pqc/external/flint-2.4.3/arith/pi_chudnovsky.c

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2014-05-18 22:03:37 +00:00
/* Pi computation using Chudnovsky's algortithm.
* Copyright 2002, 2005 Hanhong Xue (macroxue at yahoo dot com)
* Modified 2005 by Torbjorn Granlund (tege at swox dot com) to allow more than
2G digits to be computed. Modified 2009 by Torbjorn Granlund for GMPbench.
* Modified 2011 by Fredrik Johansson to make reentrant and adapt for
use in FLINT.
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
* 1. Redistributions of source code must retain the above copyright notice,
* this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright notice,
* this list of conditions and the following disclaimer in the documentation
* and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHORS ``AS IS'' AND ANY EXPRESS OR
* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
* MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO
* EVENT SHALL THE AUTHORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
* PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS;
* OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY,
* WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR
* OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF
* ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
#include <assert.h>
#include <math.h>
#include <stdlib.h>
#include <string.h>
#include <gmp.h>
#include <mpfr.h>
#include "arith.h"
#define A 13591409
#define B 545140134
#define C 640320
#define D 12
#define BITS_PER_DIGIT 3.32192809488736234787
#define DIGITS_PER_ITER 14.1816474627254776555
#define DOUBLE_PREC 53
#define min(x,y) ((x)<(y)?(x):(y))
#define max(x,y) ((x)>(y)?(x):(y))
typedef struct {
ulong max_facs;
ulong num_facs;
ulong *fac;
ulong *pow;
} fac_t[1];
typedef struct {
slong fac;
slong pow;
slong nxt;
} sieve_t;
typedef struct
{
sieve_t *sieve;
slong sieve_size;
fac_t ftmp, fmul;
mpz_t gcd;
mpz_t *pstack, *qstack, *gstack;
fac_t *fpstack, *fgstack;
slong top;
}
pi_state_struct;
typedef pi_state_struct pi_state[1];
#define INIT_FACS 32
#define p1 (state->pstack[state->top])
#define q1 (state->qstack[state->top])
#define g1 (state->gstack[state->top])
#define fp1 (state->fpstack[state->top])
#define fg1 (state->fgstack[state->top])
#define p2 (state->pstack[state->top+1])
#define q2 (state->qstack[state->top+1])
#define g2 (state->gstack[state->top+1])
#define fp2 (state->fpstack[state->top+1])
#define fg2 (state->fgstack[state->top+1])
/* r = sqrt(x) */
void
my_sqrt_ui(pi_state state, mpf_t t1, mpf_t t2, mpf_t r, ulong x)
{
ulong prec, bits, prec0;
prec0 = mpf_get_prec(r);
if (prec0 <= DOUBLE_PREC)
{
mpf_set_d(r, sqrt(x));
return;
}
bits = 0;
for (prec = prec0; prec > DOUBLE_PREC; )
{
int bit = prec&1;
prec = (prec+bit)/2;
bits = bits*2+bit;
}
mpf_set_prec_raw(t1, DOUBLE_PREC);
mpf_set_d(t1, 1/sqrt(x));
while (prec < prec0)
{
prec *=2;
if (prec < prec0)
{
/* t1 = t1+t1*(1-x*t1*t1)/2; */
mpf_set_prec_raw(t2, prec);
mpf_mul(t2, t1, t1); /* half x half -> full */
