /* 0.625 */
 170     {
 171       t = x - 0.5625;
 172       p = ((((((((((rS10 * t
 173                     + rS9) * t
 174                    + rS8) * t
 175                   + rS7) * t
 176                  + rS6) * t
 177                 + rS5) * t
 178                + rS4) * t
 179               + rS3) * t
 180              + rS2) * t
 181             + rS1) * t
 182            + rS0) * t;
 183
 184       q = ((((((((( t
 185                     + sS9) * t
 186                   + sS8) * t
 187                  + sS7) * t
 188                 + sS6) * t
 189                + sS5) * t
 190               + sS4) * t
 191              + sS3) * t
 192             + sS2) * t
 193            + sS1) * t
 194         + sS0;
 195
 196       return (pio2_hi - asinr5625) - (p / q - pio2_lo);
 197     }
 198   else
 199     return 2 * asinl (sqrtl ((1 - x) / 2));
 200 }
 201
 202 #endif
 203
 204 #if 0
 205 int
 206 main (void)
 207 {
 208   printf ("%.18Lg %.18Lg\n",
 209           acosl (1.0L),
 210           1.5707963267948966192313216916397514420984L -
 211           1.5707963267948966192313216916397514420984L);
 212   printf ("%.18Lg %.18Lg\n",
 213           acosl (0.7071067811865475244008443621048490392848L),
 214           1.5707963267948966192313216916397514420984L -
 215           0.7853981633974483096156608458198757210492L);
 216   printf ("%.18Lg %.18Lg\n",
 217           acosl (0.5L),
 218           1.5707963267948966192313216916397514420984L -
 219           0.5235987755982988730771072305465838140328L);
 220   printf ("%.18Lg %.18Lg\n",
 221           acosl (0.3090169943749474241022934171828190588600L),
 222           1.5707963267948966192313216916397514420984L -
 223           0.3141592653589793238462643383279502884196L);
 224   printf ("%.18Lg %.18Lg\n",
 225           acosl (-1.0L),
 226           1.5707963267948966192313216916397514420984L -
 227           -1.5707963267948966192313216916397514420984L);
 228   printf ("%.18Lg %.18Lg\n",
 229           acosl (-0.7071067811865475244008443621048490392848L),
 230           1.5707963267948966192313216916397514420984L -
 231           -0.7853981633974483096156608458198757210492L);
 232   printf ("%.18Lg %.18Lg\n",
 233           acosl (-0.5L),
 234           1.5707963267948966192313216916397514420984L -
 235           -0.5235987755982988730771072305465838140328L);
 236   printf ("%.18Lg %.18Lg\n",
 237           acosl (-0.3090169943749474241022934171828190588600L),
 238           1.5707963267948966192313216916397514420984L -
 239           -0.3141592653589793238462643383279502884196L);
 240 }
 241 #endif