1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
|
/* mpfr_expm1 -- Compute exp(x)-1
Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free Software Foundation, Inc.
Contributed by the Arenaire and Caramel projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.
The GNU MPFR Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
/* The computation of expm1 is done by
expm1(x)=exp(x)-1
*/
int
mpfr_expm1 (mpfr_ptr y, mpfr_srcptr x , mpfr_rnd_t rnd_mode)
{
int inexact;
mpfr_exp_t ex;
MPFR_SAVE_EXPO_DECL (expo);
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* check for inf or -inf (expm1(-inf)=-1) */
else if (MPFR_IS_INF (x))
{
if (MPFR_IS_POS (x))
{
MPFR_SET_INF (y);
MPFR_SET_POS (y);
MPFR_RET (0);
}
else
return mpfr_set_si (y, -1, rnd_mode);
}
else
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (y); /* expm1(+/- 0) = +/- 0 */
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
}
ex = MPFR_GET_EXP (x);
if (ex < 0)
{
/* For -1 < x < 0, abs(expm1(x)-x) < x^2/2.
For 0 < x < 1, abs(expm1(x)-x) < x^2. */
if (MPFR_IS_POS (x))
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 0, 1, rnd_mode, {});
else
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 1, 0, rnd_mode, {});
}
MPFR_SAVE_EXPO_MARK (expo);
if (MPFR_IS_NEG (x) && ex > 5) /* x <= -32 */
{
mpfr_t minus_one, t;
mpfr_exp_t err;
mpfr_init2 (minus_one, 2);
mpfr_init2 (t, 64);
mpfr_set_si (minus_one, -1, MPFR_RNDN);
mpfr_const_log2 (t, MPFR_RNDU); /* round upward since x is negative */
mpfr_div (t, x, t, MPFR_RNDU); /* > x / ln(2) */
err = mpfr_cmp_si (t, MPFR_EMIN_MIN >= -LONG_MAX ?
MPFR_EMIN_MIN : -LONG_MAX) <= 0 ?
- (MPFR_EMIN_MIN >= -LONG_MAX ? MPFR_EMIN_MIN : -LONG_MAX) :
- mpfr_get_si (t, MPFR_RNDU);
/* exp(x) = 2^(x/ln(2))
<= 2^max(MPFR_EMIN_MIN,-LONG_MAX,ceil(x/ln(2)+epsilon))
with epsilon > 0 */
mpfr_clear (t);
MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, minus_one, err, 0, 0, rnd_mode,
expo, { mpfr_clear (minus_one); });
mpfr_clear (minus_one);
}
/* General case */
{
/* Declaration of the intermediary variable */
mpfr_t t;
/* Declaration of the size variable */
mpfr_prec_t Ny = MPFR_PREC(y); /* target precision */
mpfr_prec_t Nt; /* working precision */
mpfr_exp_t err, exp_te; /* error */
MPFR_ZIV_DECL (loop);
/* compute the precision of intermediary variable */
/* the optimal number of bits : see algorithms.tex */
Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 6;
/* if |x| is smaller than 2^(-e), we will loose about e bits in the
subtraction exp(x) - 1 */
if (ex < 0)
Nt += - ex;
/* initialize auxiliary variable */
mpfr_init2 (t, Nt);
/* First computation of expm1 */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
MPFR_BLOCK_DECL (flags);
/* exp(x) may overflow and underflow */
MPFR_BLOCK (flags, mpfr_exp (t, x, MPFR_RNDN));
if (MPFR_OVERFLOW (flags))
{
inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN_POS);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
break;
}
else if (MPFR_UNDERFLOW (flags))
{
inexact = mpfr_set_si (y, -1, rnd_mode);
MPFR_ASSERTD (inexact == 0);
inexact = -1;
if (MPFR_IS_LIKE_RNDZ (rnd_mode, 1))
{
inexact = 1;
mpfr_nexttozero (y);
}
break;
}
exp_te = MPFR_GET_EXP (t); /* FIXME: exp(x) may overflow! */
mpfr_sub_ui (t, t, 1, MPFR_RNDN); /* exp(x)-1 */
/* error estimate */
/*err=Nt-(__gmpfr_ceil_log2(1+pow(2,MPFR_EXP(te)-MPFR_EXP(t))));*/
err = Nt - (MAX (exp_te - MPFR_GET_EXP (t), 0) + 1);
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
{
inexact = mpfr_set (y, t, rnd_mode);
break;
}
/* increase the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
}
MPFR_ZIV_FREE (loop);
mpfr_clear (t);
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
|
