Parallel GMP-Chudnovsky using OpenMP
David Carver
dcarver at tacc.utexas.edu
Mon Oct 13 23:03:37 CEST 2008
Hi!
Since GNU GCC 4.3 now supports OpenMP and most new AMD and Intel processors have more than one core, I thought a parallel version of the gmp-chudnovsky program would be an interesting exercise. This simple parallel version of gmp-chudnovsky has the factorization performance enhancement removed, so running with one core will not be as fast as the original version. I hope to post a parallel version with the factorization performance enhancement in the near future.
Later,
David Carver
/* Pi computation using Chudnovsky's algortithm.
* Copyright 2002, 2005 Hanhong Xue (macroxue at yahoo dot com)
* Slightly modified 2005 by Torbjorn Granlund (tege at swox dot com) to allow
more than 2G digits to be computed.
* Modifed 2008 by David Carver (dcarver at tacc dot utexas dot edu) to enable
multi-threading using the algorithm from "Computation of High-Precision
Mathematical Constants in a Combined Cluster and Grid Environment" by
Daisuke Takahashi, Mitsuhisa Sato, and Taisuke Boku.
For GNU gcc 4.3
gcc -fopenmp -Wall -O2 -o pgmp-chudnovsky pchudnovsky.c -lgmp -lm
For Intel 10.1 compiler
icc -openmp -O2 -o pgmp-chudnovsky pgmp-chudnovsky.c -lgmp -lm
For AIX xlc
xlc_r -qsmp=omp -O2 -o pgmp-chudnovsky pchudnovsky.c -lgmp -lm
* 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 <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <time.h>
#include "gmp.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
char *prog_name;
#if CHECK_MEMUSAGE
#undef CHECK_MEMUSAGE
#define CHECK_MEMUSAGE \
do { \
char buf[100]; \
snprintf (buf, 100, \
"ps aguxw | grep '[%c]%s'", prog_name[0], prog_name+1); \
system (buf); \
} while (0)
#else
#undef CHECK_MEMUSAGE
#define CHECK_MEMUSAGE
#endif
////////////////////////////////////////////////////////////////////////////
mpf_t t1, t2;
// r = sqrt(x)
void
my_sqrt_ui(mpf_t r, unsigned long x)
{
unsigned long 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 {
prec = prec0;
/* t2=x*t1, t1 = t2+t1*(x-t2*t2)/2; */
mpf_set_prec_raw(t2, prec/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);
break;
}
prec -= (bits&1);
bits /=2;
}
}
// r = y/x WARNING: r cannot be the same as y.
void
my_div(mpf_t r, mpf_t y, mpf_t x)
{
unsigned long 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;
}
}
////////////////////////////////////////////////////////////////////////////
int out=0;
mpz_t **pstack, **qstack, **gstack;
long int cores=1, depth, cores_depth;
double progress=0, percent;
// binary splitting
void
sum(unsigned long i, unsigned long j, unsigned long gflag)
{
mpz_mul(pstack[i][0], pstack[i][0], pstack[j][0]);
mpz_mul(qstack[i][0], qstack[i][0], pstack[j][0]);
mpz_mul(qstack[j][0], qstack[j][0], gstack[i][0]);
mpz_add(qstack[i][0], qstack[i][0], qstack[j][0]);
if (gflag) {
mpz_mul(gstack[i][0], gstack[i][0], gstack[j][0]);
}
}
void
bs(unsigned long a, unsigned long b, unsigned long gflag, unsigned long level, unsigned long index, unsigned long top)
{
unsigned long mid;
int ccc;
if (out&2) {
fprintf(stderr,"bs: a = %ld b = %ld gflag = %ld index = %ld level = %ld top = %ld \n", a,b,gflag,index,level,top);
fflush(stderr);
}
if ((b > a) && (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).
*/
mpz_set_ui(pstack[index][top], b);
mpz_mul_ui(pstack[index][top], pstack[index][top], b);
mpz_mul_ui(pstack[index][top], pstack[index][top], b);
mpz_mul_ui(pstack[index][top], pstack[index][top], (C/24)*(C/24));
mpz_mul_ui(pstack[index][top], pstack[index][top], C*24);
mpz_set_ui(gstack[index][top], 2*b-1);
mpz_mul_ui(gstack[index][top], gstack[index][top], 6*b-1);
mpz_mul_ui(gstack[index][top], gstack[index][top], 6*b-5);
mpz_set_ui(qstack[index][top], b);
mpz_mul_ui(qstack[index][top], qstack[index][top], B);
mpz_add_ui(qstack[index][top], qstack[index][top], A);
mpz_mul (qstack[index][top], qstack[index][top], gstack[index][top]);
if (b%2)
mpz_neg(qstack[index][top], qstack[index][top]);
if (b>(int)(progress)) {
fprintf(stderr,"."); fflush(stderr);
progress += percent*2;
}
} 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(a, mid, 1, level+1, index, top);
bs(mid, b, gflag, level+1, index, top+1);
ccc = level == 0;
if (ccc) CHECK_MEMUSAGE;
mpz_mul(pstack[index][top], pstack[index][top], pstack[index][top+1]);
