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 /* @(#)log.c 8.2 (Berkeley) 11/30/93 */ 33 #include <sys/cdefs.h> 34 __FBSDID("$FreeBSD$"); 35 36 #include <math.h> 37 38 #include "mathimpl.h" 39 40 /* Table-driven natural logarithm. 41 * 42 * This code was derived, with minor modifications, from: 43 * Peter Tang, "Table-Driven Implementation of the 44 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans. 45 * Math Software, vol 16. no 4, pp 378-400, Dec 1990). 46 * 47 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256, 48 * where F = j/128 for j an integer in [0, 128]. 49 * 50 * log(2^m) = log2_hi*m + log2_tail*m 51 * since m is an integer, the dominant term is exact. 52 * m has at most 10 digits (for subnormal numbers), 53 * and log2_hi has 11 trailing zero bits. 54 * 55 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h 56 * logF_hi[] + 512 is exact. 57 * 58 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ... 59 * the leading term is calculated to extra precision in two 60 * parts, the larger of which adds exactly to the dominant 61 * m and F terms. 62 * There are two cases: 63 * 1. when m, j are non-zero (m | j), use absolute 64 * precision for the leading term. 65 * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1). 66 * In this case, use a relative precision of 24 bits. 67 * (This is done differently in the original paper) 68 * 69 * Special cases: 70 * 0 return signalling -Inf 71 * neg return signalling NaN 72 * +Inf return +Inf 73 */ 74 75 #define N 128 76 77 /* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. 78 * Used for generation of extend precision logarithms. 79 * The constant 35184372088832 is 2^45, so the divide is exact. 80 * It ensures correct reading of logF_head, even for inaccurate 81 * decimal-to-binary conversion routines. (Everybody gets the 82 * right answer for integers less than 2^53.) 83 * Values for log(F) were generated using error < 10^-57 absolute 84 * with the bc -l package. 85 */ 86 static double A1 = .08333333333333178827; 87 static double A2 = .01250000000377174923; 88 static double A3 = .002232139987919447809; 89 static double A4 = .0004348877777076145742; 90 91 static double logF_head[N+1] = { 92 0., 93 .007782140442060381246, 94 .015504186535963526694, 95 .023167059281547608406, 96 .030771658666765233647, 97 .038318864302141264488, 98 .045809536031242714670, 99 .053244514518837604555, 100 .060624621816486978786, 101 .067950661908525944454, 102 .075223421237524235039, 103 .082443669210988446138, 104 .089612158689760690322, 105 .096729626458454731618, 106 .103796793681567578460, 107 .110814366340264314203, 108 .117783035656430001836, 109 .124703478501032805070, 110 .131576357788617315236, 111 .138402322859292326029, 112 .145182009844575077295, 113 .151916042025732167530, 114 .158605030176659056451, 115 .165249572895390883786, 116 .171850256926518341060, 117 .178407657472689606947, 118 .184922338493834104156, 119 .191394852999565046047, 120 .197825743329758552135, 121 .204215541428766300668, 122 .210564769107350002741, 123 .216873938300523150246, 124 .223143551314024080056, 125 .229374101064877322642, 126 .235566071312860003672, 127 .241719936886966024758, 128 .247836163904594286577, 129 .253915209980732470285, 130 .259957524436686071567, 131 .265963548496984003577, 132 .271933715484010463114, 133 .277868451003087102435, 134 .283768173130738432519, 135 .289633292582948342896, 136 .295464212893421063199, 137 .301261330578199704177, 138 .307025035294827830512, 139 .312755710004239517729, 140 .318453731118097493890, 141 .324119468654316733591, 142 .329753286372579168528, 143 .335355541920762334484, 144 .340926586970454081892, 145 .346466767346100823488, 146 .351976423156884266063, 147 .357455888922231679316, 148 .362905493689140712376, 149 .368325561158599157352, 150 .373716409793814818840, 151 .379078352934811846353, 152 .384411698910298582632, 153 .389716751140440464951, 154 .394993808240542421117, 155 .400243164127459749579, 156 .405465108107819105498, 157 .410659924985338875558, 158 .415827895143593195825, 159 .420969294644237379543, 160 .426084395310681429691, 161 .431173464818130014464, 162 .436236766774527495726, 163 .441274560805140936281, 164 .446287102628048160113, 165 .451274644139630254358, 166 .456237433481874177232, 167 .461175715122408291790, 168 .466089729924533457960, 169 .470979715219073113985, 170 .475845904869856894947, 171 .480688529345570714212, 172 .485507815781602403149, 