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
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
|
/* mpfr_const_pi -- compute Pi
Copyright 1999, 2000, 2001, 2002, 2003 Free Software Foundation.
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., 59 Temple Place - Suite 330, Boston,
MA 02111-1307, USA. */
#include "gmp.h"
#include "gmp-impl.h"
#include "longlong.h"
#include "mpfr.h"
#include "mpfr-impl.h"
static int mpfr_aux_pi _MPFR_PROTO ((mpfr_ptr, mpz_srcptr, long, int));
static int mpfr_pi_machin3 _MPFR_PROTO ((mpfr_ptr, mp_rnd_t));
#define A
#define A1 1
#define A2 2
#undef B
#define C
#define C1 3
#define C2 2
#define GENERIC mpfr_aux_pi
#define R_IS_RATIONAL
#define NO_FACTORIAL
#include "generic.c"
static int
mpfr_pi_machin3 (mpfr_ptr mylog, mp_rnd_t rnd_mode)
{
int prec, logn, prec_x;
int prec_i_want;
mpfr_t tmp1, tmp2, result, tmp3, tmp4, tmp5, tmp6;
mpz_t cst;
int inex = 0; /* here, 0 means not set */
MPFR_CLEAR_FLAGS (mylog);
prec_i_want = MPFR_PREC (mylog);
logn = __gmpfr_ceil_log2 ((double) prec_i_want);
prec_x = prec_i_want + logn + 5;
mpz_init (cst);
while (!inex)
{
prec = __gmpfr_ceil_log2 ((double) prec_x);
mpfr_init2(tmp1, prec_x);
mpfr_init2(tmp2, prec_x);
mpfr_init2(tmp3, prec_x);
mpfr_init2(tmp4, prec_x);
mpfr_init2(tmp5, prec_x);
mpfr_init2(tmp6, prec_x);
mpfr_init2(result, prec_x);
mpz_set_si(cst, -1);
mpfr_aux_pi(tmp1, cst, 268L*268L, prec - 4);
mpfr_div_ui(tmp1, tmp1, 268, GMP_RNDD);
mpfr_mul_ui(tmp1, tmp1, 61, GMP_RNDD);
mpfr_aux_pi(tmp2, cst, 343L*343L, prec - 4);
mpfr_div_ui(tmp2, tmp2, 343, GMP_RNDD);
mpfr_mul_ui(tmp2, tmp2, 122, GMP_RNDD);
mpfr_aux_pi(tmp3, cst, 557L*557L, prec - 4);
mpfr_div_ui(tmp3, tmp3, 557, GMP_RNDD);
mpfr_mul_ui(tmp3, tmp3, 115, GMP_RNDD);
mpfr_aux_pi(tmp4, cst, 1068L*1068L, prec - 4);
mpfr_div_ui(tmp4, tmp4, 1068, GMP_RNDD);
mpfr_mul_ui(tmp4, tmp4, 32, GMP_RNDD);
mpfr_aux_pi(tmp5, cst, 3458L*3458L, prec - 4);
mpfr_div_ui(tmp5, tmp5, 3458, GMP_RNDD);
mpfr_mul_ui(tmp5, tmp5, 83, GMP_RNDD);
mpfr_aux_pi(tmp6, cst, 27493L*27493L, prec - 4);
mpfr_div_ui(tmp6, tmp6, 27493, GMP_RNDD);
mpfr_mul_ui(tmp6, tmp6, 44, GMP_RNDD);
mpfr_add(result, tmp1, tmp2, GMP_RNDD);
mpfr_add(result, result, tmp3, GMP_RNDD);
mpfr_sub(result, result, tmp4, GMP_RNDD);
mpfr_add(result, result, tmp5, GMP_RNDD);
mpfr_add(result, result, tmp6, GMP_RNDD);
mpfr_mul_2ui(result, result, 2, GMP_RNDD);
mpfr_clear(tmp1);
mpfr_clear(tmp2);
mpfr_clear(tmp3);
mpfr_clear(tmp4);
mpfr_clear(tmp5);
mpfr_clear(tmp6);
if (mpfr_can_round (result, prec_x - 5, GMP_RNDD, GMP_RNDZ,
prec_i_want + (rnd_mode == GMP_RNDN)))
{
inex = mpfr_set (mylog, result, rnd_mode);
MPFR_ASSERTN (inex != 0);
}
else
{
prec_x += logn;
}
mpfr_clear (result);
}
mpz_clear (cst);
return inex;
}
/*
Set x to the value of Pi to precision MPFR_PREC(x) rounded to direction rnd_mode.
