1 /* mpfr_acosh -- inverse hyperbolic cosine
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 acosh is done by *
27 * acosh= ln(x + sqrt(x^2-1)) */
28
29 int
30 mpfr_acosh (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
31 {
32 MPFR_SAVE_EXPO_DECL (expo);
33 int inexact;
34 int comp;
35
36 MPFR_LOG_FUNC (
37 ("x[%Pd]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
38 ("y[%Pd]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y,
39 inexact));
40
41 /* Deal with special cases */
42 if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
43 {
44 /* Nan, or zero or -Inf */
45 if (MPFR_IS_INF (x) && MPFR_IS_POS (x))
46 {
47 MPFR_SET_INF (y);
48 MPFR_SET_POS (y);
49 MPFR_RET (0);
50 }
51 else /* Nan, or zero or -Inf */
52 {
53 MPFR_SET_NAN (y);
54 MPFR_RET_NAN;
55 }
56 }
57 comp = mpfr_cmp_ui (x, 1);
58 if (MPFR_UNLIKELY (comp < 0))
59 {
60 MPFR_SET_NAN (y);
61 MPFR_RET_NAN;
62 }
63 else if (MPFR_UNLIKELY (comp == 0))
64 {
65 MPFR_SET_ZERO (y); /* acosh(1) = +0 */
66 MPFR_SET_POS (y);
67 MPFR_RET (0);
68 }
69 MPFR_SAVE_EXPO_MARK (expo);
70
71 /* General case */
72 {
73 /* Declaration of the intermediary variables */
74 mpfr_t t;
75 /* Declaration of the size variables */
76 mpfr_prec_t Ny = MPFR_PREC(y); /* Precision of output variable */
77 mpfr_prec_t Nt; /* Precision of the intermediary variable */
78 mpfr_exp_t err, exp_te, d; /* Precision of error */
79 MPFR_ZIV_DECL (loop);
80
81 /* compute the precision of intermediary variable */
82 /* the optimal number of bits : see algorithms.tex */
83 Nt = Ny + 4 + MPFR_INT_CEIL_LOG2 (Ny);
84
85 /* initialization of intermediary variables */
86 mpfr_init2 (t, Nt);
87
88 /* First computation of acosh */
89 MPFR_ZIV_INIT (loop, Nt);
90 for (;;)
91 {
92 MPFR_BLOCK_DECL (flags);
93
94 /* compute acosh */
95 MPFR_BLOCK (flags, mpfr_sqr (t, x, MPFR_RNDD)); /* x^2 */
96 if (MPFR_OVERFLOW (flags))
97 {
98 mpfr_t ln2;
99 mpfr_prec_t pln2;
100
101 /* As x is very large and the precision is not too large, we
102 assume that we obtain the same result by evaluating ln(2x).
103 We need to compute ln(x) + ln(2) as 2x can overflow. TODO:
104 write a proof and add an MPFR_ASSERTN. */
105 mpfr_log (t, x, MPFR_RNDN); /* err(log) < 1/2 ulp(t) */
106 pln2 = Nt - MPFR_PREC_MIN < MPFR_GET_EXP (t) ?
107 MPFR_PREC_MIN : Nt - MPFR_GET_EXP (t);
108 mpfr_init2 (ln2, pln2);
109 mpfr_const_log2 (ln2, MPFR_RNDN); /* err(ln2) < 1/2 ulp(t) */
110 mpfr_add (t, t, ln2, MPFR_RNDN); /* err <= 3/2 ulp(t) */
111 mpfr_clear (ln2);
112 err = 1;
113 }
114 else
115 {
116 exp_te = MPFR_GET_EXP (t);
117 mpfr_sub_ui (t, t, 1, MPFR_RNDD); /* x^2-1 */
118 if (MPFR_UNLIKELY (MPFR_IS_ZERO (t)))
119 {
120 /* This means that x is very close to 1: x = 1 + t with
121 t < 2^(-Nt). We have: acosh(x) = sqrt(2t) (1 - eps(t))
122 with 0 < eps(t) < t / 12. */
123 mpfr_sub_ui (t, x, 1, MPFR_RNDD); /* t = x - 1 */
124 mpfr_mul_2ui (t, t, 1, MPFR_RNDN); /* 2t */
125 mpfr_sqrt (t, t, MPFR_RNDN); /* sqrt(2t) */
126 err = 1;
127 }
128 else
129 {
130 d = exp_te - MPFR_GET_EXP (t);
131 mpfr_sqrt (t, t, MPFR_RNDN); /* sqrt(x^2-1) */
132 mpfr_add (t, t, x, MPFR_RNDN); /* sqrt(x^2-1)+x */
133 mpfr_log (t, t, MPFR_RNDN); /* ln(sqrt(x^2-1)+x) */
134
135 /* error estimate -- see algorithms.tex */
136 err = 3 + MAX (1, d) - MPFR_GET_EXP (t);
137 /* error is bounded by 1/2 + 2^err <= 2^(max(0,1+err)) */
138 err = MAX (0, 1 + err);
139 }
140 }
141
142 if (MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - err, Ny, rnd_mode)))
143 break;
144
145 /* reactualisation of the precision */
146 MPFR_ZIV_NEXT (loop, Nt);
147 mpfr_set_prec (t, Nt);
148 }
149 MPFR_ZIV_FREE (loop);
150
151 inexact = mpfr_set (y, t, rnd_mode);
152
153 mpfr_clear (t);
154 }
155
156 MPFR_SAVE_EXPO_FREE (expo);
157 return mpfr_check_range (y, inexact, rnd_mode);
158 }