sysdeps/libm-ieee754/e_exp.c

   1 /* Double-precision floating point e^x.
   2    Copyright (C) 1997, 1998 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 Library General Public License as
   8    published by the Free Software Foundation; either version 2 of the
   9    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    Library General Public License for more details.
  15
  16    You should have received a copy of the GNU Library General Public
  17    License along with the GNU C Library; see the file COPYING.LIB.  If not,
  18    write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330,
  19    Boston, MA 02111-1307, USA.  */
  20
  21 /* How this works:
  22    The basic design here is from
  23    Shmuel Gal and Boris Bachelis, "An Accurate Elementary Mathematical
  24    Library for the IEEE Floating Point Standard", ACM Trans. Math. Soft.,
  25    17 (1), March 1991, pp. 26-45.
  26
  27    The input value, x, is written as
  28
  29    x = n * ln(2)_0 + t/512 + delta[t] + x + n * ln(2)_1
  30
  31    where:
  32    - n is an integer, 1024 >= n >= -1075;
  33    - ln(2)_0 is the first 43 bits of ln(2), and ln(2)_1 is the remainder, so
  34      that |ln(2)_1| < 2^-32;
  35    - t is an integer, 177 >= t >= -177
  36    - delta is based on a table entry, delta[t] < 2^-28
  37    - x is whatever is left, |x| < 2^-10
  38
  39    Then e^x is approximated as
  40
  41    e^x = 2^n_1 ( 2^n_0 e^(t/512 + delta[t])
  42                + ( 2^n_0 e^(t/512 + delta[t])
  43                    * ( p(x + n * ln(2)_1)
  44                        - n*ln(2)_1
  45                        - n*ln(2)_1 * p(x + n * ln(2)_1) ) ) )
  46
  47    where
  48    - p(x) is a polynomial approximating e(x)-1;
  49    - e^(t/512 + delta[t]) is obtained from a table;
  50    - n_1 + n_0 = n, so that |n_0| < DBL_MIN_EXP-1.
  51
  52    If it happens that n_1 == 0 (this is the usual case), that multiplication
  53    is omitted.
  54    */
  55 #ifndef _GNU_SOURCE
  56 #define _GNU_SOURCE
  57 #endif
  58 #include <float.h>
  59 #include <ieee754.h>
  60 #include <math.h>
  61 #include <fenv.h>
  62 #include <inttypes.h>
  63 #include <math_private.h>
  64
  65 extern const float __exp_deltatable[178];
  66 extern const double __exp_atable[355] /* __attribute__((mode(DF))) */;
  67
  68 static const volatile double TWO1023 = 8.988465674311579539e+307;
  69 static const volatile double TWOM1000 = 9.3326361850321887899e-302;
  70
  71 double
  72 __ieee754_exp (double x)
  73 {
  74   static const double himark = 709.7827128933840868;
  75   static const double lomark = -745.1332191019412221;
  76   /* Check for usual case.  */
  77   if (isless (x, himark) && isgreater (x, lomark))
  78     {
  79       static const double THREEp42 = 13194139533312.0;
  80       static const double THREEp51 = 6755399441055744.0;
  81       /* 1/ln(2).  */
  82       static const double M_1_LN2 = 1.442695040888963387;
  83       /* ln(2), part 1 */
  84       static const double M_LN2_0 = .6931471805598903302;
  85       /* ln(2), part 2 */
  86       static const double M_LN2_1 = 5.497923018708371155e-14;
  87
  88       int tval, unsafe, n_i;
  89       double x22, n, t, dely, result;
  90       union ieee754_double ex2_u, scale_u;
  91       fenv_t oldenv;
  92
  93       feholdexcept (&oldenv);
  94 #ifdef FE_TONEAREST
  95       fesetround (FE_TONEAREST);
  96 #endif
  97
  98       /* Calculate n.  */
  99       n = x * M_1_LN2 + THREEp51;
 100       n -= THREEp51;
 101       x = x - n*M_LN2_0;
 102
 103       /* Calculate t/512.  */
 104       t = x + THREEp42;
 105       t -= THREEp42;
 106       x -= t;
 107
 108       /* Compute tval = t.  */
 109       tval = (int) (t * 512.0);
 110
 111       if (t >= 0)
 112         x -= __exp_deltatable[tval];
 113       else
 114         x += __exp_deltatable[-tval];
 115
 116       /* Now, the variable x contains x + n*ln(2)_1.  */
 117       dely = n*M_LN2_1;
 118
 119       /* Compute ex2 = 2^n_0 e^(t/512+delta[t]).  */
 120       ex2_u.d = __exp_atable[tval+177];
 121       n_i = (int)n;
 122       /* 'unsafe' is 1 iff n_1 != 0.  */
 123       unsafe = abs(n_i) >= -DBL_MIN_EXP - 1;
 124       ex2_u.ieee.exponent += n_i >> unsafe;
 125
 126       /* Compute scale = 2^n_1.  */
 127       scale_u.d = 1.0;
 128       scale_u.ieee.exponent += n_i - (n_i >> unsafe);
 129
 130       /* Approximate e^x2 - 1, using a fourth-degree polynomial,
 131          with maximum error in [-2^-10-2^-28,2^-10+2^-28]
 132          less than 4.9e-19.  */
 133       x22 = (((0.04166666898464281565
 134                * x + 0.1666666766008501610)
 135               * x + 0.499999999999990008)
 136              * x + 0.9999999999999976685) * x;
 137       /* Allow for impact of dely.  */
 138       x22 -= dely + dely*x22;
 139
 140       /* Return result.  */
 141       fesetenv (&oldenv);
 142
 143       result = x22 * ex2_u.d + ex2_u.d;
 144       if (!unsafe)
 145         return result;
 146       else
 147         return result * scale_u.d;
 148     }
 149   /* Exceptional cases:  */
 150   else if (isless (x, himark))
 151     {
 152       if (__isinf (x))
 153         /* e^-inf == 0, with no error.  */
 154         return 0;
 155       else
 156         /* Underflow */
 157         return TWOM1000 * TWOM1000;
 158     }
 159   else
 160     /* Return x, if x is a NaN or Inf; or overflow, otherwise.  */
 161     return TWO1023*x;
 162 }