/* gauss.c - gaussian random numbers, using the Ziggurat method
*
* Copyright (C) 2005 Jochen Voss.
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 3 of the License, or
* (at your option) any later version.
*
* This program 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 General Public License for more details.
*
* You should have received a copy of the GNU General Public License along
* with this library; if not, write to the Free Software Foundation, Inc.,
* 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
/*
* This routine is based on the following article, with a couple of
* modifications which simplify the implementation.
*
* George Marsaglia, Wai Wan Tsang
* The Ziggurat Method for Generating Random Variables
* Journal of Statistical Software, vol. 5 (2000), no. 8
* http://www.jstatsoft.org/v05/i08/
*
* The modifications are:
*
* 1) use 128 steps instead of 256 to decrease the amount of static
* data necessary.
*
* 2) use an acceptance sampling from an exponential wedge
* exp(-R*(x-R/2)) for the tail of the base strip to simplify the
* implementation. The area of exponential wedge is used in
* calculating 'v' and the coefficients in ziggurat table, so the
* coefficients differ slightly from those in the Marsaglia and Tsang
* paper.
*
* See also Leong et al, "A Comment on the Implementation of the
* Ziggurat Method", Journal of Statistical Software, vol 5 (2005), no 7.
*
*/
#include <config.h>
#include <math.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
/* position of right-most step */
#define PARAM_R 3.44428647676
/* tabulated values for the heigt of the Ziggurat levels */
static const double ytab[128] = {
1, 0.963598623011, 0.936280813353, 0.913041104253,
0.892278506696, 0.873239356919, 0.855496407634, 0.838778928349,
0.822902083699, 0.807732738234, 0.793171045519, 0.779139726505,
0.765577436082, 0.752434456248, 0.739669787677, 0.727249120285,
0.715143377413, 0.703327646455, 0.691780377035, 0.68048276891,
0.669418297233, 0.65857233912, 0.647931876189, 0.637485254896,
0.62722199145, 0.617132611532, 0.607208517467, 0.597441877296,
0.587825531465, 0.578352913803, 0.569017984198, 0.559815170911,
0.550739320877, 0.541785656682, 0.532949739145, 0.524227434628,
0.515614886373, 0.507108489253, 0.498704867478, 0.490400854812,
0.482193476986, 0.47407993601, 0.466057596125, 0.458123971214,
0.450276713467, 0.442513603171, 0.434832539473, 0.427231532022,
0.419708693379, 0.41226223212, 0.404890446548, 0.397591718955,
0.390364510382, 0.383207355816, 0.376118859788, 0.369097692334,
0.362142585282, 0.355252328834, 0.348425768415, 0.341661801776,
0.334959376311, 0.328317486588, 0.321735172063, 0.31521151497,
0.308745638367, 0.302336704338, 0.29598391232, 0.289686497571,
0.283443729739, 0.27725491156, 0.271119377649, 0.265036493387,
0.259005653912, 0.253026283183, 0.247097833139, 0.241219782932,
0.235391638239, 0.229612930649, 0.223883217122, 0.218202079518,
0.212569124201, 0.206983981709, 0.201446306496, 0.195955776745,
0.190512094256, 0.185114984406, 0.179764196185, 0.174459502324,
0.169200699492, 0.1639876086, 0.158820075195, 0.153697969964,
0.148621189348, 0.143589656295, 0.138603321143, 0.133662162669,
0.128766189309, 0.123915440582, 0.119109988745, 0.114349940703,
0.10963544023, 0.104966670533, 0.100343857232, 0.0957672718266,
0.0912372357329, 0.0867541250127, 0.082318375932, 0.0779304915295,
0.0735910494266, 0.0693007111742, 0.065060233529, 0.0608704821745,
0.056732448584, 0.05264727098, 0.0486162607163, 0.0446409359769,
0.0407230655415, 0.0368647267386, 0.0330683839378, 0.0293369977411,
0.0256741818288, 0.0220844372634, 0.0185735200577, 0.0151490552854,
0.0118216532614, 0.00860719483079, 0.00553245272614, 0.00265435214565
};
/* tabulated values for 2^24 times x[i]/x[i+1],
* used to accept for U*x[i+1]<=x[i] without any floating point operations */
static const unsigned long ktab[128] = {
0, 12590644, 14272653, 14988939,
15384584, 15635009, 15807561, 15933577,
16029594, 16105155, 16166147, 16216399,
16258508, 16294295, 16325078, 16351831,
16375291, 16396026, 16414479, 16431002,
16445880, 16459343, 16471578, 16482744,
16492970, 16502368, 16511031, 16519039,
16526459, 16533352, 16539769, 16545755,
16551348, 16556584, 16561493, 16566101,
16570433, 16574511, 16578353, 16581977,
16585398, 16588629, 16591685, 16594575,
16597311, 16599901, 16602354, 16604679,