mpf_mul_ui(t2, t2, x);
mpf_ui_sub(t2, 1, t2);
mpf_set_prec_raw(t2, prec/2);
mpf_div_2exp(t2, t2, 1);
mpf_mul(t2, t2, t1); /* half x half -> half */
mpf_set_prec_raw(t1, prec);
mpf_add(t1, t1, t2);
}
else
{
break;
}
prec -= (bits&1);
bits /=2;
}
/* t2=x*t1, t1 = t2+t1*(x-t2*t2)/2; */
mpf_set_prec_raw(t2, prec0/2);
mpf_mul_ui(t2, t1, x);
mpf_mul(r, t2, t2); /* half x half -> full */
mpf_ui_sub(r, x, r);
mpf_mul(t1, t1, r); /* half x half -> half */
mpf_div_2exp(t1, t1, 1);
mpf_add(r, t1, t2);
}
/* r = y/x WARNING: r cannot be the same as y. */
void
my_div(pi_state state, mpf_t t1, mpf_t t2, mpf_t r, mpf_t y, mpf_t x)
{
ulong prec, bits, prec0;
prec0 = mpf_get_prec(r);
if (prec0 <= DOUBLE_PREC)
{
mpf_set_d(r, mpf_get_d(y) / mpf_get_d(x));
return;
}
bits = 0;
for (prec = prec0; prec > DOUBLE_PREC; )
{
int bit = prec & 1;
prec = (prec + bit) / 2;
bits = bits*2 + bit;
}
mpf_set_prec_raw(t1, DOUBLE_PREC);
mpf_ui_div(t1, 1, x);
while (prec < prec0)
{
prec *= 2;
if (prec < prec0)
{
/* t1 = t1+t1*(1-x*t1); */
mpf_set_prec_raw(t2, prec);
mpf_mul(t2, x, t1); /* full x half -> full */
mpf_ui_sub(t2, 1, t2);
mpf_set_prec_raw(t2, prec/2);
mpf_mul(t2, t2, t1); /* half x half -> half */
mpf_set_prec_raw(t1, prec);
mpf_add(t1, t1, t2);
}
else
{
prec = prec0;
/* t2=y*t1, t1 = t2+t1*(y-x*t2); */
mpf_set_prec_raw(t2, prec / 2);
mpf_mul(t2, t1, y); /* half x half -> half */
mpf_mul(r, x, t2); /* full x half -> full */
mpf_sub(r, y, r);
mpf_mul(t1, t1, r); /* half x half -> half */
mpf_add(r, t1, t2);
break;
}
prec -= (bits & 1);
bits /= 2;
}
}
/*///////////////////////////////////////////////////////////////////////////*/
static __inline__ void
fac_reset(fac_t f)
{
f[0].num_facs = 0;
}
static __inline__ void
fac_init_size(fac_t f, slong s)
{
if (s < INIT_FACS)
s = INIT_FACS;
f[0].fac = flint_malloc(s*sizeof(ulong)*2);
f[0].pow = f[0].fac + s;
f[0].max_facs = s;
fac_reset(f);
}
static __inline__ void
fac_init(fac_t f)
{
fac_init_size(f, INIT_FACS);
}
static __inline__ void
fac_clear(fac_t f)
{
flint_free(f[0].fac);
}
static __inline__ void
fac_resize(fac_t f, slong s)
{
if (f[0].max_facs < s)
{
fac_clear(f);
fac_init_size(f, s);
}
}
/* f = base^pow */
static __inline__ void
fac_set_bp(pi_state state, fac_t f, ulong base, slong pow)
{
slong i;
assert(base<state->sieve_size);
for (i=0, base/=2; base>0; i++, base = state->sieve[base].nxt)
{
f[0].fac[i] = state->sieve[base].fac;
f[0].pow[i] = state->sieve[base].pow*pow;
}
f[0].num_facs = i;
assert(i<=f[0].max_facs);
}
/* r = f*g */
static __inline__ void
fac_mul2(pi_state state, fac_t r, fac_t f, fac_t g)
{
slong i, j, k;
for (i=j=k=0; i<f[0].num_facs && j<g[0].num_facs; k++)
{
if (f[0].fac[i] == g[0].fac[j])
{
r[0].fac[k] = f[0].fac[i];
r[0].pow[k] = f[0].pow[i] + g[0].pow[j];
i++; j++;
}
else if (f[0].fac[i] < g[0].fac[j])
{
r[0].fac[k] = f[0].fac[i];
r[0].pow[k] = f[0].pow[i];
i++;
}
else
{
r[0].fac[k] = g[0].fac[j];
r[0].pow[k] = g[0].pow[j];
j++;
}
}
for (; i<f[0].num_facs; i++, k++)
{
r[0].fac[k] = f[0].fac[i];
r[0].pow[k] = f[0].pow[i];
}
