/* mpfr_nrandom (rop, state, rnd_mode) -- Generate a normal deviate with mean 0 and variance 1 and round it to the precision of rop according to the given rounding mode. Copyright 2013-2021 Free Software Foundation, Inc. Contributed by Charles Karney , SRI International. 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. */ /* * Sampling from the normal distribution with zero mean and unit variance. * This uses Algorithm N given in: * Charles F. F. Karney, * "Sampling exactly from the normal distribution", * ACM Trans. Math. Software 42(1), 3:1-14 (Jan. 2016). * https://dx.doi.org/10.1145/2710016 * https://arxiv.org/abs/1303.6257 * * The implementation here closely follows the C++ one given in the paper * above. However, here, C is simplified by using gmp_urandomm_ui; the initial * rejection step in H just tests the leading bit of p; and the assignment of * the sign to the deviate using gmp_urandomb_ui. Finally, the C++ * implementation benefits from caching temporary random deviates across calls. * This isn't possible in C without additional machinery which would complicate * the interface. * * There are a few "weasel words" regarding the accuracy of this * implementation. The algorithm produces exactly rounded normal deviates * provided that gmp's random number engine delivers truly random bits. If it * did, the algorithm would be perfect; however, this implementation would have * problems, e.g., in that the integer part of the normal deviate is * represented by an unsigned long, whereas in reality the integer part in * unbounded. In this implementation, asserts catch overflow in the integer * part and similar (very, very) unlikely events. In reality, of course, gmp's * random number engine has a finite internal state (19937 bits in the case of * the MT19937 method). This means that these unlikely events in fact won't * occur. If the asserts are triggered, then this is an indication that the * random number engine is defective. (Even if a hardware random number * generator were used, the most likely explanation for the triggering of the * asserts would be that the hardware generator was broken.) */ #include "random_deviate.h" /* Algorithm H: true with probability exp(-1/2). */ static int H (gmp_randstate_t r, mpfr_random_deviate_t p, mpfr_random_deviate_t q) { /* p and q are temporaries */ mpfr_random_deviate_reset (p); if (mpfr_random_deviate_tstbit (p, 1, r)) return 1; for (;;) { mpfr_random_deviate_reset (q); if (!mpfr_random_deviate_less (q, p, r)) return 0; mpfr_random_deviate_reset (p); if (!mpfr_random_deviate_less (p, q, r)) return 1; } } /* Step N1: return n >= 0 with prob. exp(-n/2) * (1 - exp(-1/2)). */ static unsigned long G (gmp_randstate_t r, mpfr_random_deviate_t p, mpfr_random_deviate_t q) { /* p and q are temporaries */ unsigned long n = 0; while (H (r, p, q)) { ++n; /* Catch n wrapping around to 0; for a 32-bit unsigned long, the * probability of this is exp(-2^30)). */ MPFR_ASSERTN (n != 0UL); } return n; } /* Step N2: true with probability exp(-m*n/2). */ static int P (unsigned long m, unsigned long n, gmp_randstate_t r, mpfr_random_deviate_t p, mpfr_random_deviate_t q) { /* p and q are temporaries. m*n is passed as two separate parameters to deal * with the case where m*n overflows an unsigned long. This may be called * with m = 0 and n = (unsigned long)(-1) and, because m in handled in to the * outer loop, this routine will correctly return 1. */ while (m--) { unsigned long k = n; while (k--) { if (!H (r, p, q)) return 0; } } return 1; } /* Algorithm C: return (-1, 0, 1) with prob (1/m, 1/m, 1-2/m). */ static int C (unsigned long m, gmp_randstate_t r) { unsigned long n = gmp_urandomm_ui (r, m); return n == 0 ? -1 : (n == 1 ? 0 : 1); } /* Algorithm B: true with prob exp(-x * (2*k + x) / (2*k + 2)). */ static int B (unsigned long k, mpfr_random_deviate_t x, gmp_randstate_t r, mpfr_random_deviate_t p, mpfr_random_deviate_t q) { /* p and q are temporaries */ unsigned long m = 2 * k + 2; /* n tracks the parity of the loop; s == 1 on first trip through loop. */ unsigned n = 0, s = 1; int f; /* Check if 2 * k + 2 would overflow; for a 32-bit unsigned long, the * probability of this is exp(-2^61)). */ MPFR_ASSERTN (k < ((unsigned long)(-1) >> 1)); for (;; ++n, s = 0) /* overflow of n is innocuous */ { if ( ((f = k ? 0 : C (m, r)) < 0) || (mpfr_random_deviate_reset (q), !mpfr_random_deviate_less (q, s ? x : p, r)) || ((f = k ? C (m, r) : f) < 0) || (f == 0 && (mpfr_random_deviate_reset (p), !mpfr_random_deviate_less (p, x, r))) ) break; mpfr_random_deviate_swap (p, q); /* an efficient way of doing p = q */ } return (n & 1U) == 0; } /* return a normal random deviate with mean 0 and variance 1 as a MPFR */ int mpfr_nrandom (mpfr_ptr z, gmp_randstate_t r, mpfr_rnd_t rnd) { mpfr_random_deviate_t x, p, q; int inex; unsigned long k, j; mpfr_random_deviate_init (x); mpfr_random_deviate_init (p); mpfr_random_deviate_init (q); for (;;) { k = G (r, p, q); /* step 1 */ if (!P (k, k - 1, r, p, q)) continue; /* step 2 */ mpfr_random_deviate_reset (x); /* step 3 */ for (j = 0; j <= k && B (k, x, r, p, q); ++j); /* step 4 */ if (j > k) break; } mpfr_random_deviate_clear (q); mpfr_random_deviate_clear (p); /* steps 5, 6, 7 */ inex = mpfr_random_deviate_value (gmp_urandomb_ui (r, 1), k, x, z, r, rnd); mpfr_random_deviate_clear (x); return inex; }