compiler/stdc/math/ld128/e_log10l.c

   1 /*      $OpenBSD: e_log10l.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 /*                                                      log10l.c
  20  *
  21  *      Common logarithm, 128-bit long double precision
  22  *
  23  *
  24  *
  25  * SYNOPSIS:
  26  *
  27  * long double x, y, log10l();
  28  *
  29  * y = log10l( x );
  30  *
  31  *
  32  *
  33  * DESCRIPTION:
  34  *
  35  * Returns the base 10 logarithm of x.
  36  *
  37  * The argument is separated into its exponent and fractional
  38  * parts.  If the exponent is between -1 and +1, the logarithm
  39  * of the fraction is approximated by
  40  *
  41  *     log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
  42  *
  43  * Otherwise, setting  z = 2(x-1)/x+1),
  44  *
  45  *     log(x) = z + z^3 P(z)/Q(z).
  46  *
  47  *
  48  *
  49  * ACCURACY:
  50  *
  51  *                      Relative error:
  52  * arithmetic   domain     # trials      peak         rms
  53  *    IEEE      0.5, 2.0     30000      2.3e-34     4.9e-35
  54  *    IEEE     exp(+-10000)  30000      1.0e-34     4.1e-35
  55  *
  56  * In the tests over the interval exp(+-10000), the logarithms
  57  * of the random arguments were uniformly distributed over
  58  * [-10000, +10000].
  59  *
  60  */
  61
  62 #include "math.h"
  63
  64 #include "math_private.h"
  65
  66 /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
  67  * 1/sqrt(2) <= x < sqrt(2)
  68  * Theoretical peak relative error = 5.3e-37,
  69  * relative peak error spread = 2.3e-14
  70  */
  71 static const long double P[13] =
  72 {
  73   1.313572404063446165910279910527789794488E4L,
  74   7.771154681358524243729929227226708890930E4L,
  75   2.014652742082537582487669938141683759923E5L,
  76   3.007007295140399532324943111654767187848E5L,
  77   2.854829159639697837788887080758954924001E5L,
  78   1.797628303815655343403735250238293741397E5L,
  79   7.594356839258970405033155585486712125861E4L,
  80   2.128857716871515081352991964243375186031E4L,
  81   3.824952356185897735160588078446136783779E3L,
  82   4.114517881637811823002128927449878962058E2L,
  83   2.321125933898420063925789532045674660756E1L,
  84   4.998469661968096229986658302195402690910E-1L,
  85   1.538612243596254322971797716843006400388E-6L
  86 };
  87 static const long double Q[12] =
  88 {
  89   3.940717212190338497730839731583397586124E4L,
  90   2.626900195321832660448791748036714883242E5L,
  91   7.777690340007566932935753241556479363645E5L,
  92   1.347518538384329112529391120390701166528E6L,
  93   1.514882452993549494932585972882995548426E6L,
  94   1.158019977462989115839826904108208787040E6L,
  95   6.132189329546557743179177159925690841200E5L,
  96   2.248234257620569139969141618556349415120E5L,
  97   5.605842085972455027590989944010492125825E4L,
  98   9.147150349299596453976674231612674085381E3L,
  99   9.104928120962988414618126155557301584078E2L,
 100   4.839208193348159620282142911143429644326E1L
 101 /* 1.000000000000000000000000000000000000000E0L, */
 102 };
 103
 104 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
 105  * where z = 2(x-1)/(x+1)
 106  * 1/sqrt(2) <= x < sqrt(2)
 107  * Theoretical peak relative error = 1.1e-35,
 108  * relative peak error spread 1.1e-9
 109  */
 110 static const long double R[6] =
 111 {
 112   1.418134209872192732479751274970992665513E5L,
 113  -8.977257995689735303686582344659576526998E4L,
 114   2.048819892795278657810231591630928516206E4L,
