source/libs/mpfr/mpfr-3.1.3/src/expm1.c

   1 /* mpfr_expm1 -- Compute exp(x)-1
   2
   3 Copyright 2001-2015 Free Software Foundation, Inc.
   4 Contributed by the AriC and Caramel projects, INRIA.
   5
   6 This file is part of the GNU MPFR Library.
   7
   8 The GNU MPFR Library is free software; you can redistribute it and/or modify
   9 it under the terms of the GNU Lesser General Public License as published by
  10 the Free Software Foundation; either version 3 of the License, or (at your
  11 option) any later version.
  12
  13 The GNU MPFR Library is distributed in the hope that it will be useful, but
  14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
  15 or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
  16 License for more details.
  17
  18 You should have received a copy of the GNU Lesser General Public License
  19 along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
  20 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
  21 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
  22
  23 #define MPFR_NEED_LONGLONG_H
  24 #include "mpfr-impl.h"
  25
  26  /* The computation of expm1 is done by
  27     expm1(x)=exp(x)-1
  28  */
  29
  30 int
  31 mpfr_expm1 (mpfr_ptr y, mpfr_srcptr x , mpfr_rnd_t rnd_mode)
  32 {
  33   int inexact;
  34   mpfr_exp_t ex;
  35   MPFR_SAVE_EXPO_DECL (expo);
  36
  37   MPFR_LOG_FUNC
  38     (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
  39      ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y,
  40       inexact));
  41
  42   if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
  43     {
  44       if (MPFR_IS_NAN (x))
  45         {
  46           MPFR_SET_NAN (y);
  47           MPFR_RET_NAN;
  48         }
  49       /* check for inf or -inf (expm1(-inf)=-1) */
  50       else if (MPFR_IS_INF (x))
  51         {
  52           if (MPFR_IS_POS (x))
  53             {
  54               MPFR_SET_INF (y);
  55               MPFR_SET_POS (y);
  56               MPFR_RET (0);
  57             }
  58           else
  59             return mpfr_set_si (y, -1, rnd_mode);
  60         }
  61       else
  62         {
  63           MPFR_ASSERTD (MPFR_IS_ZERO (x));
  64           MPFR_SET_ZERO (y);   /* expm1(+/- 0) = +/- 0 */
  65           MPFR_SET_SAME_SIGN (y, x);
  66           MPFR_RET (0);
  67         }
  68     }
  69
  70   ex = MPFR_GET_EXP (x);
  71   if (ex < 0)
  72     {
  73       /* For -1 < x < 0, abs(expm1(x)-x) < x^2/2.
  74          For 0 < x < 1,  abs(expm1(x)-x) < x^2. */
  75       if (MPFR_IS_POS (x))
  76         MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 0, 1, rnd_mode, {});
  77       else
  78         MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 1, 0, rnd_mode, {});
  79     }
  80
  81   MPFR_SAVE_EXPO_MARK (expo);
  82
  83   if (MPFR_IS_NEG (x) && ex > 5)  /* x <= -32 */
  84     {
  85       mpfr_t minus_one, t;
  86       mpfr_exp_t err;
  87
  88       mpfr_init2 (minus_one, 2);
  89       mpfr_init2 (t, 64);
  90       mpfr_set_si (minus_one, -1, MPFR_RNDN);
  91       mpfr_const_log2 (t, MPFR_RNDU); /* round upward since x is negative */
  92       mpfr_div (t, x, t, MPFR_RNDU); /* > x / ln(2) */
  93       err = mpfr_cmp_si (t, MPFR_EMIN_MIN >= -LONG_MAX ?
  94                          MPFR_EMIN_MIN : -LONG_MAX) <= 0 ?
  95         - (MPFR_EMIN_MIN >= -LONG_MAX ? MPFR_EMIN_MIN : -LONG_MAX) :
  96         - mpfr_get_si (t, MPFR_RNDU);
  97       /* exp(x) = 2^(x/ln(2))
  98                <= 2^max(MPFR_EMIN_MIN,-LONG_MAX,ceil(x/ln(2)+epsilon))
  99          with epsilon > 0 */
 100       mpfr_clear (t);
 101       MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, minus_one, err, 0, 0, rnd_mode,
 102                                         expo, { mpfr_clear (minus_one); });
 103       mpfr_clear (minus_one);
 104     }
 105
 106   /* General case */
 107   {
 108     /* Declaration of the intermediary variable */
 109     mpfr_t t;
 110     /* Declaration of the size variable */
 111     mpfr_prec_t Ny = MPFR_PREC(y);   /* target precision */
 112     mpfr_prec_t Nt;                  /* working precision */
 113     mpfr_exp_t err, exp_te;          /* error */
 114     MPFR_ZIV_DECL (loop);
 115
 116     /* compute the precision of intermediary variable */
 117     /* the optimal number of bits : see algorithms.tex */
 118     Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 6;
 119
 120     /* if |x| is smaller than 2^(-e), we will loose about e bits in the
 121        subtraction exp(x) - 1 */
 122     if (ex < 0)
 123       Nt += - ex;
 124
 125     /* initialize auxiliary variable */
 126     mpfr_init2 (t, Nt);
 127
 128     /* First computation of expm1 */
 129     MPFR_ZIV_INIT (loop, Nt);
 130     for (;;)
 131       {
 132         MPFR_BLOCK_DECL (flags);
 133
 134         /* exp(x) may overflow and underflow */
 135         MPFR_BLOCK (flags, mpfr_exp (t, x, MPFR_RNDN));
 136         if (MPFR_OVERFLOW (flags))
 137           {
 138             inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN_POS);
 139             MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
 140             break;
 141           }
 142         else if (MPFR_UNDERFLOW (flags))
 143           {
 144             inexact = mpfr_set_si (y, -1, rnd_mode);
 145             MPFR_ASSERTD (inexact == 0);
 146             inexact = -1;
 147             if (MPFR_IS_LIKE_RNDZ (rnd_mode, 1))
 148               {
 149                 inexact = 1;
 150                 mpfr_nexttozero (y);
 151               }
 152             break;
 153           }
 154
 155         exp_te = MPFR_GET_EXP (t);         /* FIXME: exp(x) may overflow! */
 156         mpfr_sub_ui (t, t, 1, MPFR_RNDN);   /* exp(x)-1 */
 157
 158         /* error estimate */
 159         /*err=Nt-(__gmpfr_ceil_log2(1+pow(2,MPFR_EXP(te)-MPFR_EXP(t))));*/
 160         err = Nt - (MAX (exp_te - MPFR_GET_EXP (t), 0) + 1);
 161
 162         if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
 163           {
 164             inexact = mpfr_set (y, t, rnd_mode);
 165             break;
 166           }
 167
 168         /* increase the precision */
 169         MPFR_ZIV_NEXT (loop, Nt);
 170         mpfr_set_prec (t, Nt);
 171       }
 172     MPFR_ZIV_FREE (loop);
 173
 174     mpfr_clear (t);
 175   }
 176
 177   MPFR_SAVE_EXPO_FREE (expo);
 178   return mpfr_check_range (y, inexact, rnd_mode);
 179 }