1 /* mpfr_const_catalan -- compute Catalan's constant.
2
3 Copyright 2005-2023 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 /* Declare the cache */
27 MPFR_DECL_INIT_CACHE (__gmpfr_cache_const_catalan, mpfr_const_catalan_internal)
28
29 /* Set User Interface */
30 #undef mpfr_const_catalan
31 int
32 mpfr_const_catalan (mpfr_ptr x, mpfr_rnd_t rnd_mode) {
33 return mpfr_cache (x, __gmpfr_cache_const_catalan, rnd_mode);
34 }
35
36 /* return T, Q such that T/Q = sum(k!^2/(2k)!/(2k+1)^2, k=n1..n2-1) */
37 static void
38 S (mpz_t T, mpz_t P, mpz_t Q, unsigned long n1, unsigned long n2)
39 {
40 if (n2 == n1 + 1)
41 {
42 if (n1 == 0)
43 {
44 mpz_set_ui (P, 1);
45 mpz_set_ui (Q, 1);
46 }
47 else
48 {
49 mpz_set_ui (P, 2 * n1 - 1);
50 mpz_mul_ui (P, P, n1);
51 mpz_ui_pow_ui (Q, 2 * n1 + 1, 2);
52 mpz_mul_2exp (Q, Q, 1);
53 }
54 mpz_set (T, P);
55 }
56 else
57 {
58 unsigned long m = (n1 + n2) / 2;
59 mpz_t T2, P2, Q2;
60 S (T, P, Q, n1, m);
61 mpz_init (T2);
62 mpz_init (P2);
63 mpz_init (Q2);
64 S (T2, P2, Q2, m, n2);
65 mpz_mul (T, T, Q2);
66 mpz_mul (T2, T2, P);
67 mpz_add (T, T, T2);
68 mpz_mul (P, P, P2);
69 mpz_mul (Q, Q, Q2);
70 mpz_clear (T2);
71 mpz_clear (P2);
72 mpz_clear (Q2);
73 }
74 }
75
76 /* Don't need to save/restore exponent range: the cache does it.
77 Catalan's constant is G = sum((-1)^k/(2*k+1)^2, k=0..infinity).
78 We compute it using formula (31) of Victor Adamchik's page
79 "33 representations for Catalan's constant"
80 https://web.archive.org/web/20090624123133/http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm
81
82 G = Pi/8*log(2+sqrt(3)) + 3/8*sum(k!^2/(2k)!/(2k+1)^2,k=0..infinity)
83 */
84 int
85 mpfr_const_catalan_internal (mpfr_ptr g, mpfr_rnd_t rnd_mode)
86 {
87 mpfr_t x, y, z;
88 mpz_t T, P, Q;
89 mpfr_prec_t pg, p;
90 int inex;
91 MPFR_ZIV_DECL (loop);
92 MPFR_GROUP_DECL (group);
93
94 MPFR_LOG_FUNC (("rnd_mode=%d", rnd_mode),
95 ("g[%Pd]=%.*Rg inex=%d", mpfr_get_prec (g), mpfr_log_prec, g, inex));
96
97 /* Here are the WC (max prec = 100.000.000)
98 Once we have found a chain of 11, we only look for bigger chain.
99 Found 3 '1' at 0
100 Found 5 '1' at 9
101 Found 6 '0' at 34
102 Found 9 '1' at 176
103 Found 11 '1' at 705
104 Found 12 '0' at 913
105 Found 14 '1' at 12762
106 Found 15 '1' at 152561
107 Found 16 '0' at 171725
108 Found 18 '0' at 525355
109 Found 20 '0' at 529245
110 Found 21 '1' at 6390133
111 Found 22 '0' at 7806417
112 Found 25 '1' at 11936239
113 Found 27 '1' at 51752950
114 */
115 pg = MPFR_PREC (g);
116 p = pg + MPFR_INT_CEIL_LOG2 (pg) + 7;
117
118 MPFR_GROUP_INIT_3 (group, p, x, y, z);
119 mpz_init (T);
120 mpz_init (P);
121 mpz_init (Q);
122
123 MPFR_ZIV_INIT (loop, p);
124 for (;;) {
125 mpfr_sqrt_ui (x, 3, MPFR_RNDU);
126 mpfr_add_ui (x, x, 2, MPFR_RNDU);
127 mpfr_log (x, x, MPFR_RNDU);
128 mpfr_const_pi (y, MPFR_RNDU);
129 mpfr_mul (x, x, y, MPFR_RNDN);
130 S (T, P, Q, 0, (p - 1) / 2);
131 mpz_mul_ui (T, T, 3);
132 mpfr_set_z (y, T, MPFR_RNDU);
133 mpfr_set_z (z, Q, MPFR_RNDD);
134 mpfr_div (y, y, z, MPFR_RNDN);
135 mpfr_add (x, x, y, MPFR_RNDN);
136 mpfr_div_2ui (x, x, 3, MPFR_RNDN);
137
138 if (MPFR_LIKELY (MPFR_CAN_ROUND (x, p - 5, pg, rnd_mode)))
139 break;
140
141 MPFR_ZIV_NEXT (loop, p);
142 MPFR_GROUP_REPREC_3 (group, p, x, y, z);
143 }
144 MPFR_ZIV_FREE (loop);
145 inex = mpfr_set (g, x, rnd_mode);
146
147 MPFR_GROUP_CLEAR (group);
148 mpz_clear (T);
149 mpz_clear (P);
150 mpz_clear (Q);
151
152 return inex;
153 }