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/* mpfr_const_pi -- compute Pi
Copyright 1999, 2000, 2001, 2002, 2003, 2004 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. */
#define MPFR_NEED_LONGLONG_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;
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)
*/
int
(mpfr_const_pi) (mpfr_ptr x, mp_rnd_t rnd_mode)
{
int N, oldN, n;
mpfr_prec_t prec;
mpz_t pi, num, den, d3, d2, tmp;
int inex;
mpfr_save_emin_emax ();
prec = MPFR_PREC(x);
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);
#ifdef WANT_ASSERT
{
mpfr_t y;
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_clear (y);
}
#endif
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);
}
else
inex = mpfr_pi_machin3 (x, rnd_mode);
mpfr_restore_emin_emax ();
return mpfr_check_range(x, inex, rnd_mode);
}
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