1/* mpfr_expm1 -- Compute exp(x)-1
2
3Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
4Contributed by the AriC and Caramel projects, INRIA.
5
6This file is part of the GNU MPFR Library.
7
8The GNU MPFR Library is free software; you can redistribute it and/or modify
9it under the terms of the GNU Lesser General Public License as published by
10the Free Software Foundation; either version 3 of the License, or (at your
11option) any later version.
12
13The GNU MPFR Library is distributed in the hope that it will be useful, but
14WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
15or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
16License for more details.
17
18You should have received a copy of the GNU Lesser General Public License
19along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
20http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
2151 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
30int
31mpfr_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}
180