173 .490303988045525329653, 174 .495077266798034543171, 175 .499827869556611403822, 176 .504556010751912253908, 177 .509261901790523552335, 178 .513945751101346104405, 179 .518607764208354637958, 180 .523248143765158602036, 181 .527867089620485785417, 182 .532464798869114019908, 183 .537041465897345915436, 184 .541597282432121573947, 185 .546132437597407260909, 186 .550647117952394182793, 187 .555141507540611200965, 188 .559615787935399566777, 189 .564070138285387656651, 190 .568504735352689749561, 191 .572919753562018740922, 192 .577315365035246941260, 193 .581691739635061821900, 194 .586049045003164792433, 195 .590387446602107957005, 196 .594707107746216934174, 197 .599008189645246602594, 198 .603290851438941899687, 199 .607555250224322662688, 200 .611801541106615331955, 201 .616029877215623855590, 202 .620240409751204424537, 203 .624433288012369303032, 204 .628608659422752680256, 205 .632766669570628437213, 206 .636907462236194987781, 207 .641031179420679109171, 208 .645137961373620782978, 209 .649227946625615004450, 210 .653301272011958644725, 211 .657358072709030238911, 212 .661398482245203922502, 213 .665422632544505177065, 214 .669430653942981734871, 215 .673422675212350441142, 216 .677398823590920073911, 217 .681359224807238206267, 218 .685304003098281100392, 219 .689233281238557538017, 220 .693147180560117703862 221 }; 222 223 static double logF_tail[N+1] = { 224 0., 225 -.00000000000000543229938420049, 226 .00000000000000172745674997061, 227 -.00000000000001323017818229233, 228 -.00000000000001154527628289872, 229 -.00000000000000466529469958300, 230 .00000000000005148849572685810, 231 -.00000000000002532168943117445, 232 -.00000000000005213620639136504, 233 -.00000000000001819506003016881, 234 .00000000000006329065958724544, 235 .00000000000008614512936087814, 236 -.00000000000007355770219435028, 237 .00000000000009638067658552277, 238 .00000000000007598636597194141, 239 .00000000000002579999128306990, 240 -.00000000000004654729747598444, 241 -.00000000000007556920687451336, 242 .00000000000010195735223708472, 243 -.00000000000017319034406422306, 244 -.00000000000007718001336828098, 245 .00000000000010980754099855238, 246 -.00000000000002047235780046195, 247 -.00000000000008372091099235912, 248 .00000000000014088127937111135, 249 .00000000000012869017157588257, 250 .00000000000017788850778198106, 251 .00000000000006440856150696891, 252 .00000000000016132822667240822, 253 -.00000000000007540916511956188, 254 -.00000000000000036507188831790, 255 .00000000000009120937249914984, 256 .00000000000018567570959796010, 257 -.00000000000003149265065191483, 258 -.00000000000009309459495196889, 259 .00000000000017914338601329117, 260 -.00000000000001302979717330866, 261 .00000000000023097385217586939, 262 .00000000000023999540484211737, 263 .00000000000015393776174455408, 264 -.00000000000036870428315837678, 265 .00000000000036920375082080089, 266 -.00000000000009383417223663699, 267 .00000000000009433398189512690, 268 .00000000000041481318704258568, 269 -.00000000000003792316480209314, 270 .00000000000008403156304792424, 271 -.00000000000034262934348285429, 272 .00000000000043712191957429145, 273 -.00000000000010475750058776541, 274 -.00000000000011118671389559323, 275 .00000000000037549577257259853, 276 .00000000000013912841212197565, 277 .00000000000010775743037572640, 278 .00000000000029391859187648000, 279 -.00000000000042790509060060774, 280 .00000000000022774076114039555, 281 .00000000000010849569622967912, 282 -.00000000000023073801945705758, 283 .00000000000015761203773969435, 284 .00000000000003345710269544082, 285 -.00000000000041525158063436123, 286 .00000000000032655698896907146, 287 -.00000000000044704265010452446, 288 .00000000000034527647952039772, 289 -.00000000000007048962392109746, 290 .00000000000011776978751369214, 291 -.00000000000010774341461609578, 292 .00000000000021863343293215910, 293 .00000000000024132639491333131, 294 .00000000000039057462209830700, 295 -.00000000000026570679203560751, 296 .00000000000037135141919592021, 297 -.00000000000017166921336082431, 298 -.00000000000028658285157914353, 299 -.00000000000023812542263446809, 300 .00000000000006576659768580062, 301 -.00000000000028210143846181267, 302 .00000000000010701931762114254, 303 .00000000000018119346366441110, 304 .00000000000009840465278232627, 305 -.00000000000033149150282752542, 306 -.00000000000018302857356041668, 307 -.00000000000016207400156744949, 308 .00000000000048303314949553201, 309 -.00000000000071560553172382115, 310 .00000000000088821239518571855, 311 -.00000000000030900580513238244, 312 -.00000000000061076551972851496, 313 .00000000000035659969663347830, 314 .00000000000035782396591276383, 315 -.00000000000046226087001544578, 316 .00000000000062279762917225156, 317 .00000000000072838947272065741, 318 .00000000000026809646615211673, 319 -.00000000000010960825046059278, 320 .00000000000002311949383800537, 321 -.00000000000058469058005299247, 322 -.00000000000002103748251144494, 323 -.00000000000023323182945587408, 324 -.00000000000042333694288141916, 325 -.00000000000043933937969737844, 326 .00000000000041341647073835565, 327 .00000000000006841763641591466, 328 .00000000000047585534004430641, 329 .00000000000083679678674757695, 330 -.00000000000085763734646658640, 331 .00000000000021913281229340092, 332 -.00000000000062242842536431148, 333 -.00000000000010983594325438430, 334 .00000000000065310431377633651, 335 -.00000000000047580199021710769, 336 -.00000000000037854251265457040, 337 .00000000000040939233218678664, 338 .00000000000087424383914858291, 339 .00000000000025218188456842882, 340 -.00000000000003608131360422557, 341 -.00000000000050518555924280902, 342 .00000000000078699403323355317, 343 -.00000000000067020876961949060, 344 .00000000000016108575753932458, 345 .00000000000058527188436251509, 346 -.00000000000035246757297904791, 347 -.00000000000018372084495629058, 348 .00000000000088606689813494916, 349 .00000000000066486268071468700, 350 .00000000000063831615170646519, 351 .00000000000025144230728376072, 352 -.00000000000017239444525614834 353 }; 354 355 #if 0 356 double 357 #ifdef _ANSI_SOURCE 358 log(double x) 359 #else 360 log(x) double x; 361 #endif 362 { 363 int m, j; 364 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0; 365 volatile double u1; 366 367 /* Catch special cases */ 368 if (x <= 0) 369 if (x == zero) /* log(0) = -Inf */ 370 return (-one/zero); 371 else /* log(neg) = NaN */ 372 return (zero/zero); 373 else if (!finite(x)) 374 return (x+x); /* x = NaN, Inf */ 375 376 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 377 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 378 379 m = logb(x); 380 g = ldexp(x, -m); 381 if (m == -1022) { 382 j = logb(g), m += j; 383 g = ldexp(g, -j); 384 } 385 j = N*(g-1) + .5; 386 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */ 387 f = g - F; 388 389 /* Approximate expansion for log(1+f/F) ~= u + q */ 390 g = 1/(2*F+f); 391 u = 2*f*g; 392 v = u*u; 393 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 394 395 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8, 396 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits. 397 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750 398 */ 399 if (m | j) 400 u1 = u + 513, u1 -= 513; 401 402 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero; 403 * u1 = u to 24 bits. 404 */ 405 else 406 u1 = u, TRUNC(u1); 407 u2 = (2.0*(f - F*u1) - u1*f) * g; 408 /* u1 + u2 = 2f/(2F+f) to extra precision. */ 409 410 /* log(x) = log(2^m*F*(1+f/F)) = */ 411 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */ 412 /* (exact) + (tiny) */ 413 414 u1 += m*logF_head[N] + logF_head[j]; /* exact */ 415 u2 = (u2 + logF_tail[j]) + q; /* tiny */ 416 u2 += logF_tail[N]*m; 417 return (u1 + u2); 418 } 419 #endif 420 421 /* 422 * Extra precision variant, returning struct {double a, b;}; 423 * log(x) = a+b to 63 bits, with a rounded to 26 bits. 424 */ 425 struct Double 426 #ifdef _ANSI_SOURCE 427 __log__D(double x) 428 #else 429 __log__D(x) double x; 430 #endif 431 { 432 int m, j; 433 double F, f, g, q, u, v, u2; 434 volatile double u1; 435 struct Double r; 436 437 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 438 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 439 440 m = logb(x); 441 g = ldexp(x, -m); 442 if (m == -1022) { 443 j = logb(g), m += j; 444 g = ldexp(g, -j); 445 } 446 j = N*(g-1) + .5; 447 F = (1.0/N) * j + 1; 448 f = g - F; 449 450 g = 1/(2*F+f); 451 u = 2*f*g; 452 v = u*u; 453 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 454 if (m | j) 455 u1 = u + 513, u1 -= 513; 456 else 457 u1 = u, TRUNC(u1); 458 u2 = (2.0*(f - F*u1) - u1*f) * g; 459 460 u1 += m*logF_head[N] + logF_head[j]; 461 462 u2 += logF_tail[j]; u2 += q; 463 u2 += logF_tail[N]*m; 464 r.a = u1 + u2; /* Only difference is here */ 465 TRUNC(r.a); 466 r.b = (u1 - r.a) + u2; 467 return (r); 468 } 469