 115  -2.024301798136027039250415126250455056397E3L,
 116   8.057002716646055371965756206836056074715E1L,
 117  -8.828896441624934385266096344596648080902E-1L
 118 };
 119 static const long double S[6] =
 120 {
 121   1.701761051846631278975701529965589676574E6L,
 122  -1.332535117259762928288745111081235577029E6L,
 123   4.001557694070773974936904547424676279307E5L,
 124  -5.748542087379434595104154610899551484314E4L,
 125   3.998526750980007367835804959888064681098E3L,
 126  -1.186359407982897997337150403816839480438E2L
 127 /* 1.000000000000000000000000000000000000000E0L, */
 128 };
 129
 130 static const long double
 131 /* log10(2) */
 132 L102A = 0.3125L,
 133 L102B = -1.14700043360188047862611052755069732318101185E-2L,
 134 /* log10(e) */
 135 L10EA = 0.5L,
 136 L10EB = -6.570551809674817234887108108339491770560299E-2L,
 137 /* sqrt(2)/2 */
 138 SQRTH = 7.071067811865475244008443621048490392848359E-1L;
 139
 140
 141
 142 /* Evaluate P[n] x^n  +  P[n-1] x^(n-1)  +  ...  +  P[0] */
 143
 144 static long double
 145 neval (long double x, const long double *p, int n)
 146 {
 147   long double y;
 148
 149   p += n;
 150   y = *p--;
 151   do
 152     {
 153       y = y * x + *p--;
 154     }
 155   while (--n > 0);
 156   return y;
 157 }
 158
 159
 160 /* Evaluate x^n+1  +  P[n] x^(n)  +  P[n-1] x^(n-1)  +  ...  +  P[0] */
 161
 162 static long double
 163 deval (long double x, const long double *p, int n)
 164 {
 165   long double y;
 166
 167   p += n;
 168   y = x + *p--;
 169   do
 170     {
 171       y = y * x + *p--;
 172     }
 173   while (--n > 0);
 174   return y;
 175 }
 176
 177
 178
 179 long double
 180 log10l(long double x)
 181 {
 182   long double z;
 183   long double y;
 184   int e;
 185   int64_t hx, lx;
 186
 187 /* Test for domain */
 188   GET_LDOUBLE_WORDS64 (hx, lx, x);
 189   if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
 190     return (-1.0L / (x - x));
 191   if (hx < 0)
 192     return (x - x) / (x - x);
 193   if (hx >= 0x7fff000000000000LL)
 194     return (x + x);
 195
 196 /* separate mantissa from exponent */
 197
 198 /* Note, frexp is used so that denormal numbers
 199  * will be handled properly.
 200  */
 201   x = frexpl (x, &e);
 202
 203
 204 /* logarithm using log(x) = z + z**3 P(z)/Q(z),
 205  * where z = 2(x-1)/x+1)
 206  */
 207   if ((e > 2) || (e < -2))
 208     {
 209       if (x < SQRTH)
 210         {                       /* 2( 2x-1 )/( 2x+1 ) */
 211           e -= 1;
 212           z = x - 0.5L;
 213           y = 0.5L * z + 0.5L;
 214         }
 215       else
 216         {                       /*  2 (x-1)/(x+1)   */
 217           z = x - 0.5L;
 218           z -= 0.5L;
 219           y = 0.5L * x + 0.5L;
 220         }
 221       x = z / y;
 222       z = x * x;
 223       y = x * (z * neval (z, R, 5) / deval (z, S, 5));
 224       goto done;
 225     }
 226
 227
 228 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
 229
 230   if (x < SQRTH)
 231     {
 232       e -= 1;
 233       x = 2.0 * x - 1.0L;       /*  2x - 1  */
 234     }
 235   else
 236     {
 237       x = x - 1.0L;
 238     }
 239   z = x * x;
 240   y = x * (z * neval (x, P, 12) / deval (x, Q, 11));
 241   y = y - 0.5 * z;
 242
 243 done:
 244
 245   /* Multiply log of fraction by log10(e)
 246    * and base 2 exponent by log10(2).
 247    */
 248   z = y * L10EB;
 249   z += x * L10EB;
 250   z += e * L102B;
 251   z += y * L10EA;
 252   z += x * L10EA;
 253   z += e * L102A;
 254   return (z);
 255 }