1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
|
/* mpfr_const_pi -- compute Pi
Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006 Free Software Foundation, Inc.
This file is part of the MPFR Library.
The MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 2.1 of the License, or (at your
option) any later version.
The MPFR Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the MPFR Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
MA 02110-1301, USA. */
#include "mpfr-impl.h"
/* Declare the cache */
MPFR_DECL_INIT_CACHE(__gmpfr_cache_const_pi, mpfr_const_pi_internal);
/* Set User Interface */
#undef mpfr_const_pi
int
mpfr_const_pi (mpfr_ptr x, mp_rnd_t rnd_mode) {
return mpfr_cache (x, __gmpfr_cache_const_pi, rnd_mode);
}
/* Don't need to save/restore exponent range: the cache does it */
int
mpfr_const_pi_internal (mpfr_ptr x, mp_rnd_t rnd_mode)
{
mpfr_t a, A, B, D, S;
mp_prec_t px, p, cancel, k, kmax;
MPFR_ZIV_DECL (loop);
int inex;
MPFR_LOG_FUNC (("rnd_mode=%d", rnd_mode), ("x[%#R]=%R inex=%d", x, x, inex));
px = MPFR_PREC (x);
/* we need 9*2^kmax - 4 >= px+2*kmax+8 */
for (kmax = 2; ((px + 2 * kmax + 12) / 9) >> kmax; kmax ++);
p = px + 3 * kmax + 14; /* guarantees no recomputation for px <= 10000 */
mpfr_init2 (a, p);
mpfr_init2 (A, p);
mpfr_init2 (B, p);
mpfr_init2 (D, p);
mpfr_init2 (S, p);
MPFR_ZIV_INIT (loop, p);
for (;;) {
mpfr_set_ui (a, 1, GMP_RNDN); /* a = 1 */
mpfr_set_ui (A, 1, GMP_RNDN); /* A = a^2 = 1 */
mpfr_set_ui_2exp (B, 1, -1, GMP_RNDN); /* B = b^2 = 1/2 */
mpfr_set_ui_2exp (D, 1, -2, GMP_RNDN); /* D = 1/4 */
#define b B
#define ap a
#define Ap A
#define Bp B
for (k = 0, cancel = 0; ; k++)
{
/* invariant: 1/2 <= B <= A <= a < 1 */
mpfr_add (S, A, B, GMP_RNDN); /* 1 <= S <= 2 */
mpfr_div_2exp (S, S, 2, GMP_RNDN); /* exact, 1/4 <= S <= 1/2 */
mpfr_sqrt (b, B, GMP_RNDN); /* 1/2 <= b <= 1 */
mpfr_add (ap, a, b, GMP_RNDN); /* 1 <= ap <= 2 */
mpfr_div_2exp (ap, ap, 1, GMP_RNDN); /* exact, 1/2 <= ap <= 1 */
mpfr_mul (Ap, ap, ap, GMP_RNDN); /* 1/4 <= Ap <= 1 */
mpfr_sub (Bp, Ap, S, GMP_RNDN); /* -1/4 <= Bp <= 3/4 */
mpfr_mul_2exp (Bp, Bp, 1, GMP_RNDN); /* -1/2 <= Bp <= 3/2 */
mpfr_sub (S, Ap, Bp, GMP_RNDN);
MPFR_ASSERTN (mpfr_cmp_ui (S, 1) < 0);
cancel = mpfr_cmp_ui (S, 0) ? (mpfr_uexp_t) -mpfr_get_exp(S) : p;
/* MPFR_ASSERTN (cancel >= px || cancel >= 9 * (1 << k) - 4); */
mpfr_mul_2exp (S, S, k, GMP_RNDN);
mpfr_sub (D, D, S, GMP_RNDN);
/* stop when |A_k - B_k| <= 2^(k-p) i.e. cancel >= p-k */
if (cancel + k >= p)
break;
}
#undef b
#undef ap
#undef Ap
#undef Bp
mpfr_div (A, B, D, GMP_RNDN);
/* MPFR_ASSERTN(p >= 2 * k + 8); */
if (MPFR_LIKELY (MPFR_CAN_ROUND (A, p - 2 * k - 8, px, rnd_mode)))
break;
p += kmax;
MPFR_ZIV_NEXT (loop, p);
mpfr_set_prec (a, p);
mpfr_set_prec (A, p);
mpfr_set_prec (B, p);
mpfr_set_prec (D, p);
mpfr_set_prec (S, p);
}
MPFR_ZIV_FREE (loop);
inex = mpfr_set (x, A, rnd_mode);
mpfr_clear (a);
mpfr_clear (A);
mpfr_clear (B);
mpfr_clear (D);
mpfr_clear (S);
return inex;
}
|