for (; j<g[0].num_facs; j++, k++)
{
r[0].fac[k] = g[0].fac[j];
r[0].pow[k] = g[0].pow[j];
}
r[0].num_facs = k;
assert(k<=r[0].max_facs);
}
/* f *= g */
static __inline__ void
fac_mul(pi_state state, fac_t f, fac_t g)
{
fac_t tmp;
fac_resize(state->fmul, f[0].num_facs + g[0].num_facs);
fac_mul2(state, state->fmul, f, g);
tmp[0] = f[0];
f[0] = state->fmul[0];
state->fmul[0] = tmp[0];
}
/* f *= base^pow */
static __inline__ void
fac_mul_bp(pi_state state, fac_t f, ulong base, ulong pow)
{
fac_set_bp(state, state->ftmp, base, pow);
fac_mul(state, f, state->ftmp);
}
/* remove factors of power 0 */
static __inline__ void
fac_compact(fac_t f)
{
slong i, j;
for (i=0, j=0; i<f[0].num_facs; i++)
{
if (f[0].pow[i]>0)
{
if (j < i)
{
f[0].fac[j] = f[0].fac[i];
f[0].pow[j] = f[0].pow[i];
}
j++;
}
}
f[0].num_facs = j;
}
/* convert factorized form to number */
void
bs_mul(pi_state state, mpz_t r, slong a, slong b)
{
slong i, j;
if (b-a<=32)
{
flint_mpz_set_ui(r, 1);
for (i=a; i<b; i++)
for (j=0; j<state->fmul[0].pow[i]; j++)
flint_mpz_mul_ui(r, r, state->fmul[0].fac[i]);
}
else
{
mpz_t r2;
mpz_init(r2);
bs_mul(state, r2, a, (a+b)/2);
bs_mul(state, r, (a+b)/2, b);
mpz_mul(r, r, r2);
mpz_clear(r2);
}
}
/* f /= gcd(f,g), g /= gcd(f,g) */
void
fac_remove_gcd(pi_state state, mpz_t p, fac_t fp, mpz_t g, fac_t fg)
{
slong i, j, k, c;
fac_resize(state->fmul, min(fp->num_facs, fg->num_facs));
for (i=j=k=0; i < fp->num_facs && j < fg->num_facs; )
{
if (fp->fac[i] == fg->fac[j])
{
c = min(fp->pow[i], fg->pow[j]);
fp->pow[i] -= c;
fg->pow[j] -= c;
state->fmul->fac[k] = fp->fac[i];
state->fmul->pow[k] = c;
i++; j++; k++;
}
else if (fp->fac[i] < fg->fac[j])
{
i++;
}
else
{
j++;
}
}
state->fmul->num_facs = k;
assert(k <= state->fmul->max_facs);
if (state->fmul->num_facs)
{
bs_mul(state, state->gcd, 0, state->fmul->num_facs);
mpz_divexact(p, p, state->gcd);
mpz_divexact(g, g, state->gcd);
fac_compact(fp);
fac_compact(fg);
}
}
/*///////////////////////////////////////////////////////////////////////////*/
/* binary splitting */
void
bs(pi_state state, ulong a, ulong b,
unsigned gflag, slong level)
{
ulong i, mid;
if (b - a == 1)
{
/*
g(b-1,b) = (6b-5)(2b-1)(6b-1)
p(b-1,b) = b^3 * C^3 / 24
q(b-1,b) = (-1)^b*g(b-1,b)*(A+Bb).
*/
flint_mpz_set_ui(p1, b);
flint_mpz_mul_ui(p1, p1, b);
flint_mpz_mul_ui(p1, p1, b);
flint_mpz_mul_ui(p1, p1, (C/24)*(C/24));
flint_mpz_mul_ui(p1, p1, C*24);
flint_mpz_set_ui(g1, 2*b-1);
flint_mpz_mul_ui(g1, g1, 6*b-1);
flint_mpz_mul_ui(g1, g1, 6*b-5);
flint_mpz_set_ui(q1, b);
flint_mpz_mul_ui(q1, q1, B);
flint_mpz_add_ui(q1, q1, A);
mpz_mul (q1, q1, g1);
if (b%2)
mpz_neg(q1, q1);
i=b;
while ((i&1)==0) i>>=1;
fac_set_bp(state, fp1, i, 3); /* b^3 */
fac_mul_bp(state, fp1, 3*5*23*29, 3);
fp1[0].pow[0]--;
fac_set_bp(state, fg1, 2*b-1, 1); /* 2b-1 */
fac_mul_bp(state, fg1, 6*b-1, 1); /* 6b-1 */
fac_mul_bp(state, fg1, 6*b-5, 1); /* 6b-5 */
}
else
{
/*
p(a,b) = p(a,m) * p(m,b)
g(a,b) = g(a,m) * g(m,b)
q(a,b) = q(a,m) * p(m,b) + q(m,b) * g(a,m)
*/
mid = a+(b-a)*0.5224; /* tuning parameter */
