Mercurial > octave
annotate liboctave/numeric/randmtzig.cc @ 27923:bd51beb6205e
update formatting of copyright notices
* Use <https://octave.org/copyright/> instead of
<https://octave.org/COPYRIGHT.html/>.
* For consistency with other comments in the Octave sources, use
C++-style comments for copyright blocks in C and C++ files.
* Use delimiters above and below copyright blocks that are appropriate
for the language used in the file.
* Eliminate extra spacing inside copyright blocks.
* lex.ll (looks_like_copyright): Also allow newlines and carriage
returns before the word "Copyright".
* scripts/mk-doc.pl (gethelp): Also skip empty comment lines.
* bp-table.cc, type.m: Adjust tests.
author | John W. Eaton <jwe@octave.org> |
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date | Wed, 08 Jan 2020 11:59:41 -0500 |
parents | 1891570abac8 |
children | 863ae57eee69 |
rev | line source |
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1 //////////////////////////////////////////////////////////////////////// |
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2 // |
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3 // Copyright (C) 2006-2020 The Octave Project Developers |
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4 // |
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5 // See the file COPYRIGHT.md in the top-level directory of this |
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6 // distribution or <https://octave.org/copyright/>. |
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7 // |
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8 // This file is part of Octave. |
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9 // |
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10 // Octave is free software: you can redistribute it and/or modify it |
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11 // under the terms of the GNU General Public License as published by |
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12 // the Free Software Foundation, either version 3 of the License, or |
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13 // (at your option) any later version. |
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14 // |
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15 // Octave is distributed in the hope that it will be useful, but |
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16 // WITHOUT ANY WARRANTY; without even the implied warranty of |
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17 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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18 // GNU General Public License for more details. |
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19 // |
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20 // You should have received a copy of the GNU General Public License |
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21 // along with Octave; see the file COPYING. If not, see |
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22 // <https://www.gnu.org/licenses/>. |
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23 // |
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24 //////////////////////////////////////////////////////////////////////// |
7019 | 25 |
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26 /* |
5742 | 27 A C-program for MT19937, with initialization improved 2002/2/10. |
28 Coded by Takuji Nishimura and Makoto Matsumoto. | |
29 This is a faster version by taking Shawn Cokus's optimization, | |
30 Matthe Bellew's simplification, Isaku Wada's real version. | |
31 David Bateman added normal and exponential distributions following | |
32 Marsaglia and Tang's Ziggurat algorithm. | |
33 | |
34 Copyright (C) 1997 - 2002, Makoto Matsumoto and Takuji Nishimura, | |
35 Copyright (C) 2004, David Bateman | |
36 All rights reserved. | |
37 | |
38 Redistribution and use in source and binary forms, with or without | |
39 modification, are permitted provided that the following conditions | |
40 are met: | |
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41 |
5742 | 42 1. Redistributions of source code must retain the above copyright |
43 notice, this list of conditions and the following disclaimer. | |
44 | |
45 2. Redistributions in binary form must reproduce the above copyright | |
46 notice, this list of conditions and the following disclaimer in the | |
47 documentation and/or other materials provided with the distribution. | |
48 | |
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49 3. The names of its contributors may not be used to endorse or promote |
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50 products derived from this software without specific prior written |
5742 | 51 permission. |
52 | |
53 THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS | |
54 "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT | |
55 LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR | |
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56 A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER |
5742 | 57 OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, |
58 EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, | |
59 PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR | |
60 PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF | |
61 LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING | |
62 NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS | |
63 SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. | |
64 | |
65 | |
66 Any feedback is very welcome. | |
67 http://www.math.keio.ac.jp/matumoto/emt.html | |
68 email: matumoto@math.keio.ac.jp | |
69 | |
70 * 2004-01-19 Paul Kienzle | |
71 * * comment out main | |
72 * add init_by_entropy, get_state, set_state | |
73 * * converted to allow compiling by C++ compiler | |
74 * | |
75 * 2004-01-25 David Bateman | |
76 * * Add Marsaglia and Tsang Ziggurat code | |
77 * | |
78 * 2004-07-13 Paul Kienzle | |
79 * * make into an independent library with some docs. | |
80 * * introduce new main and test code. | |
81 * | |
82 * 2004-07-28 Paul Kienzle & David Bateman | |
83 * * add -DALLBITS flag for 32 vs. 53 bits of randomness in mantissa | |
84 * * make the naming scheme more uniform | |
85 * * add -DHAVE_X86 for faster support of 53 bit mantissa on x86 arch. | |
86 * | |
87 * 2005-02-23 Paul Kienzle | |
88 * * fix -DHAVE_X86_32 flag and add -DUSE_X86_32=0|1 for explicit control | |
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89 * |
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90 * 2006-04-01 David Bateman |
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91 * * convert for use in octave, declaring static functions only used |
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92 * here and adding oct_ to functions visible externally |
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93 * * inverse sense of ALLBITS |
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94 * |
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95 * 2012-05-18 David Bateman |
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96 * * Remove randu64 and ALLBIT option |
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97 * * Add the single precision generators |
5742 | 98 */ |
99 | |
100 /* | |
101 === Build instructions === | |
102 | |
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103 Compile with -DHAVE_GETTIMEOFDAY if the gettimeofday function is |
5742 | 104 available. This is not necessary if your architecture has |
105 /dev/urandom defined. | |
106 | |
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107 Uses implicit -Di386 or explicit -DHAVE_X86_32 to determine if CPU=x86. |
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108 You can force X86 behavior with -DUSE_X86_32=1, or suppress it with |
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109 -DUSE_X86_32=0. You should also consider -march=i686 or similar for |
5742 | 110 extra performance. Check whether -DUSE_X86_32=0 is faster on 64-bit |
111 x86 architectures. | |
112 | |
113 If you want to replace the Mersenne Twister with another | |
114 generator then redefine randi32 appropriately. | |
115 | |
116 === Usage instructions === | |
117 Before using any of the generators, initialize the state with one of | |
