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/* mpfr_grandom (rop1, rop2, state, rnd_mode) -- Generate up to two
pseudorandom real numbers according to a standard normal Gaussian
distribution and round it to the precision of rop1, rop2 according
to the given rounding mode.
Copyright 2011-2021 Free Software Foundation, Inc.
Contributed by the AriC and Caramba projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU 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 3 of the License, or (at your
option) any later version.
The GNU 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 GNU MPFR Library; see the file COPYING.LESSER. If not, see
https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
/* #define MPFR_NEED_LONGLONG_H */
#include "mpfr-impl.h"
int
mpfr_grandom (mpfr_ptr rop1, mpfr_ptr rop2, gmp_randstate_t rstate,
mpfr_rnd_t rnd)
{
int inex1, inex2, s1, s2;
mpz_t x, y, xp, yp, t, a, b, s;
mpfr_t sfr, l, r1, r2;
mpfr_prec_t tprec, tprec0;
inex2 = inex1 = 0;
if (rop2 == NULL) /* only one output requested. */
tprec0 = MPFR_PREC (rop1);
else
tprec0 = MAX (MPFR_PREC (rop1), MPFR_PREC (rop2));
tprec0 += 11;
/* We use "Marsaglia polar method" here (cf.
George Marsaglia, Normal (Gaussian) random variables for supercomputers
The Journal of Supercomputing, Volume 5, Number 1, 49–55
DOI: 10.1007/BF00155857).
First we draw uniform x and y in [0,1] using mpz_urandomb (in
fixed precision), and scale them to [-1, 1].
*/
mpz_init (xp);
mpz_init (yp);
mpz_init (x);
mpz_init (y);
mpz_init (t);
mpz_init (s);
mpz_init (a);
mpz_init (b);
mpfr_init2 (sfr, MPFR_PREC_MIN);
mpfr_init2 (l, MPFR_PREC_MIN);
mpfr_init2 (r1, MPFR_PREC_MIN);
if (rop2 != NULL)
mpfr_init2 (r2, MPFR_PREC_MIN);
mpz_set_ui (xp, 0);
mpz_set_ui (yp, 0);
for (;;)
{
tprec = tprec0;
do
{
mpz_urandomb (xp, rstate, tprec);
mpz_urandomb (yp, rstate, tprec);
mpz_mul (a, xp, xp);
mpz_mul (b, yp, yp);
mpz_add (s, a, b);
}
while (mpz_sizeinbase (s, 2) > tprec * 2);
/* now s = x^2 + y^2 < 2^{2tprec} */
for (;;)
{
/* we compute (xp+1)^2 + (yp+1)^2 as s + 2xp + 2yp + 2 */
mpz_addmul_ui (s, xp, 2);
mpz_addmul_ui (s, yp, 2);
mpz_add_ui (s, s, 2);
/* The case s = 2^(2*tprec) is not possible:
(a) if xp and yp have different parities, s is odd
(b) if xp and yp are even, (xp+1)^2 and (yp+1)^2 are 1 mod 4,
thus s = 2 mod 4 (and tprec >= 1);
(c) if xp and yp are odd, if we note x = xp+1, y = yp+1 and
p = tprec, we would have x^2 + y^2 = 2^(2p) with x and y even
0 < x, y <= 2^p, thus if x' = x/2, y' = y/2 and p'=p-1,
we would have x'^2 + y'^2 = 2^(2p') with
0 < x', y' <= 2^p', and we conclude by induction. */
if (mpz_sizeinbase (s, 2) <= 2 * tprec)
goto yeepee;
/* Extend by 32 bits: for tprec=12, the probability we get here
is 8191/13180825, i.e., about 0.000621 */
mpz_mul_2exp (xp, xp, 32);
mpz_mul_2exp (yp, yp, 32);
mpz_urandomb (x, rstate, 32);
mpz_urandomb (y, rstate, 32);
mpz_add (xp, xp, x);
mpz_add (yp, yp, y);
tprec += 32;
mpz_mul (a, xp, xp);
mpz_mul (b, yp, yp);
mpz_add (s, a, b);
if (mpz_sizeinbase (s, 2) > tprec * 2)
break;
}
}
yeepee:
/* FIXME: compute s with s -= 2x + 2y + 2 */
mpz_mul (a, xp, xp);
mpz_mul (b, yp, yp);
mpz_add (s, a, b);
/* Compute the signs of the output */
mpz_urandomb (x, rstate, 2);
s1 = mpz_tstbit (x, 0);
s2 = mpz_tstbit (x, 1);
for (;;)
{
/* s = xp^2 + yp^2 (loop invariant) */
mpfr_set_prec (sfr, 2 * tprec);
mpfr_set_prec (l, tprec);
mpfr_set_z (sfr, s, MPFR_RNDN); /* exact */
mpfr_mul_2si (sfr, sfr, -2 * tprec, MPFR_RNDN); /* exact */
mpfr_log (l, sfr, MPFR_RNDN);
mpfr_neg (l, l, MPFR_RNDN);
mpfr_mul_2si (l, l, 1, MPFR_RNDN);
mpfr_div (l, l, sfr, MPFR_RNDN);
mpfr_sqrt (l, l, MPFR_RNDN);
mpfr_set_prec (r1, tprec);
mpfr_mul_z (r1, l, xp, MPFR_RNDN);
mpfr_div_2ui (r1, r1, tprec, MPFR_RNDN); /* exact */
if (s1)
mpfr_neg (r1, r1, MPFR_RNDN);
if (MPFR_CAN_ROUND (r1, tprec - 2, MPFR_PREC (rop1), rnd))
{
if (rop2 != NULL)
{
mpfr_set_prec (r2, tprec);
mpfr_mul_z (r2, l, yp, MPFR_RNDN);
mpfr_div_2ui (r2, r2, tprec, MPFR_RNDN); /* exact */
if (s2)
mpfr_neg (r2, r2, MPFR_RNDN);
if (MPFR_CAN_ROUND (r2, tprec - 2, MPFR_PREC (rop2), rnd))
break;
}
else
break;
}
/* Extend by 32 bits */
mpz_mul_2exp (xp, xp, 32);
mpz_mul_2exp (yp, yp, 32);
mpz_urandomb (x, rstate, 32);
mpz_urandomb (y, rstate, 32);
mpz_add (xp, xp, x);
mpz_add (yp, yp, y);
tprec += 32;
mpz_mul (a, xp, xp);
mpz_mul (b, yp, yp);
mpz_add (s, a, b);
}
inex1 = mpfr_set (rop1, r1, rnd);
if (rop2 != NULL)
{
inex2 = mpfr_set (rop2, r2, rnd);
inex2 = mpfr_check_range (rop2, inex2, rnd);
}
inex1 = mpfr_check_range (rop1, inex1, rnd);
if (rop2 != NULL)
mpfr_clear (r2);
mpfr_clear (r1);
mpfr_clear (l);
mpfr_clear (sfr);
mpz_clear (b);
mpz_clear (a);
mpz_clear (s);
mpz_clear (t);
mpz_clear (y);
mpz_clear (x);
mpz_clear (yp);
mpz_clear (xp);
return INEX (inex1, inex2);
}
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