1 /*- 2 * SPDX-License-Identifier: BSD-4-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. All advertising materials mentioning features or use of this software 16 * must display the following acknowledgement: 17 * This product includes software developed by the University of 18 * California, Berkeley and its contributors. 19 * 4. Neither the name of the University nor the names of its contributors 20 * may be used to endorse or promote products derived from this software 21 * without specific prior written permission. 22 * 23 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 24 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 25 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 26 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 27 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 28 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 29 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 30 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 31 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 32 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 33 * SUCH DAMAGE. 34 */ 35 36 /* @(#)log.c 8.2 (Berkeley) 11/30/93 */ 37 #include <sys/cdefs.h> 38 __FBSDID("$FreeBSD$"); 39 40 #include <math.h> 41 42 #include "mathimpl.h" 43 44 /* Table-driven natural logarithm. 45 * 46 * This code was derived, with minor modifications, from: 47 * Peter Tang, "Table-Driven Implementation of the 48 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans. 49 * Math Software, vol 16. no 4, pp 378-400, Dec 1990). 50 * 51 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256, 52 * where F = j/128 for j an integer in [0, 128]. 53 * 54 * log(2^m) = log2_hi*m + log2_tail*m 55 * since m is an integer, the dominant term is exact. 56 * m has at most 10 digits (for subnormal numbers), 57 * and log2_hi has 11 trailing zero bits. 58 * 59 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h 60 * logF_hi[] + 512 is exact. 61 * 62 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ... 63 * the leading term is calculated to extra precision in two 64 * parts, the larger of which adds exactly to the dominant 65 * m and F terms. 66 * There are two cases: 67 * 1. when m, j are non-zero (m | j), use absolute 68 * precision for the leading term. 69 * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1). 70 * In this case, use a relative precision of 24 bits. 71 * (This is done differently in the original paper) 72 * 73 * Special cases: 74 * 0 return signalling -Inf 75 * neg return signalling NaN 76 * +Inf return +Inf 77 */ 78 79 #define N 128 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 A1 = .08333333333333178827; 91 static double A2 = .01250000000377174923; 92 static double A3 = .002232139987919447809; 93 static double A4 = .0004348877777076145742; 94 95 static double logF_head[N+1] = { 96 0., 97 .007782140442060381246, 98 .015504186535963526694, 99 .023167059281547608406, 100 .030771658666765233647, 101 .038318864302141264488, 102 .045809536031242714670, 103 .053244514518837604555, 104 .060624621816486978786, 105 .067950661908525944454, 106 .075223421237524235039, 107 .082443669210988446138, 108 .089612158689760690322, 109 .096729626458454731618, 110 .103796793681567578460, 111 .110814366340264314203, 112 .117783035656430001836, 113 .124703478501032805070, 114 .131576357788617315236, 115 .138402322859292326029, 116 .145182009844575077295, 117 .151916042025732167530, 118 .158605030176659056451, 119 .165249572895390883786, 120 .171850256926518341060, 121 .178407657472689606947, 122 .184922338493834104156, 123 .191394852999565046047, 124 .197825743329758552135, 125 .204215541428766300668, 126 .210564769107350002741, 127 .216873938300523150246, 128 .223143551314024080056, 129 .229374101064877322642, 130 .235566071312860003672, 131 .241719936886966024758, 132 .247836163904594286577, 133 .253915209980732470285, 134 .259957524436686071567, 135 .265963548496984003577, 136 .271933715484010463114, 137 .277868451003087102435, 138 .283768173130738432519, 139 .289633292582948342896, 140 .295464212893421063199, 