1 /* mpfr_const_pi -- compute Pi
2
3 Copyright 1999, 2000, 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 #include "mpfr-impl.h"
24
25 /* Declare the cache */
26 #ifndef MPFR_USE_LOGGING
27 MPFR_DECL_INIT_CACHE(__gmpfr_cache_const_pi, mpfr_const_pi_internal);
28 #else
29 MPFR_DECL_INIT_CACHE(__gmpfr_normal_pi, mpfr_const_pi_internal);
30 MPFR_DECL_INIT_CACHE(__gmpfr_logging_pi, mpfr_const_pi_internal);
31 mpfr_cache_ptr MPFR_THREAD_ATTR __gmpfr_cache_const_pi = __gmpfr_normal_pi;
32 #endif
33
34 /* Set User Interface */
35 #undef mpfr_const_pi
36 int
mpfr_const_pi(mpfr_ptr x,mpfr_rnd_t rnd_mode)37 mpfr_const_pi (mpfr_ptr x, mpfr_rnd_t rnd_mode) {
38 return mpfr_cache (x, __gmpfr_cache_const_pi, rnd_mode);
39 }
40
41 /* Don't need to save/restore exponent range: the cache does it */
42 int
mpfr_const_pi_internal(mpfr_ptr x,mpfr_rnd_t rnd_mode)43 mpfr_const_pi_internal (mpfr_ptr x, mpfr_rnd_t rnd_mode)
44 {
45 mpfr_t a, A, B, D, S;
46 mpfr_prec_t px, p, cancel, k, kmax;
47 MPFR_ZIV_DECL (loop);
48 int inex;
49
50 MPFR_LOG_FUNC
51 (("rnd_mode=%d", rnd_mode),
52 ("x[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(x), mpfr_log_prec, x, inex));
53
54 px = MPFR_PREC (x);
55
56 /* we need 9*2^kmax - 4 >= px+2*kmax+8 */
57 for (kmax = 2; ((px + 2 * kmax + 12) / 9) >> kmax; kmax ++);
58
59 p = px + 3 * kmax + 14; /* guarantees no recomputation for px <= 10000 */
60
61 mpfr_init2 (a, p);
62 mpfr_init2 (A, p);
63 mpfr_init2 (B, p);
64 mpfr_init2 (D, p);
65 mpfr_init2 (S, p);
66
67 MPFR_ZIV_INIT (loop, p);
68 for (;;) {
69 mpfr_set_ui (a, 1, MPFR_RNDN); /* a = 1 */
70 mpfr_set_ui (A, 1, MPFR_RNDN); /* A = a^2 = 1 */
71 mpfr_set_ui_2exp (B, 1, -1, MPFR_RNDN); /* B = b^2 = 1/2 */
72 mpfr_set_ui_2exp (D, 1, -2, MPFR_RNDN); /* D = 1/4 */
73
74 #define b B
75 #define ap a
76 #define Ap A
77 #define Bp B
78 for (k = 0; ; k++)
79 {
80 /* invariant: 1/2 <= B <= A <= a < 1 */
81 mpfr_add (S, A, B, MPFR_RNDN); /* 1 <= S <= 2 */
82 mpfr_div_2ui (S, S, 2, MPFR_RNDN); /* exact, 1/4 <= S <= 1/2 */
83 mpfr_sqrt (b, B, MPFR_RNDN); /* 1/2 <= b <= 1 */
84 mpfr_add (ap, a, b, MPFR_RNDN); /* 1 <= ap <= 2 */
85 mpfr_div_2ui (ap, ap, 1, MPFR_RNDN); /* exact, 1/2 <= ap <= 1 */
86 mpfr_mul (Ap, ap, ap, MPFR_RNDN); /* 1/4 <= Ap <= 1 */
87 mpfr_sub (Bp, Ap, S, MPFR_RNDN); /* -1/4 <= Bp <= 3/4 */
88 mpfr_mul_2ui (Bp, Bp, 1, MPFR_RNDN); /* -1/2 <= Bp <= 3/2 */
89 mpfr_sub (S, Ap, Bp, MPFR_RNDN);
90 MPFR_ASSERTN (mpfr_cmp_ui (S, 1) < 0);
91 cancel = mpfr_cmp_ui (S, 0) ? (mpfr_uexp_t) -mpfr_get_exp(S) : p;
92 /* MPFR_ASSERTN (cancel >= px || cancel >= 9 * (1 << k) - 4); */
93 mpfr_mul_2ui (S, S, k, MPFR_RNDN);
94 mpfr_sub (D, D, S, MPFR_RNDN);
95 /* stop when |A_k - B_k| <= 2^(k-p) i.e. cancel >= p-k */
96 if (cancel + k >= p)
97 break;
98 }
99 #undef b
100 #undef ap
101 #undef Ap
102 #undef Bp
103
104 mpfr_div (A, B, D, MPFR_RNDN);
105
106 /* MPFR_ASSERTN(p >= 2 * k + 8); */
107 if (MPFR_LIKELY (MPFR_CAN_ROUND (A, p - 2 * k - 8, px, rnd_mode)))
108 break;
109
110 p += kmax;
111 MPFR_ZIV_NEXT (loop, p);
112 mpfr_set_prec (a, p);
113 mpfr_set_prec (A, p);
114 mpfr_set_prec (B, p);
115 mpfr_set_prec (D, p);
116 mpfr_set_prec (S, p);
117 }
118 MPFR_ZIV_FREE (loop);
119 inex = mpfr_set (x, A, rnd_mode);
120
121 mpfr_clear (a);
122 mpfr_clear (A);
123 mpfr_clear (B);
124 mpfr_clear (D);
125 mpfr_clear (S);
126
127 return inex;
128 }
129