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118 the init_mersenne_twister functions. |
5742 | 119 |
120 All generators share the same state vector. | |
121 | |
122 === Mersenne Twister === | |
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123 random initial state: |
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124 void init_mersenne_twister (void) |
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125 |
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126 // 32-bit initial state: |
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127 void init_mersenne_twister (uint32_t s) |
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128 |
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129 // m*32-bit initial state: |
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130 void init_mersenne_twister (uint32_t k[],int m) |
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131 |
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132 // saves state in array: |
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133 void get_mersenne_twister_state (uint32_t save[MT_N+1]) |
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134 |
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135 // restores state from array |
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136 void set_mersenne_twister_state (uint32_t save[MT_N+1]) |
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137 |
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138 static uint32_t randmt (void) returns 32-bit unsigned int |
5742 | 139 |
140 === inline generators === | |
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141 static uint32_t randi32 (void) returns 32-bit unsigned int |
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142 static uint64_t randi53 (void) returns 53-bit unsigned int |
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143 static uint64_t randi54 (void) returns 54-bit unsigned int |
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144 static float randu24 (void) returns 24-bit uniform in (0,1) |
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145 static double randu53 (void) returns 53-bit uniform in (0,1) |
5742 | 146 |
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147 double rand_uniform (void) returns M-bit uniform in (0,1) |
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148 double rand_normal (void) returns M-bit standard normal |
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149 double rand_exponential (void) returns N-bit standard exponential |
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150 |
5742 | 151 === Array generators === |
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152 void rand_uniform (octave_idx_type, double []) |
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153 void rand_normal (octave_idx_type, double []) |
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154 void rand_exponential (octave_idx_type, double []) |
5742 | 155 */ |
156 | |
157 #if defined (HAVE_CONFIG_H) | |
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158 # include "config.h" |
5742 | 159 #endif |
160 | |
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161 #include <cmath> |
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162 #include <cstdio> |
5742 | 163 |
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164 #include <algorithm> |
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165 |
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166 #include "oct-time.h" |
5742 | 167 #include "randmtzig.h" |
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168 |
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169 /* FIXME: may want to suppress X86 if sizeof(long) > 4 */ |
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170 #if ! defined (USE_X86_32) |
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171 # if defined (i386) || defined (HAVE_X86_32) |
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172 # define USE_X86_32 1 |
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173 # else |
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174 # define USE_X86_32 0 |
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175 # endif |
5742 | 176 #endif |
177 | |
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178 namespace octave |
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179 { |
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180 /* ===== Mersenne Twister 32-bit generator ===== */ |
5742 | 181 |
182 #define MT_M 397 | |
183 #define MATRIX_A 0x9908b0dfUL /* constant vector a */ | |
184 #define UMASK 0x80000000UL /* most significant w-r bits */ | |
185 #define LMASK 0x7fffffffUL /* least significant r bits */ | |
186 #define MIXBITS(u,v) ( ((u) & UMASK) | ((v) & LMASK) ) | |
187 #define TWIST(u,v) ((MIXBITS(u,v) >> 1) ^ ((v)&1UL ? MATRIX_A : 0UL)) | |
188 | |
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189 static uint32_t *next; |
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190 static uint32_t state[MT_N]; /* the array for the state vector */ |
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191 static int left = 1; |
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192 static int initf = 0; |
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193 static int initt = 1; |
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194 static int inittf = 1; |
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196 /* initializes state[MT_N] with a seed */ |
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197 void init_mersenne_twister (const uint32_t s) |
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198 { |
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199 int j; |
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200 state[0] = s & 0xffffffffUL; |
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201 for (j = 1; j < MT_N; j++) |
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202 { |
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203 state[j] = (1812433253UL * (state[j-1] ^ (state[j-1] >> 30)) + j); |
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204 /* See Knuth TAOCP Vol2. 3rd Ed. P.106 for multiplier. */ |
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205 /* In the previous versions, MSBs of the seed affect */ |
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206 /* only MSBs of the array state[]. */ |
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207 /* 2002/01/09 modified by Makoto Matsumoto */ |
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208 state[j] &= 0xffffffffUL; /* for >32 bit machines */ |
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209 } |
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210 left = 1; |
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211 initf = 1; |
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212 } |
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214 /* initialize by an array with array-length */ |
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215 /* init_key is the array for initializing keys */ |
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216 /* key_length is its length */ |
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217 void init_mersenne_twister (const uint32_t *init_key, const int key_length) |
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218 { |
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219 int i, j, k; |
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220 init_mersenne_twister (19650218UL); |
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221 i = 1; |
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222 j = 0; |
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223 k = (MT_N > key_length ? MT_N : key_length); |
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224 for (; k; k--) |
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225 { |
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226 state[i] = (state[i] ^ ((state[i-1] ^ (state[i-1] >> 30)) * 1664525UL)) |
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227 + init_key[j] + j; /* non linear */ |
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228 state[i] &= 0xffffffffUL; /* for WORDSIZE > 32 machines */ |
