1 /* mpfr_expm1 -- Compute exp(x)-1
2
3 Copyright 2001-2023 Free Software Foundation, Inc.
4 Contributed by the AriC and Caramba 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 https://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[%Pd]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
39 ("y[%Pd]=%.*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 mp_limb_t t_limb[(64 - 1) / GMP_NUMB_BITS + 1];
86 mpfr_t t;
87 mpfr_eexp_t err;
88
89 MPFR_TMP_INIT1(t_limb, t, 64);
90 /* since x < 0, to get an upper bound on x/log(2), we need to divide
91 by an upper bound on log(2) */
92 mpfr_div (t, x, __gmpfr_const_log2_RNDU, MPFR_RNDU); /* > x / ln(2) */
93 err = mpfr_get_exp_t (t, MPFR_RNDU);
94 MPFR_ASSERTD (err < 0);
95 err = err < - MPFR_EXP_MAX ? MPFR_EXP_MAX : - err;
96 /* err = -max(ceil(t),-MPFR_EXP_MAX). */
97 MPFR_LOG_MSG (("err=%" MPFR_EXP_FSPEC "d\n", err));
98 /* exp(x) = 2^(x/ln(2))
99 <= 2^max(-MPFR_EXP_MAX,ceil(x/ln(2)+epsilon))
100 with epsilon > 0 */
101 MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, __gmpfr_mone, (mpfr_exp_t) err,
102 0, 0, rnd_mode, expo, {});
103 }
104
105 /* General case */
106 {
107 /* Declaration of the intermediary variable */
108 mpfr_t t;
109 /* Declaration of the size variable */
110 mpfr_prec_t Ny = MPFR_PREC(y); /* target precision */
111 mpfr_prec_t Nt; /* working precision */
112 mpfr_exp_t err, exp_te; /* error */
113 MPFR_ZIV_DECL (loop);
114
115 /* compute the precision of intermediary variable */
116 /* the optimal number of bits : see algorithms.tex */
117 Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 6;
118
119 /* if |x| is smaller than 2^(-e), we will loose about e bits in the
120 subtraction exp(x) - 1 */
121 if (ex < 0)
122 Nt += - ex;
123
124 /* initialize auxiliary variable */
125 mpfr_init2 (t, Nt);
126
127 /* First computation of expm1 */
128 MPFR_ZIV_INIT (loop, Nt);
129 for (;;)
130 {
131 MPFR_BLOCK_DECL (flags);
132
133 /* exp(x) may overflow and underflow */
134 MPFR_BLOCK (flags, mpfr_exp (t, x, MPFR_RNDN));
135
136 if (MPFR_OVERFLOW (flags)) /* overflow case */
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
143 /* To get an underflow in exp(x), we need exp(x) < 0.5*2^MPFR_EMIN_MIN
144 thus x/log(2) < MPFR_EMIN_MIN-1. But in that case the above
145 MPFR_SMALL_INPUT_AFTER_SAVE_EXPO() will return the result. */
146 MPFR_ASSERTD(!MPFR_UNDERFLOW (flags));
147
148 exp_te = MPFR_GET_EXP (t);
149 mpfr_sub_ui (t, t, 1, MPFR_RNDN); /* exp(x)-1 */
150
151 /* error estimate */
152 /*err=Nt-(__gmpfr_ceil_log2(1+pow(2,MPFR_EXP(te)-MPFR_EXP(t))));*/
153 err = Nt - (MAX (exp_te - MPFR_GET_EXP (t), 0) + 1);
154
155 if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
156 {
157 inexact = mpfr_set (y, t, rnd_mode);
158 break;
159 }
160
161 /* increase the precision */
162 MPFR_ZIV_NEXT (loop, Nt);
163 mpfr_set_prec (t, Nt);
164 }
165 MPFR_ZIV_FREE (loop);
166
167 mpfr_clear (t);
168 }
169
170 MPFR_SAVE_EXPO_FREE (expo);
171 return mpfr_check_range (y, inexact, rnd_mode);
172 }