16606881, 16608968, 16610945, 16612818,
16614592, 16616272, 16617861, 16619363,
16620782, 16622121, 16623383, 16624570,
16625685, 16626730, 16627708, 16628619,
16629465, 16630248, 16630969, 16631628,
16632228, 16632768, 16633248, 16633671,
16634034, 16634340, 16634586, 16634774,
16634903, 16634972, 16634980, 16634926,
16634810, 16634628, 16634381, 16634066,
16633680, 16633222, 16632688, 16632075,
16631380, 16630598, 16629726, 16628757,
16627686, 16626507, 16625212, 16623794,
16622243, 16620548, 16618698, 16616679,
16614476, 16612071, 16609444, 16606571,
16603425, 16599973, 16596178, 16591995,
16587369, 16582237, 16576520, 16570120,
16562917, 16554758, 16545450, 16534739,
16522287, 16507638, 16490152, 16468907,
16442518, 16408804, 16364095, 16301683,
16207738, 16047994, 15704248, 15472926
};
/* tabulated values of 2^{-24}*x[i] */
static const double wtab[128] = {
1.62318314817e-08, 2.16291505214e-08, 2.54246305087e-08, 2.84579525938e-08,
3.10340022482e-08, 3.33011726243e-08, 3.53439060345e-08, 3.72152672658e-08,
3.8950989572e-08, 4.05763964764e-08, 4.21101548915e-08, 4.35664624904e-08,
4.49563968336e-08, 4.62887864029e-08, 4.75707945735e-08, 4.88083237257e-08,
5.00063025384e-08, 5.11688950428e-08, 5.22996558616e-08, 5.34016475624e-08,
5.44775307871e-08, 5.55296344581e-08, 5.65600111659e-08, 5.75704813695e-08,
5.85626690412e-08, 5.95380306862e-08, 6.04978791776e-08, 6.14434034901e-08,
6.23756851626e-08, 6.32957121259e-08, 6.42043903937e-08, 6.51025540077e-08,
6.59909735447e-08, 6.68703634341e-08, 6.77413882848e-08, 6.8604668381e-08,
6.94607844804e-08, 7.03102820203e-08, 7.11536748229e-08, 7.1991448372e-08,
7.2824062723e-08, 7.36519550992e-08, 7.44755422158e-08, 7.52952223703e-08,
7.61113773308e-08, 7.69243740467e-08, 7.77345662086e-08, 7.85422956743e-08,
7.93478937793e-08, 8.01516825471e-08, 8.09539758128e-08, 8.17550802699e-08,
8.25552964535e-08, 8.33549196661e-08, 8.41542408569e-08, 8.49535474601e-08,
8.57531242006e-08, 8.65532538723e-08, 8.73542180955e-08, 8.8156298059e-08,
8.89597752521e-08, 8.97649321908e-08, 9.05720531451e-08, 9.138142487e-08,
9.21933373471e-08, 9.30080845407e-08, 9.38259651738e-08, 9.46472835298e-08,
9.54723502847e-08, 9.63014833769e-08, 9.71350089201e-08, 9.79732621669e-08,
9.88165885297e-08, 9.96653446693e-08, 1.00519899658e-07, 1.0138063623e-07,
1.02247952126e-07, 1.03122261554e-07, 1.04003996769e-07, 1.04893609795e-07,
1.05791574313e-07, 1.06698387725e-07, 1.07614573423e-07, 1.08540683296e-07,
1.09477300508e-07, 1.1042504257e-07, 1.11384564771e-07, 1.12356564007e-07,
1.13341783071e-07, 1.14341015475e-07, 1.15355110887e-07, 1.16384981291e-07,
1.17431607977e-07, 1.18496049514e-07, 1.19579450872e-07, 1.20683053909e-07,
1.21808209468e-07, 1.2295639141e-07, 1.24129212952e-07, 1.25328445797e-07,
1.26556042658e-07, 1.27814163916e-07, 1.29105209375e-07, 1.30431856341e-07,
1.31797105598e-07, 1.3320433736e-07, 1.34657379914e-07, 1.36160594606e-07,
1.37718982103e-07, 1.39338316679e-07, 1.41025317971e-07, 1.42787873535e-07,
1.44635331499e-07, 1.4657889173e-07, 1.48632138436e-07, 1.50811780719e-07,
1.53138707402e-07, 1.55639532047e-07, 1.58348931426e-07, 1.61313325908e-07,
1.64596952856e-07, 1.68292495203e-07, 1.72541128694e-07, 1.77574279496e-07,
1.83813550477e-07, 1.92166040885e-07, 2.05295471952e-07, 2.22600839893e-07
};
double
gsl_ran_gaussian_ziggurat (const gsl_rng * r, const double sigma)
{
unsigned long int i, j;
int sign;
double x, y;
const unsigned long int range = r->type->max - r->type->min;
const unsigned long int offset = r->type->min;
while (1)
{
if (range >= 0xFFFFFFFF)
{
unsigned long int k = gsl_rng_get(r) - offset;
i = (k & 0xFF);
j = (k >> 8) & 0xFFFFFF;
}
else if (range >= 0x00FFFFFF)
{
unsigned long int k1 = gsl_rng_get(r) - offset;
unsigned long int k2 = gsl_rng_get(r) - offset;
i = (k1 & 0xFF);
j = (k2 & 0x00FFFFFF);
}
else
{
i = gsl_rng_uniform_int (r, 256); /* choose the step */
j = gsl_rng_uniform_int (r, 16777216); /* sample from 2^24 */
}
sign = (i & 0x80) ? +1 : -1;
i &= 0x7f;
x = j * wtab[i];
if (j < ktab[i])
break;
if (i < 127)
{
double y0, y1, U1;
y0 = ytab[i];
y1 = ytab[i + 1];
U1 = gsl_rng_uniform (r);
y = y1 + (y0 - y1) * U1;
}
else
{
double U1, U2;
U1 = 1.0 - gsl_rng_uniform (r);
U2 = gsl_rng_uniform (r);
x = PARAM_R - log (U1) / PARAM_R;
y = exp (-PARAM_R * (x - 0.5 * PARAM_R)) * U2;
}
if (y < exp (-0.5 * x * x))
break;
}
return sign * sigma * x;
}