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229 i++; |
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230 j++; |
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231 if (i >= MT_N) |
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232 { |
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233 state[0] = state[MT_N-1]; |
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234 i = 1; |
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235 } |
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236 if (j >= key_length) |
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237 j = 0; |
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238 } |
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239 for (k = MT_N - 1; k; k--) |
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240 { |
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241 state[i] = (state[i] ^ ((state[i-1] ^ (state[i-1] >> 30)) * 1566083941UL)) |
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242 - i; /* non linear */ |
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243 state[i] &= 0xffffffffUL; /* for WORDSIZE > 32 machines */ |
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244 i++; |
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245 if (i >= MT_N) |
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246 { |
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247 state[0] = state[MT_N-1]; |
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248 i = 1; |
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249 } |
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250 } |
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252 state[0] = 0x80000000UL; /* MSB is 1; assuring nonzero initial array */ |
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253 left = 1; |
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254 initf = 1; |
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255 } |
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257 void init_mersenne_twister (void) |
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258 { |
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259 uint32_t entropy[MT_N]; |
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260 int n = 0; |
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262 /* Look for entropy in /dev/urandom */ |
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263 FILE *urandom = std::fopen ("/dev/urandom", "rb"); |
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264 if (urandom) |
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265 { |
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266 while (n < MT_N) |
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267 { |
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268 unsigned char word[4]; |
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269 if (std::fread (word, 4, 1, urandom) != 1) |
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270 break; |
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271 entropy[n++] = word[0] + (word[1]<<8) + (word[2]<<16) |
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272 + (static_cast<uint32_t> (word[3])<<24); |
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273 } |
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274 std::fclose (urandom); |
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275 } |
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277 /* If there isn't enough entropy, gather some from various sources */ |
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278 |
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279 sys::time now; |
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280 |
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281 if (n < MT_N) |
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282 entropy[n++] = now.unix_time (); /* Current time in seconds */ |
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283 |
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284 if (n < MT_N) |
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285 entropy[n++] = clock (); /* CPU time used (usec) */ |
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286 |
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287 if (n < MT_N) |
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288 entropy[n++] = now.usec (); /* Fractional part of current time */ |
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289 |
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290 /* Send all the entropy into the initial state vector */ |
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291 init_mersenne_twister (entropy,n); |
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292 } |
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294 void set_mersenne_twister_state (const uint32_t *save) |
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295 { |
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296 std::copy_n (save, MT_N, state); |
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297 left = save[MT_N]; |
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298 next = state + (MT_N - left + 1); |
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299 } |
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301 void get_mersenne_twister_state (uint32_t *save) |
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302 { |
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303 std::copy_n (state, MT_N, save); |
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304 save[MT_N] = left; |
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305 } |
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307 static void next_state (void) |
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308 { |
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309 uint32_t *p = state; |
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310 int j; |
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312 /* if init_by_int() has not been called, */ |
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313 /* a default initial seed is used */ |
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314 /* if (initf==0) init_by_int(5489UL); */ |
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315 /* Or better yet, a random seed! */ |
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316 if (initf == 0) |
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317 init_mersenne_twister (); |
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319 left = MT_N; |
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320 next = state; |
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321 |
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322 for (j = MT_N - MT_M + 1; --j; p++) |
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323 *p = p[MT_M] ^ TWIST(p[0], p[1]); |
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325 for (j = MT_M; --j; p++) |
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326 *p = p[MT_M-MT_N] ^ TWIST(p[0], p[1]); |
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328 *p = p[MT_M-MT_N] ^ TWIST(p[0], state[0]); |
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329 } |
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330 |
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331 /* generates a random number on [0,0xffffffff]-interval */ |
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332 static uint32_t randmt (void) |
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333 { |
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334 uint32_t y; |
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336 if (--left == 0) |
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337 next_state (); |
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338 y = *next++; |
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340 /* Tempering */ |
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341 y ^= (y >> 11); |
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342 y ^= (y << 7) & 0x9d2c5680UL; |
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343 y ^= (y << 15) & 0xefc60000UL; |
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344 return (y ^ (y >> 18)); |
