lib/exp2l.c

   1 /* Exponential base 2 function.
   2    Copyright (C) 2011-2018 Free Software Foundation, Inc.
   3
   4    This program is free software: you can redistribute it and/or modify
   5    it under the terms of the GNU General Public License as published by
   6    the Free Software Foundation; either version 3 of the License, or
   7    (at your option) any later version.
   8
   9    This program is distributed in the hope that it will be useful,
  10    but WITHOUT ANY WARRANTY; without even the implied warranty of
  11    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  12    GNU General Public License for more details.
  13
  14    You should have received a copy of the GNU General Public License
  15    along with this program.  If not, see <https://www.gnu.org/licenses/>.  */
  16
  17 #include <config.h>
  18
  19 /* Specification.  */
  20 #include <math.h>
  21
  22 #if HAVE_SAME_LONG_DOUBLE_AS_DOUBLE
  23
  24 long double
  25 exp2l (long double x)
  26 {
  27   return exp2 (x);
  28 }
  29
  30 #else
  31
  32 # include <float.h>
  33
  34 /* gl_expl_table[i] = exp((i - 128) * log(2)/256).  */
  35 extern const long double gl_expl_table[257];
  36
  37 /* Best possible approximation of log(2) as a 'long double'.  */
  38 #define LOG2 0.693147180559945309417232121458176568075L
  39
  40 /* Best possible approximation of 1/log(2) as a 'long double'.  */
  41 #define LOG2_INVERSE 1.44269504088896340735992468100189213743L
  42
  43 /* Best possible approximation of log(2)/256 as a 'long double'.  */
  44 #define LOG2_BY_256 0.00270760617406228636491106297444600221904L
  45
  46 /* Best possible approximation of 256/log(2) as a 'long double'.  */
  47 #define LOG2_BY_256_INVERSE 369.329930467574632284140718336484387181L
  48
  49 long double
  50 exp2l (long double x)
  51 {
  52   /* exp2(x) = exp(x*log(2)).
  53      If we would compute it like this, there would be rounding errors for
  54      integer or near-integer values of x.  To avoid these, we inline the
  55      algorithm for exp(), and the multiplication with log(2) cancels a
  56      division by log(2).  */
  57
  58   if (isnanl (x))
  59     return x;
  60
  61   if (x > (long double) LDBL_MAX_EXP)
  62     /* x > LDBL_MAX_EXP
  63        hence exp2(x) > 2^LDBL_MAX_EXP, overflows to Infinity.  */
  64     return HUGE_VALL;
  65
  66   if (x < (long double) (LDBL_MIN_EXP - 1 - LDBL_MANT_DIG))
  67     /* x < (LDBL_MIN_EXP - 1 - LDBL_MANT_DIG)
  68        hence exp2(x) < 2^(LDBL_MIN_EXP-1-LDBL_MANT_DIG),
  69        underflows to zero.  */
  70     return 0.0L;
  71
  72   /* Decompose x into
  73        x = n + m/256 + y/log(2)
  74      where
  75        n is an integer,
  76        m is an integer, -128 <= m <= 128,
  77        y is a number, |y| <= log(2)/512 + epsilon = 0.00135...
  78      Then
  79        exp2(x) = 2^n * exp(m * log(2)/256) * exp(y)
  80      The first factor is an ldexpl() call.
  81      The second factor is a table lookup.
  82      The third factor is computed
  83      - either as sinh(y) + cosh(y)
  84        where sinh(y) is computed through the power series:
  85          sinh(y) = y + y^3/3! + y^5/5! + ...
  86        and cosh(y) is computed as hypot(1, sinh(y)),
  87      - or as exp(2*z) = (1 + tanh(z)) / (1 - tanh(z))
  88        where z = y/2
  89        and tanh(z) is computed through its power series:
  90          tanh(z) = z
  91                    - 1/3 * z^3
  92                    + 2/15 * z^5
  93                    - 17/315 * z^7
  94                    + 62/2835 * z^9
  95                    - 1382/155925 * z^11
  96                    + 21844/6081075 * z^13
  97                    - 929569/638512875 * z^15
  98                    + ...
  99        Since |z| <= log(2)/1024 < 0.0007, the relative contribution of the
 100        z^13 term is < 0.0007^12 < 2^-120 <= 2^-LDBL_MANT_DIG, therefore we
 101        can truncate the series after the z^11 term.  */
 102
 103   {
 104     long double nm = roundl (x * 256.0L); /* = 256 * n + m */
 105     long double z = (x * 256.0L - nm) * (LOG2_BY_256 * 0.5L);
 106
 107 /* Coefficients of the power series for tanh(z).  */
 108 #define TANH_COEFF_1   1.0L
 109 #define TANH_COEFF_3  -0.333333333333333333333333333333333333334L
 110 #define TANH_COEFF_5   0.133333333333333333333333333333333333334L
 111 #define TANH_COEFF_7  -0.053968253968253968253968253968253968254L
 112 #define TANH_COEFF_9   0.0218694885361552028218694885361552028218L
 113 #define TANH_COEFF_11 -0.00886323552990219656886323552990219656886L
 114 #define TANH_COEFF_13  0.00359212803657248101692546136990581435026L
 115 #define TANH_COEFF_15 -0.00145583438705131826824948518070211191904L
 116
 117     long double z2 = z * z;
 118     long double tanh_z =
 119       (((((TANH_COEFF_11
 120            * z2 + TANH_COEFF_9)
 121           * z2 + TANH_COEFF_7)
 122          * z2 + TANH_COEFF_5)
 123         * z2 + TANH_COEFF_3)
 124        * z2 + TANH_COEFF_1)
 125       * z;
 126
 127     long double exp_y = (1.0L + tanh_z) / (1.0L - tanh_z);
 128
 129     int n = (int) roundl (nm * (1.0L / 256.0L));
 130     int m = (int) nm - 256 * n;
 131
 132     return ldexpl (gl_expl_table[128 + m] * exp_y, n);
 133   }
 134 }
 135
 136 #endif