if (ccc) CHECK_MEMUSAGE;
mpz_mul(qstack[index][top], qstack[index][top], pstack[index][top+1]);
if (ccc) CHECK_MEMUSAGE;
mpz_mul(qstack[index][top+1], qstack[index][top+1], gstack[index][top]);
if (ccc) CHECK_MEMUSAGE;
mpz_add(qstack[index][top], qstack[index][top], qstack[index][top+1]);
if (gflag) {
mpz_mul(gstack[index][top], gstack[index][top], gstack[index][top+1]);
}
}
}
int
main(int argc, char *argv[])
{
mpf_t pi, qi;
long int d=100, terms, i, j, k, cores_size;
unsigned long psize, qsize, mid;
clock_t begin, mid0, mid1, mid2, mid3, mid4, end;
prog_name = argv[0];
if (argc==1) {
fprintf(stderr,"\nSyntax: %s <digits> <option> <cores>\n",prog_name);
fprintf(stderr," <digits> digits of pi to output\n");
fprintf(stderr," <option> 0 - just run (default)\n");
fprintf(stderr," 1 - output digits\n");
fprintf(stderr," 2 - debug\n");
fprintf(stderr," <cores> number of cores (default 1)\n");
exit(1);
}
if (argc>1)
d = strtoul(argv[1], 0, 0);
if (argc>2)
out = atoi(argv[2]);
if (argc>3)
cores = atoi(argv[3]);
terms = d/DIGITS_PER_ITER;
depth = 0;
while ((1L<<depth)<terms)
depth++;
depth++;
if (cores < 1) {
fprintf(stderr,"Number of cores reset from %ld to 1\n",cores);
fflush(stderr);
cores = 1;
}
if ((terms > 0) && (terms < cores)) {
fprintf(stderr,"Number of cores reset from %ld to %ld\n",cores,terms);
fflush(stderr);
cores = terms;
}
cores_depth = 0;
while ((1L<<cores_depth)<cores)
cores_depth++;
cores_size=pow(2,cores_depth);
percent = terms/100.0;
fprintf(stderr,"#terms=%ld, depth=%ld, cores=%ld\n", terms, depth, cores);
begin = mid0 = clock();
pstack = malloc(sizeof(mpz_t)*cores);
qstack = malloc(sizeof(mpz_t)*cores);
gstack = malloc(sizeof(mpz_t)*cores);
/* allocate stacks */
for (j = 0; j < cores_size; j++) {
pstack[j] = malloc(sizeof(mpz_t)*depth);
qstack[j] = malloc(sizeof(mpz_t)*depth);
gstack[j] = malloc(sizeof(mpz_t)*depth);
for (i = 0; i < depth; i++) {
mpz_init(pstack[j][i]);
mpz_init(qstack[j][i]);
mpz_init(gstack[j][i]);
}
}
/* begin binary splitting process */
if (terms<=0) {
mpz_set_ui(pstack[0][0],1);
mpz_set_ui(qstack[0][0],0);
mpz_set_ui(gstack[0][0],1);
} else {
mid0 = clock();
mid = terms / cores;
#ifdef _OPENMP
#pragma omp parallel for
#endif
for (i = 0; i < cores; i++) {
if (i < (cores-1))
bs(i*mid, (i+1)*mid, 1, 0, i, 0);
else
bs(i*mid, terms, 1, 0, i, 0);
}
for (k = 1; k < cores_size; k*=2) {
#ifdef _OPENMP
#pragma omp parallel for
#endif
for (i = 0; i < cores; i=i+2*k) {
if (i+k < cores) {
sum( i, i+k, 1);
}
}
}
for (j=0; j<cores_size; j++) {
free(gstack[j]);
}
free(gstack);
}
mid1 = clock();
fprintf(stderr,"\nbs time = %6.3f\n", (double)(mid1-mid0)/CLOCKS_PER_SEC);
/* prepare to convert integers to floats */
mpf_set_default_prec((long int)(d*BITS_PER_DIGIT+16));
/*
p*(C/D)*sqrt(C)
pi = -----------------
(q+A*p)
*/
psize = mpz_sizeinbase(pstack[0][0],10);
qsize = mpz_sizeinbase(qstack[0][0],10);
mpz_addmul_ui(qstack[0][0], pstack[0][0], A);
mpz_mul_ui(pstack[0][0], pstack[0][0], C/D);
mpf_init(pi);
mpf_set_z(pi, pstack[0][0]);
mpz_clear(pstack[0][0]);
mpf_init(qi);
mpf_set_z(qi, qstack[0][0]);
mpz_clear(qstack[0][0]);
free(pstack[0]);
free(qstack[0]);
free(pstack);
free(qstack);
mid2 = clock();
//fprintf(stderr,"time = %6.3f\n", (double)(mid2-mid1)/CLOCKS_PER_SEC);
/* initialize temp float variables for sqrt & div */
mpf_init(t1);
mpf_init(t2);
//mpf_set_prec_raw(t1, mpf_get_prec(pi));
/* final step */
fprintf(stderr,"div "); fflush(stderr);
my_div(qi, pi, qi);
mid3 = clock();
fprintf(stderr,"time = %6.3f\n", (double)(mid3-mid2)/CLOCKS_PER_SEC);
fprintf(stderr,"sqrt "); fflush(stderr);
my_sqrt_ui(pi, C);
mid4 = clock();
fprintf(stderr,"time = %6.3f\n", (double)(mid4-mid3)/CLOCKS_PER_SEC);
fprintf(stderr,"mul "); fflush(stderr);
mpf_mul(qi, qi, pi);
end = clock();
fprintf(stderr,"time = %6.3f\n", (double)(end-mid4)/CLOCKS_PER_SEC);
fprintf(stderr,"total time = %6.3f\n", (double)(end-begin)/CLOCKS_PER_SEC);
fflush(stderr);
fprintf(stderr," P size=%ld digits (%f)\n"
" Q size=%ld digits (%f)\n",
psize, (double)psize/d, qsize, (double)qsize/d);
/* output Pi and timing statistics */
if (out&1) {
fprintf(stdout,"pi(0,%ld)=\n", terms);
mpf_out_str(stdout, 10, d+2, qi);
fprintf(stdout,"\n");
}
/* free float resources */
mpf_clear(pi);
mpf_clear(qi);
mpf_clear(t1);
mpf_clear(t2);
exit (0);
}
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