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345 } |
5742 | 346 |
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347 /* ===== Uniform generators ===== */ |
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349 /* Select which 32 bit generator to use */ |
5742 | 350 #define randi32 randmt |
351 | |
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352 static uint64_t randi53 (void) |
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353 { |
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354 const uint32_t lo = randi32 (); |
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355 const uint32_t hi = randi32 () & 0x1FFFFF; |
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356 #if defined (HAVE_X86_32) |
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357 uint64_t u; |
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358 uint32_t *p = (uint32_t *)&u; |
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359 p[0] = lo; |
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360 p[1] = hi; |
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361 return u; |
5742 | 362 #else |
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363 return ((static_cast<uint64_t> (hi) << 32) | lo); |
5742 | 364 #endif |
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365 } |
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367 static uint64_t randi54 (void) |
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368 { |
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369 const uint32_t lo = randi32 (); |
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370 const uint32_t hi = randi32 () & 0x3FFFFF; |
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371 #if defined (HAVE_X86_32) |
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372 uint64_t u; |
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373 uint32_t *p = static_cast<uint32_t *> (&u); |
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374 p[0] = lo; |
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375 p[1] = hi; |
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376 return u; |
5742 | 377 #else |
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378 return ((static_cast<uint64_t> (hi) << 32) | lo); |
5742 | 379 #endif |
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380 } |
5742 | 381 |
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382 /* generates a random number on (0,1)-real-interval */ |
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383 static float randu24 (void) |
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384 { |
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385 uint32_t i; |
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386 |
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387 do |
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388 { |
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389 i = randi32 () & static_cast<uint32_t> (0xFFFFFF); |
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390 } |
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391 while (i == 0); |
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392 |
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393 return i * (1.0f / 16777216.0f); |
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394 } |
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395 |
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396 /* generates a random number on (0,1) with 53-bit resolution */ |
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397 static double randu53 (void) |
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398 { |
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399 int32_t a, b; |
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400 |
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401 do |
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402 { |
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403 a = randi32 () >> 5; |
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404 b = randi32 () >> 6; |
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405 } |
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406 while (a == 0 && b == 0); |
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407 |
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408 return (a*67108864.0 + b) * (1.0/9007199254740992.0); |
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409 } |
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410 |
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411 /* Determine mantissa for uniform doubles */ |
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412 template <> |
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413 double |
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414 rand_uniform<double> (void) |
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415 { |
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416 return randu53 (); |
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417 } |
5742 | 418 |
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419 /* Determine mantissa for uniform floats */ |
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420 template <> |
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421 float |
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422 rand_uniform<float> (void) |
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423 { |
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424 return randu24 (); |
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425 } |
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426 |
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427 /* ===== Ziggurat normal and exponential generators ===== */ |
5742 | 428 |
429 #define ZIGGURAT_TABLE_SIZE 256 | |
430 | |
431 #define ZIGGURAT_NOR_R 3.6541528853610088 | |
432 #define ZIGGURAT_NOR_INV_R 0.27366123732975828 | |
433 #define NOR_SECTION_AREA 0.00492867323399 | |
434 | |
435 #define ZIGGURAT_EXP_R 7.69711747013104972 | |
436 #define ZIGGURAT_EXP_INV_R 0.129918765548341586 | |
437 #define EXP_SECTION_AREA 0.0039496598225815571993 | |
438 | |
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439 #define ZIGINT uint64_t |
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440 #define EMANTISSA 9007199254740992.0 /* 53 bit mantissa */ |
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441 #define ERANDI randi53() /* 53 bits for mantissa */ |
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442 #define NMANTISSA EMANTISSA |
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443 #define NRANDI randi54() /* 53 bits for mantissa + 1 bit sign */ |
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444 #define RANDU randu53() |
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445 |
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446 static ZIGINT ki[ZIGGURAT_TABLE_SIZE]; |
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447 static double wi[ZIGGURAT_TABLE_SIZE], fi[ZIGGURAT_TABLE_SIZE]; |
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448 static ZIGINT ke[ZIGGURAT_TABLE_SIZE]; |
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449 static double we[ZIGGURAT_TABLE_SIZE], fe[ZIGGURAT_TABLE_SIZE]; |
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450 |
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451 /* |
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452 This code is based on the paper Marsaglia and Tsang, "The ziggurat method |
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453 for generating random variables", Journ. Statistical Software. Code was |
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454 presented in this paper for a Ziggurat of 127 levels and using a 32 bit |
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455 integer random number generator. This version of the code, uses the |
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456 Mersenne Twister as the integer generator and uses 256 levels in the |
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457 Ziggurat. This has several advantages. |
5742 | 458 |
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459 1) As Marsaglia and Tsang themselves states, the more levels the few |
