1/* $OpenBSD: b_log__D.c,v 1.4 2009/10/27 23:59:29 deraadt Exp $ */ 2/* 3 * Copyright (c) 1992, 1993 4 * The Regents of the University of California. All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 3. Neither the name of the University nor the names of its contributors 15 * may be used to endorse or promote products derived from this software 16 * without specific prior written permission. 17 * 18 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 19 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 20 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 21 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 22 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 23 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 24 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 25 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 26 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 27 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 28 * SUCH DAMAGE. 29 */ 30 31#include "math.h" 32#include "math_private.h" 33 34/* Table-driven natural logarithm. 35 * 36 * This code was derived, with minor modifications, from: 37 * Peter Tang, "Table-Driven Implementation of the 38 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans. 39 * Math Software, vol 16. no 4, pp 378-400, Dec 1990). 40 * 41 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256, 42 * where F = j/128 for j an integer in [0, 128]. 43 * 44 * log(2^m) = log2_hi*m + log2_tail*m 45 * since m is an integer, the dominant term is exact. 46 * m has at most 10 digits (for subnormal numbers), 47 * and log2_hi has 11 trailing zero bits. 48 * 49 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h 50 * logF_hi[] + 512 is exact. 51 * 52 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ... 53 * the leading term is calculated to extra precision in two 54 * parts, the larger of which adds exactly to the dominant 55 * m and F terms. 56 * There are two cases: 57 * 1. when m, 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/* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. 72 * Used for generation of extend precision logarithms. 73 * The constant 35184372088832 is 2^45, so the divide is exact. 74 * It ensures correct reading of logF_head, even for inaccurate 75 * decimal-to-binary conversion routines. (Everybody gets the 76 * right answer for integers less than 2^53.) 77 * Values for log(F) were generated using error < 10^-57 absolute 78 * with the bc -l package. 79*/ 80static const double A1 = .08333333333333178827; 81static const double A2 = .01250000000377174923; 82static const double A3 = .002232139987919447809; 83static const double A4 = .0004348877777076145742; 84 85static const double logF_head[N+1] = { 86 0., 87 .007782140442060381246, 88 .015504186535963526694, 89 .023167059281547608406, 90 .030771658666765233647, 91 .038318864302141264488, 92 .045809536031242714670, 93 .053244514518837604555, 94 .060624621816486978786, 95 .067950661908525944454, 96 .075223421237524235039, 97 .082443669210988446138, 98 .089612158689760690322, 99 .096729626458454731618, 100 .103796793681567578460, 101 .110814366340264314203, 102 .117783035656430001836, 103 .124703478501032805070, 104 .131576357788617315236, 105 .138402322859292326029, 106 .145182009844575077295, 107 .151916042025732167530, 108 .158605030176659056451, 109 .165249572895390883786, 110 .171850256926518341060, 111 .178407657472689606947, 112 .184922338493834104156, 113 .191394852999565046047, 114 .197825743329758552135, 115 .204215541428766300668, 116 .210564769107350002741, 117 .216873938300523150246, 118 .223143551314024080056, 119 .229374101064877322642, 120 .235566071312860003672, 121 .241719936886966024758, 122 .247836163904594286577, 123 .253915209980732470285, 124 .259957524436686071567, 125 .265963548496984003577, 126 .271933715484010463114, 127 .277868451003087102435, 128 .283768173130738432519, 129 .289633292582948342896, 130 .295464212893421063199, 131 .301261330578199704177, 132 .307025035294827830512, 133 .312755710004239517729, 134 .318453731118097493890, 135 .324119468654316733591, 136 .329753286372579168528, 137 .335355541920762334484, 138 .340926586970454081892, 139 .346466767346100823488, 140 .351976423156884266063, 141 .357455888922231679316, 142 .362905493689140712376, 143 .368325561158599157352, 144 .373716409793814818840, 145 .379078352934811846353, 146 .384411698910298582632, 147 .389716751140440464951, 148 .394993808240542421117, 149 .400243164127459749579, 150 .405465108107819105498, 151 .410659924985338875558, 152 .415827895143593195825, 153 .420969294644237379543, 154 .426084395310681429691, 155 .431173464818130014464, 156 .436236766774527495726, 157 .441274560805140936281, 158 .446287102628048160113, 159 .451274644139630254358, 160 .456237433481874177232, 161 .461175715122408291790, 162 .466089729924533457960, 163 .470979715219073113985, 164 .475845904869856894947, 165 .480688529345570714212, 166 .485507815781602403149, 167 .490303988045525329653, 168 .495077266798034543171, 169 .499827869556611403822, 170 .504556010751912253908, 171 .509261901790523552335, 172 .513945751101346104405, 173 .518607764208354637958, 174 .523248143765158602036, 175 .527867089620485785417, 176 .532464798869114019908, 177 .537041465897345915436, 178 .541597282432121573947, 179 .546132437597407260909, 180 .550647117952394182793, 181 .555141507540611200965, 182 .559615787935399566777, 183 .564070138285387656651, 184 .568504735352689749561, 185 .572919753562018740922, 186 .577315365035246941260, 187 .581691739635061821900, 188 .586049045003164792433, 189 .590387446602107957005, 190 .594707107746216934174, 191 .599008189645246602594, 192 .603290851438941899687, 193 .607555250224322662688, 194 .611801541106615331955, 195 .616029877215623855590, 196 .620240409751204424537, 197 .624433288012369303032, 198 .628608659422752680256, 199 .632766669570628437213, 200 .636907462236194987781, 201 .641031179420679109171, 202 .645137961373620782978, 203 .649227946625615004450, 204 .653301272011958644725, 205 .657358072709030238911, 206 .661398482245203922502, 207 .665422632544505177065, 208 .669430653942981734871, 209 .673422675212350441142, 210 .677398823590920073911, 211 .681359224807238206267, 212 .685304003098281100392, 213 .689233281238557538017, 214 .693147180560117703862 215}; 216 217static const double logF_tail[N+1] = { 218 0., 219 -.00000000000000543229938420049, 220 .00000000000000172745674997061, 221 -.00000000000001323017818229233, 222 -.00000000000001154527628289872, 223 -.00000000000000466529469958300, 224 .00000000000005148849572685810, 225 -.00000000000002532168943117445, 226 -.00000000000005213620639136504, 227 -.00000000000001819506003016881, 228 .00000000000006329065958724544, 229 .00000000000008614512936087814, 230 -.00000000000007355770219435028, 231 .00000000000009638067658552277, 232 .00000000000007598636597194141, 233 .00000000000002579999128306990, 234 -.00000000000004654729747598444, 235 -.00000000000007556920687451336, 236 .00000000000010195735223708472, 237 -.00000000000017319034406422306, 238 -.00000000000007718001336828098, 239 .00000000000010980754099855238, 240 -.00000000000002047235780046195, 241 -.00000000000008372091099235912, 242 .00000000000014088127937111135, 243 .00000000000012869017157588257, 244 .00000000000017788850778198106, 245 .00000000000006440856150696891, 246 .00000000000016132822667240822, 247 -.00000000000007540916511956188, 248 -.00000000000000036507188831790, 249 .00000000000009120937249914984, 250 .00000000000018567570959796010, 251 -.00000000000003149265065191483, 252 -.00000000000009309459495196889, 253 .00000000000017914338601329117, 254 -.00000000000001302979717330866, 255 .00000000000023097385217586939, 256 .00000000000023999540484211737, 257 .00000000000015393776174455408, 258 -.00000000000036870428315837678, 259 .00000000000036920375082080089, 260 -.00000000000009383417223663699, 261 .00000000000009433398189512690, 262 .00000000000041481318704258568, 263 -.00000000000003792316480209314, 264 .00000000000008403156304792424, 265 -.00000000000034262934348285429, 266 .00000000000043712191957429145, 267 -.00000000000010475750058776541, 268 -.00000000000011118671389559323, 269 .00000000000037549577257259853, 270 .00000000000013912841212197565, 271 .00000000000010775743037572640, 272 .00000000000029391859187648000, 273 -.00000000000042790509060060774, 274 .00000000000022774076114039555, 275 .00000000000010849569622967912, 276 -.00000000000023073801945705758, 277 .00000000000015761203773969435, 278 .00000000000003345710269544082, 279 -.00000000000041525158063436123, 280 .00000000000032655698896907146, 281 -.00000000000044704265010452446, 282 .00000000000034527647952039772, 283 -.00000000000007048962392109746, 284 .00000000000011776978751369214, 285 -.00000000000010774341461609578, 286 .00000000000021863343293215910, 287 .00000000000024132639491333131, 288 .00000000000039057462209830700, 289 -.00000000000026570679203560751, 290 .00000000000037135141919592021, 291 -.00000000000017166921336082431, 292 -.00000000000028658285157914353, 293 -.00000000000023812542263446809, 294 .00000000000006576659768580062, 295 -.00000000000028210143846181267, 296 .00000000000010701931762114254, 297 .00000000000018119346366441110, 298 .00000000000009840465278232627, 299 -.00000000000033149150282752542, 300 -.00000000000018302857356041668, 301 -.00000000000016207400156744949, 302 .00000000000048303314949553201, 303 -.00000000000071560553172382115, 304 .00000000000088821239518571855, 305 -.00000000000030900580513238244, 306 -.00000000000061076551972851496, 307 .00000000000035659969663347830, 308 .00000000000035782396591276383, 309 -.00000000000046226087001544578, 310 .00000000000062279762917225156, 311 .00000000000072838947272065741, 312 .00000000000026809646615211673, 313 -.00000000000010960825046059278, 314 .00000000000002311949383800537, 315 -.00000000000058469058005299247, 316 -.00000000000002103748251144494, 317 -.00000000000023323182945587408, 318 -.00000000000042333694288141916, 319 -.00000000000043933937969737844, 320 .00000000000041341647073835565, 321 .00000000000006841763641591466, 322 .00000000000047585534004430641, 323 .00000000000083679678674757695, 324 -.00000000000085763734646658640, 325 .00000000000021913281229340092, 326 -.00000000000062242842536431148, 327 -.00000000000010983594325438430, 328 .00000000000065310431377633651, 329 -.00000000000047580199021710769, 330 -.00000000000037854251265457040, 331 .00000000000040939233218678664, 332 .00000000000087424383914858291, 333 .00000000000025218188456842882, 334 -.00000000000003608131360422557, 335 -.00000000000050518555924280902, 336 .00000000000078699403323355317, 337 -.00000000000067020876961949060, 338 .00000000000016108575753932458, 339 .00000000000058527188436251509, 340 -.00000000000035246757297904791, 341 -.00000000000018372084495629058, 342 .00000000000088606689813494916, 343 .00000000000066486268071468700, 344 .00000000000063831615170646519, 345 .00000000000025144230728376072, 346 -.00000000000017239444525614834 347}; 348 349/* 350 * Extra precision variant, returning struct {double a, b;}; 351 * log(x) = a+b to 63 bits, with a rounded to 26 bits. 352 */ 353struct Double 354__log__D(double x) 355{ 356 int m, j; 357 double F, f, g, q, u, v, u2; 358 volatile double u1; 359 struct Double r; 360 361 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 362 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 363 364 m = logb(x); 365 g = ldexp(x, -m); 366 if (m == -1022) { 367 j = logb(g); 368 m += j; 369 g = ldexp(g, -j); 370 } 371 j = N*(g-1) + .5; 372 F = (1.0/N) * j + 1; 373 f = g - F; 374 375 g = 1/(2*F+f); 376 u = 2*f*g; 377 v = u*u; 378 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 379 if (m | j) { 380 u1 = u + 513; 381 u1 -= 513; 382 } 383 else { 384 u1 = u; 385 TRUNC(u1); 386 } 387 u2 = (2.0*(f - F*u1) - u1*f) * g; 388 389 u1 += m*logF_head[N] + logF_head[j]; 390 391 u2 += logF_tail[j]; u2 += q; 392 u2 += logF_tail[N]*m; 393 r.a = u1 + u2; /* Only difference is here */ 394 TRUNC(r.a); 395 r.b = (u1 - r.a) + u2; 396 return (r); 397} 398