bs(state, a, mid, 1, level+1);
state->top++;
bs(state, mid, b, gflag, level+1);
state->top--;
if (level>=4) { /* tuning parameter */
fac_remove_gcd(state, p2, fp2, g1, fg1);
}
mpz_mul(p1, p1, p2);
mpz_mul(q1, q1, p2);
mpz_mul(q2, q2, g1);
mpz_add(q1, q1, q2);
fac_mul(state, fp1, fp2);
if (gflag)
{
mpz_mul(g1, g1, g2);
fac_mul(state, fg1, fg2);
}
}
}
void
build_sieve(pi_state state, slong n, sieve_t *s)
{
slong m, i, j, k;
state->sieve_size = n;
m = (slong)sqrt(n);
memset(s, 0, sizeof(sieve_t)*n/2);
s[1/2].fac = 1;
s[1/2].pow = 1;
for (i=3; i<=n; i+=2)
{
if (s[i/2].fac == 0)
{
s[i/2].fac = i;
s[i/2].pow = 1;
if (i <= m)
{
for (j=i*i, k=i/2; j<=n; j+=i+i, k++)
{
if (s[j/2].fac==0)
{
s[j/2].fac = i;
if (s[k].fac == i)
{
s[j/2].pow = s[k].pow + 1;
s[j/2].nxt = s[k].nxt;
}
else
{
s[j/2].pow = 1;
s[j/2].nxt = k;
}
}
}
}
}
}
}
void
mpfr_pi_chudnovsky(mpfr_t res, mpfr_rnd_t rnd)
{
mpf_t pi, qi, t1, t2;
mpfr_prec_t prec;
slong i, depth=1, terms;
pi_state state;
prec = mpfr_get_prec(res) + 64;
terms = prec / (BITS_PER_DIGIT * DIGITS_PER_ITER);
while ((WORD(1)<<depth)<terms)
depth++;
depth++;
state->top = 0;
state->sieve_size = max(3*5*23*29+1, terms*6);
state->sieve = (sieve_t *)flint_malloc(sizeof(sieve_t)*(state->sieve_size)/2);
build_sieve(state, state->sieve_size, state->sieve);
/* allocate stacks */
state->pstack = flint_malloc(sizeof(mpz_t)*depth);
state->qstack = flint_malloc(sizeof(mpz_t)*depth);
state->gstack = flint_malloc(sizeof(mpz_t)*depth);
state->fpstack = flint_malloc(sizeof(fac_t)*depth);
state->fgstack = flint_malloc(sizeof(fac_t)*depth);
for (i=0; i<depth; i++)
{
mpz_init(state->pstack[i]);
mpz_init(state->qstack[i]);
mpz_init(state->gstack[i]);
fac_init(state->fpstack[i]);
fac_init(state->fgstack[i]);
}
mpz_init(state->gcd);
fac_init(state->ftmp);
fac_init(state->fmul);
/* begin binary splitting process */
if (terms<=0)
{
flint_mpz_set_ui(p2,1);
flint_mpz_set_ui(q2,0);
flint_mpz_set_ui(g2,1);
}
else
{
bs(state, 0,terms,0,0);
}
/* free some resources */
flint_free(state->sieve);
mpz_clear(state->gcd);
fac_clear(state->ftmp);
fac_clear(state->fmul);
for (i=1; i<depth; i++)
{
mpz_clear(state->pstack[i]);
mpz_clear(state->qstack[i]);
mpz_clear(state->gstack[i]);
fac_clear(state->fpstack[i]);
fac_clear(state->fgstack[i]);
}
mpz_clear(state->gstack[0]);
fac_clear(state->fpstack[0]);
fac_clear(state->fgstack[0]);
flint_free(state->gstack);
flint_free(state->fpstack);
flint_free(state->fgstack);
/*
p*(C/D)*sqrt(C)
pi = -----------------
(q+A*p)
*/
flint_mpz_addmul_ui(q1, p1, A);
flint_mpz_mul_ui(p1, p1, C/D);
mpf_init2(pi, prec);
mpf_set_z(pi, p1);
mpz_clear(p1);
mpf_init2(qi, prec);
mpf_set_z(qi, q1);
mpz_clear(q1);
flint_free(state->pstack);
flint_free(state->qstack);
/* initialize temp float variables for sqrt & div */
mpf_init2(t1, prec);
mpf_init2(t2, prec);
/* final step */
my_div(state, t1, t2, qi, pi, qi);
my_sqrt_ui(state, t1, t2, pi, C);
mpf_mul(qi, qi, pi);
mpfr_set_f(res, qi, rnd);
/* free float resources */
mpf_clear(pi);
mpf_clear(qi);
mpf_clear(t1);
mpf_clear(t2);
}