1/* mpfr_expm1 -- Compute exp(x)-1
2
3Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc.
4Contributed by the Arenaire and Cacao 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  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
38    {
39      if (MPFR_IS_NAN (x))
40        {
41          MPFR_SET_NAN (y);
42          MPFR_RET_NAN;
43        }
44      /* check for inf or -inf (expm1(-inf)=-1) */
45      else if (MPFR_IS_INF (x))
46        {
47          if (MPFR_IS_POS (x))
48            {
49              MPFR_SET_INF (y);
50              MPFR_SET_POS (y);
51              MPFR_RET (0);
52            }
53          else
54            return mpfr_set_si (y, -1, rnd_mode);
55        }
56      else
57        {
58          MPFR_ASSERTD (MPFR_IS_ZERO (x));
59          MPFR_SET_ZERO (y);   /* expm1(+/- 0) = +/- 0 */
60          MPFR_SET_SAME_SIGN (y, x);
61          MPFR_RET (0);
62        }
63    }
64
65  ex = MPFR_GET_EXP (x);
66  if (ex < 0)
67    {
68      /* For -1 < x < 0, abs(expm1(x)-x) < x^2/2.
69         For 0 < x < 1,  abs(expm1(x)-x) < x^2. */
70      if (MPFR_IS_POS (x))
71        MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 0, 1, rnd_mode, {});
72      else
73        MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 1, 0, rnd_mode, {});
74    }
75
76  MPFR_SAVE_EXPO_MARK (expo);
77
78  if (MPFR_IS_NEG (x) && ex > 5)  /* x <= -32 */
79    {
80      mpfr_t minus_one, t;
81      mpfr_exp_t err;
82
83      mpfr_init2 (minus_one, 2);
84      mpfr_init2 (t, 64);
85      mpfr_set_si (minus_one, -1, MPFR_RNDN);
86      mpfr_const_log2 (t, MPFR_RNDU); /* round upward since x is negative */
87      mpfr_div (t, x, t, MPFR_RNDU); /* > x / ln(2) */
88      err = mpfr_cmp_si (t, MPFR_EMIN_MIN >= -LONG_MAX ?
89                         MPFR_EMIN_MIN : -LONG_MAX) <= 0 ?
90        - (MPFR_EMIN_MIN >= -LONG_MAX ? MPFR_EMIN_MIN : -LONG_MAX) :
91        - mpfr_get_si (t, MPFR_RNDU);
92      /* exp(x) = 2^(x/ln(2))
93               <= 2^max(MPFR_EMIN_MIN,-LONG_MAX,ceil(x/ln(2)+epsilon))
94         with epsilon > 0 */
95      mpfr_clear (t);
96      MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, minus_one, err, 0, 0, rnd_mode,
97                                        expo, { mpfr_clear (minus_one); });
98      mpfr_clear (minus_one);
99    }
100
101  /* General case */
102  {
103    /* Declaration of the intermediary variable */
104    mpfr_t t;
105    /* Declaration of the size variable */
106    mpfr_prec_t Ny = MPFR_PREC(y);   /* target precision */
107    mpfr_prec_t Nt;                  /* working precision */
108    mpfr_exp_t err, exp_te;          /* error */
109    MPFR_ZIV_DECL (loop);
110
111    /* compute the precision of intermediary variable */
112    /* the optimal number of bits : see algorithms.tex */
113    Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 6;
114
115    /* if |x| is smaller than 2^(-e), we will loose about e bits in the
116       subtraction exp(x) - 1 */
117    if (ex < 0)
118      Nt += - ex;
119
120    /* initialize auxiliary variable */
121    mpfr_init2 (t, Nt);
122
123    /* First computation of expm1 */
124    MPFR_ZIV_INIT (loop, Nt);
125    for (;;)
126      {
127        MPFR_BLOCK_DECL (flags);
128
129        /* exp(x) may overflow and underflow */
130        MPFR_BLOCK (flags, mpfr_exp (t, x, MPFR_RNDN));
131        if (MPFR_OVERFLOW (flags))
132          {
133            inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN_POS);
134            MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
135            break;
136          }
137        else if (MPFR_UNDERFLOW (flags))
138          {
139            inexact = mpfr_set_si (y, -1, rnd_mode);
140            MPFR_ASSERTD (inexact == 0);
141            inexact = -1;
142            if (MPFR_IS_LIKE_RNDZ (rnd_mode, 1))
143              {
144                inexact = 1;
145                mpfr_nexttozero (y);
146              }
147            break;
148          }
149
150        exp_te = MPFR_GET_EXP (t);         /* FIXME: exp(x) may overflow! */
151        mpfr_sub_ui (t, t, 1, MPFR_RNDN);   /* exp(x)-1 */
152
153        /* error estimate */
154        /*err=Nt-(__gmpfr_ceil_log2(1+pow(2,MPFR_EXP(te)-MPFR_EXP(t))));*/
155        err = Nt - (MAX (exp_te - MPFR_GET_EXP (t), 0) + 1);
156
157        if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
158          {
159            inexact = mpfr_set (y, t, rnd_mode);
160            break;
161          }
162
163        /* increase the precision */
164        MPFR_ZIV_NEXT (loop, Nt);
165        mpfr_set_prec (t, Nt);
166      }
167    MPFR_ZIV_FREE (loop);
168
169    mpfr_clear (t);
170  }
171
172  MPFR_SAVE_EXPO_FREE (expo);
173  return mpfr_check_range (y, inexact, rnd_mode);
174}
175