Use the formula giving the binary representation of Pi found by Simon Plouffe
David Bailey and Peter Borwein.
infinity 4 2 1 1
----- ------- - ------- - ------- - -------
\ 8 n + 1 8 n + 4 8 n + 5 8 n + 6
Pi = ) -------------------------------------
/ n
----- 16
n = 0
i.e. Pi*16^N = S(N) + R(N) where
S(N) = sum(16^(N-n)*(4/(8*n+1)-2/(8*n+4)-1/(8*n+5)-1/(8*n+6)), n=0..N-1)
R(N) = sum((4/(8*n+1)-2/(8*n+4)-1/(8*n+5)-1/(8*n+6))/16^(n-N), n=N..infinity)
Let f(n) = 4/(8*n+1)-2/(8*n+4)-1/(8*n+5)-1/(8*n+6), we can show easily that
f(n) < 15/(64*n^2), so R(N) < sum(15/(64*n^2)/16^(n-N), n=N..infinity)
< 15/64/N^2*sum(1/16^(n-N), n=N..infinity)
= 1/4/N^2
Now let S'(N) = sum(floor(16^(N-n)*(120*n^2+151*n+47),
(512*n^4+1024*n^3+712*n^2+194*n+15)), n=0..N-1)
S(N)-S'(N) <= sum(1, n=0..N-1) = N
so Pi*16^N-S'(N) <= N+1 (as 1/4/N^2 < 1)
*/
mpfr_t __mpfr_const_pi; /* stored value of Pi */
mp_prec_t __gmpfr_const_pi_prec = 0; /* precision of stored value */
static mp_rnd_t __mpfr_const_pi_rnd; /* rounding mode of stored value */
int
mpfr_const_pi (mpfr_ptr x, mp_rnd_t rnd_mode)
{
int N, oldN, n, prec;
mpz_t pi, num, den, d3, d2, tmp;
mpfr_t y;
int inex;
prec=MPFR_PREC(x);
/* has stored value enough precision ? */
if ((prec==__gmpfr_const_pi_prec && rnd_mode==__mpfr_const_pi_rnd) ||
(prec<=__gmpfr_const_pi_prec &&
mpfr_can_round(__mpfr_const_pi, __gmpfr_const_pi_prec,
__mpfr_const_pi_rnd, GMP_RNDZ,
prec + (rnd_mode == GMP_RNDN))))
{
return mpfr_set (x, __mpfr_const_pi, rnd_mode);
}
if (prec < 20000)
{
/* need to recompute */
N=1;
do
{
oldN = N;
N = (prec+3)/4 + __gmpfr_ceil_log2((double) N + 1.0);
}
while (N != oldN);
mpz_init(pi);
mpz_init(num);
mpz_init(den);
mpz_init(d3);
mpz_init(d2);
mpz_init(tmp);
mpz_set_ui(pi, 0);
mpz_set_ui(num, 16); /* num(-1) */
mpz_set_ui(den, 21); /* den(-1) */
mpz_set_si(d3, -2454);
mpz_set_ui(d2, 14736);
/* invariants: num=120*n^2+151*n+47,
den=512*n^4+1024*n^3+712*n^2+194*n+15
d3 = 2048*n^3+400*n-6, d2 = 6144*n^2-6144*n+2448
*/
for (n=0; n<N; n++)
{
/* num(n)-num(n-1) = 240*n+31 */
mpz_add_ui(num, num, 240*n+31); /* no overflow up to MPFR_PREC=71M */
/* d2(n) - d2(n-1) = 12288*(n-1) */
if (n>0)
mpz_add_ui(d2, d2, 12288*(n-1));
else
mpz_sub_ui(d2, d2, 12288);
/* d3(n) - d3(n-1) = d2 */
mpz_add(d3, d3, d2);
/* den(n)-den(n-1) = 2048*n^3 + 400n - 6 = d3 */
mpz_add(den, den, d3);
mpz_mul_2exp(tmp, num, 4*(N-n));
mpz_fdiv_q(tmp, tmp, den);
mpz_add(pi, pi, tmp);
}
inex = mpfr_set_z (x, pi, rnd_mode);
mpfr_init2 (y, mpfr_get_prec(x));
mpz_add_ui (pi, pi, N+1);
mpfr_set_z (y, pi, rnd_mode);
MPFR_ASSERTN (mpfr_cmp (x, y) == 0);
MPFR_SET_EXP (x, MPFR_GET_EXP(x) - 4*N);
mpz_clear(pi);
mpz_clear(num);
mpz_clear(den);
mpz_clear(d3);
mpz_clear(d2);
mpz_clear(tmp);
mpfr_clear(y);
}
else
inex = mpfr_pi_machin3 (x, rnd_mode);
/* store computed value */
if (__gmpfr_const_pi_prec==0)
mpfr_init2(__mpfr_const_pi, prec);
else
mpfr_set_prec(__mpfr_const_pi, prec);
mpfr_set(__mpfr_const_pi, x, rnd_mode);
__gmpfr_const_pi_prec=prec;
__mpfr_const_pi_rnd=rnd_mode;
return inex;
}
|
