1 /* mpfr_acos -- arc-cosinus of a floating-point number
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 int
27 mpfr_acos (mpfr_ptr acos, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
28 {
29 mpfr_t xp, arcc, tmp;
30 mpfr_exp_t supplement;
31 mpfr_prec_t prec;
32 int sign, compared, inexact;
33 MPFR_SAVE_EXPO_DECL (expo);
34 MPFR_ZIV_DECL (loop);
35
36 MPFR_LOG_FUNC
37 (("x[%Pd]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode),
38 ("acos[%Pd]=%.*Rg inexact=%d",
39 mpfr_get_prec(acos), mpfr_log_prec, acos, inexact));
40
41 /* Singular cases */
42 if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
43 {
44 if (MPFR_IS_NAN (x) || MPFR_IS_INF (x))
45 {
46 MPFR_SET_NAN (acos);
47 MPFR_RET_NAN;
48 }
49 else /* necessarily x=0 */
50 {
51 MPFR_ASSERTD(MPFR_IS_ZERO(x));
52 /* acos(0)=Pi/2 */
53 MPFR_SAVE_EXPO_MARK (expo);
54 inexact = mpfr_const_pi (acos, rnd_mode);
55 mpfr_div_2ui (acos, acos, 1, rnd_mode); /* exact */
56 MPFR_SAVE_EXPO_FREE (expo);
57 return mpfr_check_range (acos, inexact, rnd_mode);
58 }
59 }
60
61 /* Set x_p=|x| */
62 sign = MPFR_SIGN (x);
63 mpfr_init2 (xp, MPFR_PREC (x));
64 mpfr_abs (xp, x, MPFR_RNDN); /* Exact */
65
66 compared = mpfr_cmp_ui (xp, 1);
67
68 if (MPFR_UNLIKELY (compared >= 0))
69 {
70 mpfr_clear (xp);
71 if (compared > 0) /* acos(x) = NaN for x > 1 */
72 {
73 MPFR_SET_NAN(acos);
74 MPFR_RET_NAN;
75 }
76 else
77 {
78 if (MPFR_IS_POS_SIGN (sign)) /* acos(+1) = +0 */
79 return mpfr_set_ui (acos, 0, rnd_mode);
80 else /* acos(-1) = Pi */
81 return mpfr_const_pi (acos, rnd_mode);
82 }
83 }
84
85 MPFR_SAVE_EXPO_MARK (expo);
86
87 /* Compute the supplement */
88 mpfr_ui_sub (xp, 1, xp, MPFR_RNDD);
89 if (MPFR_IS_POS_SIGN (sign))
90 supplement = 2 - 2 * MPFR_GET_EXP (xp);
91 else
92 supplement = 2 - MPFR_GET_EXP (xp);
93 mpfr_clear (xp);
94
95 prec = MPFR_PREC (acos);
96 prec += MPFR_INT_CEIL_LOG2(prec) + 10 + supplement;
97
98 /* VL: The following change concerning prec comes from r3145
99 "Optimize mpfr_acos by choosing a better initial precision."
100 but it doesn't seem to be correct and leads to problems (assertion
101 failure or very important inefficiency) with tiny arguments.
102 Therefore, I've disabled it. */
103 /* If x ~ 2^-N, acos(x) ~ PI/2 - x - x^3/6
104 If Prec < 2*N, we can't round since x^3/6 won't be counted. */
105 #if 0
106 if (MPFR_PREC (acos) >= MPFR_PREC (x) && MPFR_GET_EXP (x) < 0)
107 {
108 mpfr_uexp_t pmin = (mpfr_uexp_t) (-2 * MPFR_GET_EXP (x)) + 5;
109 MPFR_ASSERTN (pmin <= MPFR_PREC_MAX);
110 if (prec < pmin)
111 prec = pmin;
112 }
113 #endif
114
115 mpfr_init2 (tmp, prec);
116 mpfr_init2 (arcc, prec);
117
118 MPFR_ZIV_INIT (loop, prec);
119 for (;;)
120 {
121 /* acos(x) = Pi/2 - asin(x) = Pi/2 - atan(x/sqrt(1-x^2)) */
122 mpfr_sqr (tmp, x, MPFR_RNDN);
123 mpfr_ui_sub (tmp, 1, tmp, MPFR_RNDN);
124 mpfr_sqrt (tmp, tmp, MPFR_RNDN);
125 mpfr_div (tmp, x, tmp, MPFR_RNDN);
126 mpfr_atan (arcc, tmp, MPFR_RNDN);
127 mpfr_const_pi (tmp, MPFR_RNDN);
128 mpfr_div_2ui (tmp, tmp, 1, MPFR_RNDN);
129 mpfr_sub (arcc, tmp, arcc, MPFR_RNDN);
130
131 if (MPFR_LIKELY (MPFR_CAN_ROUND (arcc, prec - supplement,
132 MPFR_PREC (acos), rnd_mode)))
133 break;
134 MPFR_ZIV_NEXT (loop, prec);
135 mpfr_set_prec (tmp, prec);
136 mpfr_set_prec (arcc, prec);
137 }
138 MPFR_ZIV_FREE (loop);
139
140 inexact = mpfr_set (acos, arcc, rnd_mode);
141 mpfr_clear (tmp);
142 mpfr_clear (arcc);
143
144 MPFR_SAVE_EXPO_FREE (expo);
145 return mpfr_check_range (acos, inexact, rnd_mode);
146 }