libgcc-math/flt-32/e_expf.c

   1 /* Single-precision floating point e^x.
   2    Copyright (C) 1997, 1998, 2005, 2006 Free Software Foundation, Inc.
   3    This file is part of the GNU C Library.
   4    Contributed by Geoffrey Keating <geoffk@ozemail.com.au>
   5
   6    The GNU C Library is free software; you can redistribute it and/or
   7    modify it under the terms of the GNU Lesser General Public
   8    License as published by the Free Software Foundation; either
   9    version 2.1 of the License, or (at your option) any later version.
  10
  11    The GNU C Library is distributed in the hope that it will be useful,
  12    but WITHOUT ANY WARRANTY; without even the implied warranty of
  13    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  14    Lesser General Public License for more details.
  15
  16    You should have received a copy of the GNU Lesser General Public
  17    License along with the GNU C Library; if not, write to the Free
  18    Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
  19    02111-1307 USA.  */
  20
  21 /* How this works:
  22
  23    The input value, x, is written as
  24
  25    x = n * ln(2) + t/512 + delta[t] + x;
  26
  27    where:
  28    - n is an integer, 127 >= n >= -150;
  29    - t is an integer, 177 >= t >= -177
  30    - delta is based on a table entry, delta[t] < 2^-28
  31    - x is whatever is left, |x| < 2^-10
  32
  33    Then e^x is approximated as
  34
  35    e^x = 2^n ( e^(t/512 + delta[t])
  36                + ( e^(t/512 + delta[t])
  37                    * ( p(x + delta[t] + n * ln(2)) - delta ) ) )
  38
  39    where
  40    - p(x) is a polynomial approximating e(x)-1;
  41    - e^(t/512 + delta[t]) is obtained from a table.
  42
  43    The table used is the same one as for the double precision version;
  44    since we have the table, we might as well use it.
  45
  46    It turns out to be faster to do calculations in double precision than
  47    to perform an 'accurate table method' expf, because of the range reduction
  48    overhead (compare exp2f).
  49    */
  50 #ifndef _GNU_SOURCE
  51 #define _GNU_SOURCE
  52 #endif
  53 #include <float.h>
  54 #include <ieee754.h>
  55 #include <math_private.h>
  56
  57 extern const float __exp_deltatable[178];
  58 extern const double __exp_atable[355] /* __attribute__((mode(DF))) */;
  59
  60 static const volatile float TWOM100 = 7.88860905e-31;
  61 static const volatile float TWO127 = 1.7014118346e+38;
  62
  63 float
  64 __ieee754_expf (float x)
  65 {
  66   static const float himark = 88.72283935546875;
  67   static const float lomark = -103.972084045410;
  68   /* Check for usual case.  */
  69   if (__builtin_isless (x, himark) && __builtin_isgreater (x, lomark))
  70     {
  71       static const float THREEp42 = 13194139533312.0;
  72       static const float THREEp22 = 12582912.0;
  73       /* 1/ln(2).  */
  74 #undef M_1_LN2
  75       static const float M_1_LN2 = 1.44269502163f;
  76       /* ln(2) */
  77 #undef M_LN2
  78       static const double M_LN2 = .6931471805599452862;
  79
  80       int tval;
  81       double x22, t, result, dx;
  82       float n, delta;
  83       union ieee754_double ex2_u;
  84
  85       /* Calculate n.  */
  86       n = x * M_1_LN2 + THREEp22;
  87       n -= THREEp22;
  88       dx = x - n*M_LN2;
  89
  90       /* Calculate t/512.  */
  91       t = dx + THREEp42;
  92       t -= THREEp42;
  93       dx -= t;
  94
  95       /* Compute tval = t.  */
  96       tval = (int) (t * 512.0);
  97
  98       if (t >= 0)
  99         delta = - __exp_deltatable[tval];
 100       else
 101         delta = __exp_deltatable[-tval];
 102
 103       /* Compute ex2 = 2^n e^(t/512+delta[t]).  */
 104       ex2_u.d = __exp_atable[tval+177];
 105       ex2_u.ieee.exponent += (int) n;
 106
 107       /* Approximate e^(dx+delta) - 1, using a second-degree polynomial,
 108          with maximum error in [-2^-10-2^-28,2^-10+2^-28]
 109          less than 5e-11.  */
 110       x22 = (0.5000000496709180453 * dx + 1.0000001192102037084) * dx + delta;
 111
 112       /* Return result.  */
 113       result = x22 * ex2_u.d + ex2_u.d;
 114       return (float) result;
 115     }
 116   /* Exceptional cases:  */
 117   else if (__builtin_isless (x, himark))
 118     {
 119       if (__isinff (x))
 120         /* e^-inf == 0, with no error.  */
 121         return 0;
 122       else
 123         /* Underflow */
 124         return TWOM100 * TWOM100;
 125     }
 126   else
 127     /* Return x, if x is a NaN or Inf; or overflow, otherwise.  */
 128     return TWO127*x;
 129 }