compiler/stdc/math/ld128/e_log2l.c

   1 /*      $OpenBSD: e_log2l.c,v 1.1 2011/07/06 00:02:42 martynas Exp $    */
   2
   3 /*
   4  * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
   5  *
   6  * Permission to use, copy, modify, and distribute this software for any
   7  * purpose with or without fee is hereby granted, provided that the above
   8  * copyright notice and this permission notice appear in all copies.
   9  *
  10  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
  11  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
  12  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
  13  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
  14  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
  15  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
  16  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
  17  */
  18
  19 /*                                                      log2l.c
  20  *      Base 2 logarithm, 128-bit long double precision
  21  *
  22  *
  23  *
  24  * SYNOPSIS:
  25  *
  26  * long double x, y, log2l();
  27  *
  28  * y = log2l( x );
  29  *
  30  *
  31  *
  32  * DESCRIPTION:
  33  *
  34  * Returns the base 2 logarithm of x.
  35  *
  36  * The argument is separated into its exponent and fractional
  37  * parts.  If the exponent is between -1 and +1, the (natural)
  38  * logarithm of the fraction is approximated by
  39  *
  40  *     log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
  41  *
  42  * Otherwise, setting  z = 2(x-1)/x+1),
  43  *
  44  *     log(x) = z + z^3 P(z)/Q(z).
  45  *
  46  *
  47  *
  48  * ACCURACY:
  49  *
  50  *                      Relative error:
  51  * arithmetic   domain     # trials      peak         rms
  52  *    IEEE      0.5, 2.0     100,000    2.6e-34     4.9e-35
  53  *    IEEE     exp(+-10000)  100,000    9.6e-35     4.0e-35
  54  *
  55  * In the tests over the interval exp(+-10000), the logarithms
  56  * of the random arguments were uniformly distributed over
  57  * [-10000, +10000].
  58  *
  59  */
  60
  61 #include "math.h"
  62
  63 #include "math_private.h"
  64
  65 /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
  66  * 1/sqrt(2) <= x < sqrt(2)
  67  * Theoretical peak relative error = 5.3e-37,
  68  * relative peak error spread = 2.3e-14
  69  */
  70 static const long double P[13] =
  71 {
  72   1.313572404063446165910279910527789794488E4L,
  73   7.771154681358524243729929227226708890930E4L,
  74   2.014652742082537582487669938141683759923E5L,
  75   3.007007295140399532324943111654767187848E5L,
  76   2.854829159639697837788887080758954924001E5L,
  77   1.797628303815655343403735250238293741397E5L,
  78   7.594356839258970405033155585486712125861E4L,
  79   2.128857716871515081352991964243375186031E4L,
  80   3.824952356185897735160588078446136783779E3L,
  81   4.114517881637811823002128927449878962058E2L,
  82   2.321125933898420063925789532045674660756E1L,
  83   4.998469661968096229986658302195402690910E-1L,
  84   1.538612243596254322971797716843006400388E-6L
  85 };
  86 static const long double Q[12] =
  87 {
  88   3.940717212190338497730839731583397586124E4L,
  89   2.626900195321832660448791748036714883242E5L,
  90   7.777690340007566932935753241556479363645E5L,
  91   1.347518538384329112529391120390701166528E6L,
  92   1.514882452993549494932585972882995548426E6L,
  93   1.158019977462989115839826904108208787040E6L,
  94   6.132189329546557743179177159925690841200E5L,
  95   2.248234257620569139969141618556349415120E5L,
  96   5.605842085972455027590989944010492125825E4L,
  97   9.147150349299596453976674231612674085381E3L,
  98   9.104928120962988414618126155557301584078E2L,
  99   4.839208193348159620282142911143429644326E1L
 100 /* 1.000000000000000000000000000000000000000E0L, */
