1 /*- 2 * SPDX-License-Identifier: BSD-3-Clause 3 * 4 * Copyright (c) 1992, 1993 5 * The Regents of the University of California. All rights reserved. 6 * 7 * Redistribution and use in source and binary forms, with or without 8 * modification, are permitted provided that the following conditions 9 * are met: 10 * 1. Redistributions of source code must retain the above copyright 11 * notice, this list of conditions and the following disclaimer. 12 * 2. Redistributions in binary form must reproduce the above copyright 13 * notice, this list of conditions and the following disclaimer in the 14 * documentation and/or other materials provided with the distribution. 15 * 3. Neither the name of the University nor the names of its contributors 16 * may be used to endorse or promote products derived from this software 17 * without specific prior written permission. 18 * 19 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 20 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 21 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 22 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 23 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 24 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 25 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 26 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 27 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 28 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 29 * SUCH DAMAGE. 30 */ 31 32 /* Table-driven natural logarithm. 33 * 34 * This code was derived, with minor modifications, from: 35 * Peter Tang, "Table-Driven Implementation of the 36 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans. 37 * Math Software, vol 16. no 4, pp 378-400, Dec 1990). 38 * 39 * Calculates log(2^m*F*(1+f/F)), |f/F| <= 1/256, 40 * where F = j/128 for j an integer in [0, 128]. 41 * 42 * log(2^m) = log2_hi*m + log2_tail*m 43 * The leading term is exact, because m is an integer, 44 * m has at most 10 digits (for subnormal numbers), 45 * and log2_hi has 11 trailing zero bits. 46 * 47 * log(F) = logF_hi[j] + logF_lo[j] is in table below. 48 * logF_hi[] + 512 is exact. 49 * 50 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ... 51 * 52 * The leading term is calculated to extra precision in two 53 * parts, the larger of which adds exactly to the dominant 54 * m and F terms. 55 * 56 * There are two cases: 57 * 1. When m and j are non-zero (m | j), use absolute 58 * precision for the leading term. 59 * 2. When m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1). 60 * In this case, use a relative precision of 24 bits. 61 * (This is done differently in the original paper) 62 * 63 * Special cases: 64 * 0 return signalling -Inf 65 * neg return signalling NaN 66 * +Inf return +Inf 67 */ 68 69 #define N 128 70 71 /* 72 * Coefficients in the polynomial approximation of log(1+f/F). 73 * Domain of x is [0,1./256] with 2**(-64.187) precision. 74 */ 75 static const double 76 A1 = 8.3333333333333329e-02, /* 0x3fb55555, 0x55555555 */ 77 A2 = 1.2499999999943598e-02, /* 0x3f899999, 0x99991a98 */ 78 A3 = 2.2321527525957776e-03; /* 0x3f624929, 0xe24e70be */ 79 80 /* 81 * Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. 82 * Used for generation of extend precision logarithms. 83 * The constant 35184372088832 is 2^45, so the divide is exact. 84 * It ensures correct reading of logF_head, even for inaccurate 85 * decimal-to-binary conversion routines. (Everybody gets the 86 * right answer for integers less than 2^53.) 87 * Values for log(F) were generated using error < 10^-57 absolute 88 * with the bc -l package. 89 */ 90 static double logF_head[N+1] = { 91 0., 92 .007782140442060381246, 93 .015504186535963526694, 94 .023167059281547608406, 95 .030771658666765233647, 96 .038318864302141264488, 97 .045809536031242714670, 98 .053244514518837604555, 99 .060624621816486978786, 100 .067950661908525944454, 101 .075223421237524235039, 102 .082443669210988446138, 103 .089612158689760690322, 104 .096729626458454731618, 105 .103796793681567578460, 106 .110814366340264314203, 107 .117783035656430001836, 108 .124703478501032805070, 109 .131576357788617315236, 110 .138402322859292326029, 111 .145182009844575077295, 112 .151916042025732167530, 113 .158605030176659056451, 114 .165249572895390883786, 115 .171850256926518341060, 116 .178407657472689606947, 117 .184922338493834104156, 118 .191394852999565046047, 119 .197825743329758552135, 120 .204215541428766300668, 121 .210564769107350002741, 122 .216873938300523150246, 123 .223143551314024080056, 124 .229374101064877322642, 125 .235566071312860003672, 126 .241719936886966024758, 127 .247836163904594286577, 128 .253915209980732470285, 129 .259957524436686071567, 130 .265963548496984003577, 131 .271933715484010463114, 132 .277868451003087102435, 133 .283768173130738432519, 134 .289633292582948342896, 135 .295464212893421063199, 136 .301261330578199704177, 137 .307025035294827830512, 138 .312755710004239517729, 139 .318453731118097493890, 140 .324119468654316733591, 141 .329753286372579168528, 142 .335355541920762334484, 143 .340926586970454081892, 144 .346466767346100823488, 145 .351976423156884266063, 146 .357455888922231679316, 147 .362905493689140712376, 148 .368325561158599157352, 149 .373716409793814818840, 150 .379078352934811846353, 151 .384411698910298582632, 152 .389716751140440464951, 153 .394993808240542421117, 154 .400243164127459749579, 155 .405465108107819105498, 156 .410659924985338875558, 157 .415827895143593195825, 158 .420969294644237379543, 159 .426084395310681429691, 160 .431173464818130014464, 161 .436236766774527495726, 162 .441274560805140936281, 163 .446287102628048160113, 164 .451274644139630254358, 165 .456237433481874177232, 166 .461175715122408291790, 167 .466089729924533457960, 168 .470979715219073113985, 169 .475845904869856894947, 170 .480688529345570714212, 171 .485507815781602403149, 172 .490303988045525329653, 173 .495077266798034543171, 174 .499827869556611403822, 175 .504556010751912253908, 176 .509261901790523552335, 177 .513945751101346104405, 178 .518607764208354637958, 179 .523248143765158602036, 180 .527867089620485785417, 181 .532464798869114019908, 182 .537041465897345915436, 183 .541597282432121573947, 184 .546132437597407260909, 185 .550647117952394182793, 186 .555141507540611200965, 187 .559615787935399566777, 188 .564070138285387656651, 189 .568504735352689749561, 190 .572919753562018740922, 191 .577315365035246941260, 192 .581691739635061821900, 193 .586049045003164792433, 194 .590387446602107957005, 195 .594707107746216934174, 196 .599008189645246602594, 197 .603290851438941899687, 198 .607555250224322662688, 199 .611801541106615331955, 200 .616029877215623855590, 201 .620240409751204424537, 202 .624433288012369303032, 203 .628608659422752680256, 204 .632766669570628437213, 205 .636907462236194987781, 206 .641031179420679109171, 207 .645137961373620782978, 208 .649227946625615004450, 209 .653301272011958644725, 210 .657358072709030238911, 211 .661398482245203922502, 212 .665422632544505177065, 213 .669430653942981734871, 214 .673422675212350441142, 215 .677398823590920073911, 216 .681359224807238206267, 217 .685304003098281100392, 218 .689233281238557538017, 219 .693147180560117703862 220 }; 221 222 static double logF_tail[N+1] = { 223 0., 224 -.00000000000000543229938420049, 225 .00000000000000172745674997061, 226 -.00000000000001323017818229233, 227 -.00000000000001154527628289872, 228 -.00000000000000466529469958300, 229 .00000000000005148849572685810, 230 -.00000000000002532168943117445, 231 -.00000000000005213620639136504, 232 -.00000000000001819506003016881, 233 .00000000000006329065958724544, 234 .00000000000008614512936087814, 235 -.00000000000007355770219435028, 236 .00000000000009638067658552277, 237 .00000000000007598636597194141, 238 .00000000000002579999128306990, 239 -.00000000000004654729747598444, 240 -.00000000000007556920687451336, 241 .00000000000010195735223708472, 242 -.00000000000017319034406422306, 243 -.00000000000007718001336828098, 244 .00000000000010980754099855238, 245 -.00000000000002047235780046195, 246 -.00000000000008372091099235912, 247 .00000000000014088127937111135, 248 .00000000000012869017157588257, 249 .00000000000017788850778198106, 250 .00000000000006440856150696891, 251 .00000000000016132822667240822, 252 -.00000000000007540916511956188, 253 -.00000000000000036507188831790, 254 .00000000000009120937249914984, 255 .00000000000018567570959796010, 256 -.00000000000003149265065191483, 257 -.00000000000009309459495196889, 258 .00000000000017914338601329117, 259 -.00000000000001302979717330866, 260 .00000000000023097385217586939, 261 .00000000000023999540484211737, 262 .00000000000015393776174455408, 263 -.00000000000036870428315837678, 264 .00000000000036920375082080089, 265 -.00000000000009383417223663699, 266 .00000000000009433398189512690, 267 .00000000000041481318704258568, 268 -.00000000000003792316480209314, 269 .00000000000008403156304792424, 270 -.00000000000034262934348285429, 271 .00000000000043712191957429145, 272 -.00000000000010475750058776541, 273 -.00000000000011118671389559323, 274 .00000000000037549577257259853, 275 .00000000000013912841212197565, 276 .00000000000010775743037572640, 277 .00000000000029391859187648000, 278 -.00000000000042790509060060774, 279 .00000000000022774076114039555, 280 .00000000000010849569622967912, 281 -.00000000000023073801945705758, 282 .00000000000015761203773969435, 283 .00000000000003345710269544082, 284 -.00000000000041525158063436123, 285 .00000000000032655698896907146, 286 -.00000000000044704265010452446, 287 .00000000000034527647952039772, 288 -.00000000000007048962392109746, 289 .00000000000011776978751369214, 290 -.00000000000010774341461609578, 291 .00000000000021863343293215910, 292 .00000000000024132639491333131, 293 .00000000000039057462209830700, 294 -.00000000000026570679203560751, 295 .00000000000037135141919592021, 296 -.00000000000017166921336082431, 297 -.00000000000028658285157914353, 298 -.00000000000023812542263446809, 299 .00000000000006576659768580062, 300 -.00000000000028210143846181267, 301 .00000000000010701931762114254, 302 .00000000000018119346366441110, 303 .00000000000009840465278232627, 304 -.00000000000033149150282752542, 305 -.00000000000018302857356041668, 306 -.00000000000016207400156744949, 307 .00000000000048303314949553201, 308 -.00000000000071560553172382115, 309 .00000000000088821239518571855, 310 -.00000000000030900580513238244, 311 -.00000000000061076551972851496, 312 .00000000000035659969663347830, 313 .00000000000035782396591276383, 314 -.00000000000046226087001544578, 315 .00000000000062279762917225156, 316 .00000000000072838947272065741, 317 .00000000000026809646615211673, 318 -.00000000000010960825046059278, 319 .00000000000002311949383800537, 320 -.00000000000058469058005299247, 321 -.00000000000002103748251144494, 322 -.00000000000023323182945587408, 323 -.00000000000042333694288141916, 324 -.00000000000043933937969737844, 325 .00000000000041341647073835565, 326 .00000000000006841763641591466, 327 .00000000000047585534004430641, 328 .00000000000083679678674757695, 329 -.00000000000085763734646658640, 330 .00000000000021913281229340092, 331 -.00000000000062242842536431148, 332 -.00000000000010983594325438430, 333 .00000000000065310431377633651, 334 -.00000000000047580199021710769, 335 -.00000000000037854251265457040, 336 .00000000000040939233218678664, 337 .00000000000087424383914858291, 338 .00000000000025218188456842882, 339 -.00000000000003608131360422557, 340 -.00000000000050518555924280902, 341 .00000000000078699403323355317, 342 -.00000000000067020876961949060, 343 .00000000000016108575753932458, 344 .00000000000058527188436251509, 345 -.00000000000035246757297904791, 346 -.00000000000018372084495629058, 347 .00000000000088606689813494916, 348 .00000000000066486268071468700, 349 .00000000000063831615170646519, 350 .00000000000025144230728376072, 351 -.00000000000017239444525614834 352 }; 353 /* 354 * Extra precision variant, returning struct {double a, b;}; 355 * log(x) = a+b to 63 bits, with 'a' rounded to 24 bits. 356 */ 357 static struct Double 358 __log__D(double x) 359 { 360 int m, j; 361 double F, f, g, q, u, v, u1, u2; 362 struct Double r; 363 364 /* 365 * Argument reduction: 1 <= g < 2; x/2^m = g; 366 * y = F*(1 + f/F) for |f| <= 2^-8 367 */ 368 g = frexp(x, &m); 369 g *= 2; 370 m--; 371 if (m == -1022) { 372 j = ilogb(g); 373 m += j; 374 g = ldexp(g, -j); 375 } 376 j = N * (g - 1) + 0.5; 377 F = (1. / N) * j + 1; 378 f = g - F; 379 380 g = 1 / (2 * F + f); 381 u = 2 * f * g; 382 v = u * u; 383 q = u * v * (A1 + v * (A2 + v * A3)); 384 if (m | j) { 385 u1 = u + 513; 386 u1 -= 513; 387 } else { 388 u1 = (float)u; 389 } 390 u2 = (2 * (f - F * u1) - u1 * f) * g; 391 392 u1 += m * logF_head[N] + logF_head[j]; 393 394 u2 += logF_tail[j]; 395 u2 += q; 396 u2 += logF_tail[N] * m; 397 r.a = (float)(u1 + u2); /* Only difference is here. */ 398 r.b = (u1 - r.a) + u2; 399 return (r); 400 } 401