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460 times the expensive tail algorithm must be called |
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461 2) The cycle time of the generator is determined by the integer |
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462 generator, thus the use of a Mersenne Twister for the core random |
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463 generator makes this cycle extremely long. |
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464 3) The license on the original code was unclear, thus rewriting the code |
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465 from the article means we are free of copyright issues. |
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466 4) Compile flag for full 53-bit random mantissa. |
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468 It should be stated that the authors made my life easier, by the fact that |
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469 the algorithm developed in the text of the article is for a 256 level |
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470 ziggurat, even if the code itself isn't... |
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472 One modification to the algorithm developed in the article, is that it is |
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473 assumed that 0 <= x < Inf, and "unsigned long"s are used, thus resulting in |
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474 terms like 2^32 in the code. As the normal distribution is defined between |
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475 -Inf < x < Inf, we effectively only have 31 bit integers plus a sign. Thus |
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476 in Marsaglia and Tsang, terms like 2^32 become 2^31. We use NMANTISSA for |
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477 this term. The exponential distribution is one sided so we use the |
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478 full 32 bits. We use EMANTISSA for this term. |
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480 It appears that I'm slightly slower than the code in the article, this |
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481 is partially due to a better generator of random integers than they |
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482 use. But might also be that the case of rapid return was optimized by |
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483 inlining the relevant code with a #define. As the basic Mersenne |
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484 Twister is only 25% faster than this code I suspect that the main |
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485 reason is just the use of the Mersenne Twister and not the inlining, |
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486 so I'm not going to try and optimize further. |
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487 */ |
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489 void create_ziggurat_tables (void) |
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490 { |
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491 int i; |
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492 double x, x1; |
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493 |
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494 /* Ziggurat tables for the normal distribution */ |
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495 x1 = ZIGGURAT_NOR_R; |
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496 wi[255] = x1 / NMANTISSA; |
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497 fi[255] = exp (-0.5 * x1 * x1); |
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498 |
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499 /* Index zero is special for tail strip, where Marsaglia and Tsang |
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500 * defines this as |
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501 * k_0 = 2^31 * r * f(r) / v, w_0 = 0.5^31 * v / f(r), f_0 = 1, |
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502 * where v is the area of each strip of the ziggurat. |
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503 */ |
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504 ki[0] = static_cast<ZIGINT> (x1 * fi[255] / NOR_SECTION_AREA * NMANTISSA); |
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505 wi[0] = NOR_SECTION_AREA / fi[255] / NMANTISSA; |
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506 fi[0] = 1.; |
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508 for (i = 254; i > 0; i--) |
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509 { |
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510 /* New x is given by x = f^{-1}(v/x_{i+1} + f(x_{i+1})), thus |
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511 * need inverse operator of y = exp(-0.5*x*x) -> x = sqrt(-2*ln(y)) |
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512 */ |
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513 x = std::sqrt (-2. * std::log (NOR_SECTION_AREA / x1 + fi[i+1])); |
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514 ki[i+1] = static_cast<ZIGINT> (x / x1 * NMANTISSA); |
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515 wi[i] = x / NMANTISSA; |
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516 fi[i] = exp (-0.5 * x * x); |
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517 x1 = x; |
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518 } |
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520 ki[1] = 0; |
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522 /* Zigurrat tables for the exponential distribution */ |
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523 x1 = ZIGGURAT_EXP_R; |
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524 we[255] = x1 / EMANTISSA; |
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525 fe[255] = exp (-x1); |
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527 /* Index zero is special for tail strip, where Marsaglia and Tsang |
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528 * defines this as |
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529 * k_0 = 2^32 * r * f(r) / v, w_0 = 0.5^32 * v / f(r), f_0 = 1, |
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530 * where v is the area of each strip of the ziggurat. |
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531 */ |
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532 ke[0] = static_cast<ZIGINT> (x1 * fe[255] / EXP_SECTION_AREA * EMANTISSA); |
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533 we[0] = EXP_SECTION_AREA / fe[255] / EMANTISSA; |
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534 fe[0] = 1.; |
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536 for (i = 254; i > 0; i--) |
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537 { |
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538 /* New x is given by x = f^{-1}(v/x_{i+1} + f(x_{i+1})), thus |
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539 * need inverse operator of y = exp(-x) -> x = -ln(y) |
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540 */ |
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541 x = - std::log (EXP_SECTION_AREA / x1 + fe[i+1]); |
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542 ke[i+1] = static_cast<ZIGINT> (x / x1 * EMANTISSA); |
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543 we[i] = x / EMANTISSA; |
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544 fe[i] = exp (-x); |
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545 x1 = x; |
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546 } |
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547 ke[1] = 0; |
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549 initt = 0; |
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550 } |
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552 /* |
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553 * Here is the guts of the algorithm. As Marsaglia and Tsang state the |
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554 * algorithm in their paper |
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555 * |