 101 };
 102
 103 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
 104  * where z = 2(x-1)/(x+1)
 105  * 1/sqrt(2) <= x < sqrt(2)
 106  * Theoretical peak relative error = 1.1e-35,
 107  * relative peak error spread 1.1e-9
 108  */
 109 static const long double R[6] =
 110 {
 111   1.418134209872192732479751274970992665513E5L,
 112  -8.977257995689735303686582344659576526998E4L,
 113   2.048819892795278657810231591630928516206E4L,
 114  -2.024301798136027039250415126250455056397E3L,
 115   8.057002716646055371965756206836056074715E1L,
 116  -8.828896441624934385266096344596648080902E-1L
 117 };
 118 static const long double S[6] =
 119 {
 120   1.701761051846631278975701529965589676574E6L,
 121  -1.332535117259762928288745111081235577029E6L,
 122   4.001557694070773974936904547424676279307E5L,
 123  -5.748542087379434595104154610899551484314E4L,
 124   3.998526750980007367835804959888064681098E3L,
 125  -1.186359407982897997337150403816839480438E2L
 126 /* 1.000000000000000000000000000000000000000E0L, */
 127 };
 128
 129 static const long double
 130 /* log2(e) - 1 */
 131 LOG2EA = 4.4269504088896340735992468100189213742664595E-1L,
 132 /* sqrt(2)/2 */
 133 SQRTH = 7.071067811865475244008443621048490392848359E-1L;
 134
 135
 136 /* Evaluate P[n] x^n  +  P[n-1] x^(n-1)  +  ...  +  P[0] */
 137
 138 static long double
 139 neval (long double x, const long double *p, int n)
 140 {
 141   long double y;
 142
 143   p += n;
 144   y = *p--;
 145   do
 146     {
 147       y = y * x + *p--;
 148     }
 149   while (--n > 0);
 150   return y;
 151 }
 152
 153
 154 /* Evaluate x^n+1  +  P[n] x^(n)  +  P[n-1] x^(n-1)  +  ...  +  P[0] */
 155
 156 static long double
 157 deval (long double x, const long double *p, int n)
 158 {
 159   long double y;
 160
 161   p += n;
 162   y = x + *p--;
 163   do
 164     {
 165       y = y * x + *p--;
 166     }
 167   while (--n > 0);
 168   return y;
 169 }
 170
 171
 172
 173 long double
 174 log2l(long double x)
 175 {
 176   long double z;
 177   long double y;
 178   int e;
 179   int64_t hx, lx;
 180
 181 /* Test for domain */
 182   GET_LDOUBLE_WORDS64 (hx, lx, x);
 183   if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
 184     return (-1.0L / (x - x));
 185   if (hx < 0)
 186     return (x - x) / (x - x);
 187   if (hx >= 0x7fff000000000000LL)
 188     return (x + x);
 189
 190 /* separate mantissa from exponent */
 191
 192 /* Note, frexp is used so that denormal numbers
 193  * will be handled properly.
 194  */
 195   x = frexpl (x, &e);
 196
 197
 198 /* logarithm using log(x) = z + z**3 P(z)/Q(z),
 199  * where z = 2(x-1)/x+1)
 200  */
 201   if ((e > 2) || (e < -2))
 202     {
 203       if (x < SQRTH)
 204         {                       /* 2( 2x-1 )/( 2x+1 ) */
 205           e -= 1;
 206           z = x - 0.5L;
 207           y = 0.5L * z + 0.5L;
 208         }
 209       else
 210         {                       /*  2 (x-1)/(x+1)   */
 211           z = x - 0.5L;
 212           z -= 0.5L;
 213           y = 0.5L * x + 0.5L;
 214         }
 215       x = z / y;
 216       z = x * x;
 217       y = x * (z * neval (z, R, 5) / deval (z, S, 5));
 218       goto done;
 219     }
 220
 221
 222 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
 223
 224   if (x < SQRTH)
 225     {
 226       e -= 1;
 227       x = 2.0 * x - 1.0L;       /*  2x - 1  */
 228     }
 229   else
 230     {
 231       x = x - 1.0L;
 232     }
 233   z = x * x;
 234   y = x * (z * neval (x, P, 12) / deval (x, Q, 11));
 235   y = y - 0.5 * z;
 236
 237 done:
 238
 239 /* Multiply log of fraction by log2(e)
 240  * and base 2 exponent by 1
 241  */
 242   z = y * LOG2EA;
 243   z += x * LOG2EA;
 244   z += y;
 245   z += x;
 246   z += e;
 247   return (z);
 248 }