1 /* mpfr_atanh -- Inverse Hyperbolic Tangente
2
3 Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
4 Contributed by the AriC and Caramel 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 http://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 atanh is done by
27 atanh= 1/2*ln(x+1)-1/2*ln(1-x) */
28
29 int
mpfr_atanh(mpfr_ptr y,mpfr_srcptr xt,mpfr_rnd_t rnd_mode)30 mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode)
31 {
32 int inexact;
33 mpfr_t x, t, te;
34 mpfr_prec_t Nx, Ny, Nt;
35 mpfr_exp_t err;
36 MPFR_ZIV_DECL (loop);
37 MPFR_SAVE_EXPO_DECL (expo);
38
39 MPFR_LOG_FUNC
40 (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (xt), mpfr_log_prec, xt, rnd_mode),
41 ("y[%Pu]=%.*Rg inexact=%d",
42 mpfr_get_prec (y), mpfr_log_prec, y, inexact));
43
44 /* Special cases */
45 if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt)))
46 {
47 /* atanh(NaN) = NaN, and atanh(+/-Inf) = NaN since tanh gives a result
48 between -1 and 1 */
49 if (MPFR_IS_NAN (xt) || MPFR_IS_INF (xt))
50 {
51 MPFR_SET_NAN (y);
52 MPFR_RET_NAN;
53 }
54 else /* necessarily xt is 0 */
55 {
56 MPFR_ASSERTD (MPFR_IS_ZERO (xt));
57 MPFR_SET_ZERO (y); /* atanh(0) = 0 */
58 MPFR_SET_SAME_SIGN (y,xt);
59 MPFR_RET (0);
60 }
61 }
62
63 /* atanh (x) = NaN as soon as |x| > 1, and arctanh(+/-1) = +/-Inf */
64 if (MPFR_UNLIKELY (MPFR_GET_EXP (xt) > 0))
65 {
66 if (MPFR_GET_EXP (xt) == 1 && mpfr_powerof2_raw (xt))
67 {
68 MPFR_SET_INF (y);
69 MPFR_SET_SAME_SIGN (y, xt);
70 mpfr_set_divby0 ();
71 MPFR_RET (0);
72 }
73 MPFR_SET_NAN (y);
74 MPFR_RET_NAN;
75 }
76
77 /* atanh(x) = x + x^3/3 + ... so the error is < 2^(3*EXP(x)-1) */
78 MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, xt, -2 * MPFR_GET_EXP (xt), 1, 1,
79 rnd_mode, {});
80
81 MPFR_SAVE_EXPO_MARK (expo);
82
83 /* Compute initial precision */
84 Nx = MPFR_PREC (xt);
85 MPFR_TMP_INIT_ABS (x, xt);
86 Ny = MPFR_PREC (y);
87 Nt = MAX (Nx, Ny);
88 /* the optimal number of bits : see algorithms.ps */
89 Nt = Nt + MPFR_INT_CEIL_LOG2 (Nt) + 4;
90
91 /* initialise of intermediary variable */
92 mpfr_init2 (t, Nt);
93 mpfr_init2 (te, Nt);
94
95 /* First computation of cosh */
96 MPFR_ZIV_INIT (loop, Nt);
97 for (;;)
98 {
99 /* compute atanh */
100 mpfr_ui_sub (te, 1, x, MPFR_RNDU); /* (1-xt)*/
101 mpfr_add_ui (t, x, 1, MPFR_RNDD); /* (xt+1)*/
102 mpfr_div (t, t, te, MPFR_RNDN); /* (1+xt)/(1-xt)*/
103 mpfr_log (t, t, MPFR_RNDN); /* ln((1+xt)/(1-xt))*/
104 mpfr_div_2ui (t, t, 1, MPFR_RNDN); /* (1/2)*ln((1+xt)/(1-xt))*/
105
106 /* error estimate: see algorithms.tex */
107 /* FIXME: this does not correspond to the value in algorithms.tex!!! */
108 /* err=Nt-__gmpfr_ceil_log2(1+5*pow(2,1-MPFR_EXP(t)));*/
109 err = Nt - (MAX (4 - MPFR_GET_EXP (t), 0) + 1);
110
111 if (MPFR_LIKELY (MPFR_IS_ZERO (t)
112 || MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
113 break;
114
115 /* reactualisation of the precision */
116 MPFR_ZIV_NEXT (loop, Nt);
117 mpfr_set_prec (t, Nt);
118 mpfr_set_prec (te, Nt);
119 }
120 MPFR_ZIV_FREE (loop);
121
122 inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt));
123
124 mpfr_clear(t);
125 mpfr_clear(te);
126
127 MPFR_SAVE_EXPO_FREE (expo);
128 return mpfr_check_range (y, inexact, rnd_mode);
129 }
130
131