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556 * 1) Calculate a random signed integer j and let i be the index |
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557 * provided by the rightmost 8-bits of j |
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558 * 2) Set x = j * w_i. If j < k_i return x |
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559 * 3) If i = 0, then return x from the tail |
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560 * 4) If [f(x_{i-1}) - f(x_i)] * U < f(x) - f(x_i), return x |
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561 * 5) goto step 1 |
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562 * |
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563 * Where f is the functional form of the distribution, which for a normal |
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564 * distribution is exp(-0.5*x*x) |
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565 */ |
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567 |
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568 template <> double rand_normal<double> (void) |
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569 { |
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570 if (initt) |
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571 create_ziggurat_tables (); |
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573 while (1) |
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574 { |
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575 /* The following code is specialized for 32-bit mantissa. |
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576 * Compared to the arbitrary mantissa code, there is a performance |
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577 * gain for 32-bits: PPC: 2%, MIPS: 8%, x86: 40% |
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578 * There is a bigger performance gain compared to using a full |
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579 * 53-bit mantissa: PPC: 60%, MIPS: 65%, x86: 240% |
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580 * Of course, different compilers and operating systems may |
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581 * have something to do with this. |
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582 */ |
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583 # if defined (HAVE_X86_32) |
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584 /* 53-bit mantissa, 1-bit sign, x86 32-bit architecture */ |
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585 double x; |
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586 int si,idx; |
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587 uint32_t lo, hi; |
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588 int64_t rabs; |
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589 uint32_t *p = (uint32_t *)&rabs; |
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590 lo = randi32 (); |
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591 idx = lo & 0xFF; |
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592 hi = randi32 (); |
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593 si = hi & UMASK; |
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594 p[0] = lo; |
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595 p[1] = hi & 0x1FFFFF; |
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596 x = ( si ? -rabs : rabs ) * wi[idx]; |
22188 | 597 # else |
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598 /* arbitrary mantissa (selected by NRANDI, with 1 bit for sign) */ |
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599 const uint64_t r = NRANDI; |
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600 const int64_t rabs = r >> 1; |
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601 const int idx = static_cast<int> (rabs & 0xFF); |
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602 const double x = ( (r & 1) ? -rabs : rabs) * wi[idx]; |
22188 | 603 # endif |
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604 if (rabs < static_cast<int64_t> (ki[idx])) |
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605 return x; /* 99.3% of the time we return here 1st try */ |
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606 else if (idx == 0) |
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607 { |
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608 /* As stated in Marsaglia and Tsang |
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609 * |
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610 * For the normal tail, the method of Marsaglia[5] provides: |
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611 * generate x = -ln(U_1)/r, y = -ln(U_2), until y+y > x*x, |
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612 * then return r+x. Except that r+x is always in the positive |
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613 * tail!!!! Any thing random might be used to determine the |
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614 * sign, but as we already have r we might as well use it |
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615 * |
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616 * [PAK] but not the bottom 8 bits, since they are all 0 here! |
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617 */ |
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618 double xx, yy; |
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619 do |
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620 { |
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621 xx = - ZIGGURAT_NOR_INV_R * std::log (RANDU); |
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622 yy = - std::log (RANDU); |
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623 } |
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624 while ( yy+yy <= xx*xx); |
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625 return ((rabs & 0x100) ? -ZIGGURAT_NOR_R-xx : ZIGGURAT_NOR_R+xx); |
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626 } |
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627 else if ((fi[idx-1] - fi[idx]) * RANDU + fi[idx] < exp (-0.5*x*x)) |
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628 return x; |
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629 } |
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630 } |
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631 |
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632 template <> double rand_exponential<double> (void) |
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633 { |
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634 if (initt) |
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635 create_ziggurat_tables (); |
5742 | 636 |
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637 while (1) |
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638 { |
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639 ZIGINT ri = ERANDI; |
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640 const int idx = static_cast<int> (ri & 0xFF); |
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641 const double x = ri * we[idx]; |
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642 if (ri < ke[idx]) |
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643 return x; /* 98.9% of the time we return here 1st try */ |
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644 else if (idx == 0) |
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645 { |
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646 /* As stated in Marsaglia and Tsang |
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647 * |
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648 * For the exponential tail, the method of Marsaglia[5] provides: |
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649 * x = r - ln(U); |
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650 */ |
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651 return ZIGGURAT_EXP_R - std::log (RANDU); |
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652 } |
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653 else if ((fe[idx-1] - fe[idx]) * RANDU + fe[idx] < exp (-x)) |
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654 return x; |
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655 } |
