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
|
/* mpfr_atanh -- Inverse Hyperbolic Tangente
Copyright 2001-2014 Free Software Foundation, Inc.
Contributed by the AriC 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 atanh is done by
atanh= 1/2*ln(x+1)-1/2*ln(1-x) */
int
mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode)
{
int inexact;
mpfr_t x, t, te;
mpfr_prec_t Nx, Ny, Nt;
mpfr_exp_t err;
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (xt), mpfr_log_prec, xt, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (y), mpfr_log_prec, y, inexact));
/* Special cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt)))
{
/* atanh(NaN) = NaN, and atanh(+/-Inf) = NaN since tanh gives a result
between -1 and 1 */
if (MPFR_IS_NAN (xt) || MPFR_IS_INF (xt))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else /* necessarily xt is 0 */
{
MPFR_ASSERTD (MPFR_IS_ZERO (xt));
MPFR_SET_ZERO (y); /* atanh(0) = 0 */
MPFR_SET_SAME_SIGN (y,xt);
MPFR_RET (0);
}
}
/* atanh (x) = NaN as soon as |x| > 1, and arctanh(+/-1) = +/-Inf */
if (MPFR_UNLIKELY (MPFR_GET_EXP (xt) > 0))
{
if (MPFR_GET_EXP (xt) == 1 && mpfr_powerof2_raw (xt))
{
MPFR_SET_INF (y);
MPFR_SET_SAME_SIGN (y, xt);
mpfr_set_divby0 ();
MPFR_RET (0);
}
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* atanh(x) = x + x^3/3 + ... so the error is < 2^(3*EXP(x)-1) */
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, xt, -2 * MPFR_GET_EXP (xt), 1, 1,
rnd_mode, {});
MPFR_SAVE_EXPO_MARK (expo);
/* Compute initial precision */
Nx = MPFR_PREC (xt);
MPFR_TMP_INIT_ABS (x, xt);
Ny = MPFR_PREC (y);
Nt = MAX (Nx, Ny);
/* the optimal number of bits : see algorithms.ps */
Nt = Nt + MPFR_INT_CEIL_LOG2 (Nt) + 4;
/* initialize of intermediary variable */
mpfr_init2 (t, Nt);
mpfr_init2 (te, Nt);
/* First computation of cosh */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
/* compute atanh */
mpfr_ui_sub (te, 1, x, MPFR_RNDU); /* (1-xt)*/
mpfr_add_ui (t, x, 1, MPFR_RNDD); /* (xt+1)*/
mpfr_div (t, t, te, MPFR_RNDN); /* (1+xt)/(1-xt)*/
mpfr_log (t, t, MPFR_RNDN); /* ln((1+xt)/(1-xt))*/
mpfr_div_2ui (t, t, 1, MPFR_RNDN); /* (1/2)*ln((1+xt)/(1-xt))*/
/* error estimate: see algorithms.tex */
/* FIXME: this does not correspond to the value in algorithms.tex!!! */
/* err=Nt-__gmpfr_ceil_log2(1+5*pow(2,1-MPFR_EXP(t)));*/
err = Nt - (MAX (4 - MPFR_GET_EXP (t), 0) + 1);
if (MPFR_LIKELY (MPFR_IS_ZERO (t)
|| MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
break;
/* reactualisation of the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
mpfr_set_prec (te, Nt);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt));
mpfr_clear(t);
mpfr_clear(te);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
|
