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       fesetround (FE_TONEAREST);
  95
  96       /* Calculate n.  */
  97       n = x * M_1_LN2 + THREEp51;
  98       n -= THREEp51;
  99       x = x - n*M_LN2_0;
 100
 101       /* Calculate t/512.  */
 102       t = x + THREEp42;
 103       t -= THREEp42;
 104       x -= t;
 105
 106       /* Compute tval = t.  */
 107       tval = (int) (t * 512.0);
 108
 109       if (t >= 0)
 110         x -= __exp_deltatable[tval];
 111       else
 112         x += __exp_deltatable[-tval];
 113
 114       /* Now, the variable x contains x + n*ln(2)_1.  */
 115       dely = n*M_LN2_1;
 116
 117       /* Compute ex2 = 2^n_0 e^(t/512+delta[t]).  */
 118       ex2_u.d = __exp_atable[tval+177];
 119       n_i = (int)n;
 120       /* 'unsafe' is 1 iff n_1 != 0.  */
 121       unsafe = abs(n_i) >= -DBL_MIN_EXP - 1;
 122       ex2_u.ieee.exponent += n_i >> unsafe;
 123
 124       /* Compute scale = 2^n_1.  */
 125       scale_u.d = 1.0;
 126       scale_u.ieee.exponent += n_i - (n_i >> unsafe);
 127
 128       /* Approximate e^x2 - 1, using a fourth-degree polynomial,
 129          with maximum error in [-2^-10-2^-28,2^-10+2^-28]
 130          less than 4.9e-19.  */
 131       x22 = (((0.04166666898464281565
 132                * x + 0.1666666766008501610)
 133               * x + 0.499999999999990008)
 134              * x + 0.9999999999999976685) * x;
 135       /* Allow for impact of dely.  */
 136       x22 -= dely + dely*x22;
 137
 138       /* Return result.  */
 139       fesetenv (&oldenv);
 140
 141       result = x22 * ex2_u.d + ex2_u.d;
 142       if (!unsafe)
 143         return result;
 144       else
 145         return result * scale_u.d;
 146     }
 147   /* Exceptional cases:  */
 148   else if (isless (x, himark))
 149     {
 150       if (__isinf (x))
 151         /* e^-inf == 0, with no error.  */
 152         return 0;
 153       else
 154         /* Underflow */
 155         return TWOM1000 * TWOM1000;
 156     }
 157   else
 158     /* Return x, if x is a NaN or Inf; or overflow, otherwise.  */
 159     return TWO1023*x;
 160 }