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656 } |
5742 | 657 |
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658 template <> void rand_uniform<double> (octave_idx_type n, double *p) |
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659 { |
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660 std::generate_n (p, n, [](void) { return rand_uniform<double> (); }); |
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661 } |
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662 |
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663 template <> void rand_normal (octave_idx_type n, double *p) |
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664 { |
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665 std::generate_n (p, n, [](void) { return rand_normal<double> (); }); |
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666 } |
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667 |
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668 template <> void rand_exponential (octave_idx_type n, double *p) |
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669 { |
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670 std::generate_n (p, n, [](void) { return rand_exponential<double> (); }); |
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671 } |
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673 #undef ZIGINT |
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674 #undef EMANTISSA |
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675 #undef ERANDI |
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676 #undef NMANTISSA |
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677 #undef NRANDI |
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678 #undef RANDU |
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679 |
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680 #define ZIGINT uint32_t |
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681 #define EMANTISSA 4294967296.0 /* 32 bit mantissa */ |
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682 #define ERANDI randi32() /* 32 bits for mantissa */ |
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683 #define NMANTISSA 2147483648.0 /* 31 bit mantissa */ |
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684 #define NRANDI randi32() /* 31 bits for mantissa + 1 bit sign */ |
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685 #define RANDU randu24() |
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686 |
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687 static ZIGINT fki[ZIGGURAT_TABLE_SIZE]; |
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688 static float fwi[ZIGGURAT_TABLE_SIZE], ffi[ZIGGURAT_TABLE_SIZE]; |
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689 static ZIGINT fke[ZIGGURAT_TABLE_SIZE]; |
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690 static float fwe[ZIGGURAT_TABLE_SIZE], ffe[ZIGGURAT_TABLE_SIZE]; |
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691 |
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692 static void create_ziggurat_float_tables (void) |
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693 { |
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694 int i; |
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695 float x, x1; |
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696 |
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697 /* Ziggurat tables for the normal distribution */ |
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698 x1 = ZIGGURAT_NOR_R; |
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699 fwi[255] = x1 / NMANTISSA; |
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700 ffi[255] = exp (-0.5 * x1 * x1); |
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701 |
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702 /* Index zero is special for tail strip, where Marsaglia and Tsang |
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703 * defines this as |
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704 * k_0 = 2^31 * r * f(r) / v, w_0 = 0.5^31 * v / f(r), f_0 = 1, |
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705 * where v is the area of each strip of the ziggurat. |
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706 */ |
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707 fki[0] = static_cast<ZIGINT> (x1 * ffi[255] / NOR_SECTION_AREA * NMANTISSA); |
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708 fwi[0] = NOR_SECTION_AREA / ffi[255] / NMANTISSA; |
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709 ffi[0] = 1.; |
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710 |
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711 for (i = 254; i > 0; i--) |
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712 { |
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713 /* New x is given by x = f^{-1}(v/x_{i+1} + f(x_{i+1})), thus |
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714 * need inverse operator of y = exp(-0.5*x*x) -> x = sqrt(-2*ln(y)) |
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715 */ |
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716 x = std::sqrt (-2. * std::log (NOR_SECTION_AREA / x1 + ffi[i+1])); |
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717 fki[i+1] = static_cast<ZIGINT> (x / x1 * NMANTISSA); |
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718 fwi[i] = x / NMANTISSA; |
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719 ffi[i] = exp (-0.5 * x * x); |
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720 x1 = x; |
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721 } |
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722 |
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723 fki[1] = 0; |
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724 |
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725 /* Zigurrat tables for the exponential distribution */ |
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726 x1 = ZIGGURAT_EXP_R; |
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727 fwe[255] = x1 / EMANTISSA; |
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728 ffe[255] = exp (-x1); |
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729 |
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730 /* Index zero is special for tail strip, where Marsaglia and Tsang |
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731 * defines this as |
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732 * k_0 = 2^32 * r * f(r) / v, w_0 = 0.5^32 * v / f(r), f_0 = 1, |
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733 * where v is the area of each strip of the ziggurat. |
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734 */ |
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735 fke[0] = static_cast<ZIGINT> (x1 * ffe[255] / EXP_SECTION_AREA * EMANTISSA); |
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736 fwe[0] = EXP_SECTION_AREA / ffe[255] / EMANTISSA; |
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737 ffe[0] = 1.; |
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738 |
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739 for (i = 254; i > 0; i--) |
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740 { |
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741 /* New x is given by x = f^{-1}(v/x_{i+1} + f(x_{i+1})), thus |
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742 * need inverse operator of y = exp(-x) -> x = -ln(y) |
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743 */ |
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744 x = - std::log (EXP_SECTION_AREA / x1 + ffe[i+1]); |
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745 fke[i+1] = static_cast<ZIGINT> (x / x1 * EMANTISSA); |
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746 fwe[i] = x / EMANTISSA; |
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747 ffe[i] = exp (-x); |
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748 x1 = x; |
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749 } |
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750 fke[1] = 0; |
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751 |
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752 inittf = 0; |
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753 } |
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754 |
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755 /* |
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756 * Here is the guts of the algorithm. As Marsaglia and Tsang state the |
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757 * algorithm in their paper |
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758 * |
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759 * 1) Calculate a random signed integer j and let i be the index |
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760 * provided by the rightmost 8-bits of j |
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761 * 2) Set x = j * w_i. If j < k_i return x |
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762 * 3) If i = 0, then return x from the tail |
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763 * 4) If [f(x_{i-1}) - f(x_i)] * U < f(x) - f(x_i), return x |
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764 * 5) goto step 1 |
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765 * |
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766 * Where f is the functional form of the distribution, which for a normal |
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767 * distribution is exp(-0.5*x*x) |
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768 */ |
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769 |
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770 template <> float rand_normal<float> (void) |
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771 { |
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772 if (inittf) |
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773 create_ziggurat_float_tables (); |
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774 |
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775 while (1) |
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776 { |
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777 /* 32-bit mantissa */ |
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778 const uint32_t r = randi32 (); |
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779 const uint32_t rabs = r & LMASK; |
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780 const int idx = static_cast<int> (r & 0xFF); |
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781 const float x = static_cast<int32_t> (r) * fwi[idx]; |
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782 if (rabs < fki[idx]) |
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783 return x; /* 99.3% of the time we return here 1st try */ |
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784 else if (idx == 0) |
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785 { |
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786 /* As stated in Marsaglia and Tsang |
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787 * |
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788 * For the normal tail, the method of Marsaglia[5] provides: |
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789 * generate x = -ln(U_1)/r, y = -ln(U_2), until y+y > x*x, |
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790 * then return r+x. Except that r+x is always in the positive |
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791 * tail!!!! Any thing random might be used to determine the |
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792 * sign, but as we already have r we might as well use it |
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793 * |
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794 * [PAK] but not the bottom 8 bits, since they are all 0 here! |
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795 */ |
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796 float xx, yy; |
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797 do |
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798 { |
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799 xx = - ZIGGURAT_NOR_INV_R * std::log (RANDU); |
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800 yy = - std::log (RANDU); |
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801 } |
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802 while ( yy+yy <= xx*xx); |
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803 return ((rabs & 0x100) ? -ZIGGURAT_NOR_R-xx : ZIGGURAT_NOR_R+xx); |
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804 } |
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805 else if ((ffi[idx-1] - ffi[idx]) * RANDU + ffi[idx] < exp (-0.5*x*x)) |
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806 return x; |
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807 } |
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808 } |
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809 |
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810 template <> float rand_exponential<float> (void) |
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811 { |
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812 if (inittf) |
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813 create_ziggurat_float_tables (); |
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814 |
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815 while (1) |
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816 { |
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817 ZIGINT ri = ERANDI; |
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818 const int idx = static_cast<int> (ri & 0xFF); |
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819 const float x = ri * fwe[idx]; |
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820 if (ri < fke[idx]) |
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821 return x; /* 98.9% of the time we return here 1st try */ |
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822 else if (idx == 0) |
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823 { |
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824 /* As stated in Marsaglia and Tsang |
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825 * |
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826 * For the exponential tail, the method of Marsaglia[5] provides: |
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827 * x = r - ln(U); |
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828 */ |
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829 return ZIGGURAT_EXP_R - std::log (RANDU); |
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830 } |
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831 else if ((ffe[idx-1] - ffe[idx]) * RANDU + ffe[idx] < exp (-x)) |
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832 return x; |
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833 } |
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834 } |
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835 |
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836 template <> void rand_uniform (octave_idx_type n, float *p) |
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837 { |
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838 std::generate_n (p, n, [](void) { return rand_uniform<float> (); }); |
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839 } |
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840 |
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841 template <> void rand_normal (octave_idx_type n, float *p) |
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842 { |
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843 std::generate_n (p, n, [](void) { return rand_normal<float> (); }); |
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844 } |
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845 |
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846 template <> void rand_exponential (octave_idx_type n, float *p) |
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847 { |
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848 std::generate_n (p, n, [](void) { return rand_exponential<float> (); }); |
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849 } |
14655
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850 } |
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851 |