141 .301261330578199704177, 142 .307025035294827830512, 143 .312755710004239517729, 144 .318453731118097493890, 145 .324119468654316733591, 146 .329753286372579168528, 147 .335355541920762334484, 148 .340926586970454081892, 149 .346466767346100823488, 150 .351976423156884266063, 151 .357455888922231679316, 152 .362905493689140712376, 153 .368325561158599157352, 154 .373716409793814818840, 155 .379078352934811846353, 156 .384411698910298582632, 157 .389716751140440464951, 158 .394993808240542421117, 159 .400243164127459749579, 160 .405465108107819105498, 161 .410659924985338875558, 162 .415827895143593195825, 163 .420969294644237379543, 164 .426084395310681429691, 165 .431173464818130014464, 166 .436236766774527495726, 167 .441274560805140936281, 168 .446287102628048160113, 169 .451274644139630254358, 170 .456237433481874177232, 171 .461175715122408291790, 172 .466089729924533457960, 173 .470979715219073113985, 174 .475845904869856894947, 175 .480688529345570714212, 176 .485507815781602403149, 177 .490303988045525329653, 178 .495077266798034543171, 179 .499827869556611403822, 180 .504556010751912253908, 181 .509261901790523552335, 182 .513945751101346104405, 183 .518607764208354637958, 184 .523248143765158602036, 185 .527867089620485785417, 186 .532464798869114019908, 187 .537041465897345915436, 188 .541597282432121573947, 189 .546132437597407260909, 190 .550647117952394182793, 191 .555141507540611200965, 192 .559615787935399566777, 193 .564070138285387656651, 194 .568504735352689749561, 195 .572919753562018740922, 196 .577315365035246941260, 197 .581691739635061821900, 198 .586049045003164792433, 199 .590387446602107957005, 200 .594707107746216934174, 201 .599008189645246602594, 202 .603290851438941899687, 203 .607555250224322662688, 204 .611801541106615331955, 205 .616029877215623855590, 206 .620240409751204424537, 207 .624433288012369303032, 208 .628608659422752680256, 209 .632766669570628437213, 210 .636907462236194987781, 211 .641031179420679109171, 212 .645137961373620782978, 213 .649227946625615004450, 214 .653301272011958644725, 215 .657358072709030238911, 216 .661398482245203922502, 217 .665422632544505177065, 218 .669430653942981734871, 219 .673422675212350441142, 220 .677398823590920073911, 221 .681359224807238206267, 222 .685304003098281100392, 223 .689233281238557538017, 224 .693147180560117703862 225 }; 226 227 static double logF_tail[N+1] = { 228 0., 229 -.00000000000000543229938420049, 230 .00000000000000172745674997061, 231 -.00000000000001323017818229233, 232 -.00000000000001154527628289872, 233 -.00000000000000466529469958300, 234 .00000000000005148849572685810, 235 -.00000000000002532168943117445, 236 -.00000000000005213620639136504, 237 -.00000000000001819506003016881, 238 .00000000000006329065958724544, 239 .00000000000008614512936087814, 240 -.00000000000007355770219435028, 241 .00000000000009638067658552277, 242 .00000000000007598636597194141, 243 .00000000000002579999128306990, 244 -.00000000000004654729747598444, 245 -.00000000000007556920687451336, 246 .00000000000010195735223708472, 247 -.00000000000017319034406422306, 248 -.00000000000007718001336828098, 249 .00000000000010980754099855238, 250 -.00000000000002047235780046195, 251 -.00000000000008372091099235912, 252 .00000000000014088127937111135, 253 .00000000000012869017157588257, 254 .00000000000017788850778198106, 255 .00000000000006440856150696891, 256 .00000000000016132822667240822, 257 -.00000000000007540916511956188, 258 -.00000000000000036507188831790, 259 .00000000000009120937249914984, 260 .00000000000018567570959796010, 261 -.00000000000003149265065191483, 262 -.00000000000009309459495196889, 263 .00000000000017914338601329117, 264 -.00000000000001302979717330866, 265 .00000000000023097385217586939, 266 .00000000000023999540484211737, 267 .00000000000015393776174455408, 268 -.00000000000036870428315837678, 269 .00000000000036920375082080089, 270 -.00000000000009383417223663699, 271 .00000000000009433398189512690, 272 .00000000000041481318704258568, 273 -.00000000000003792316480209314, 274 .00000000000008403156304792424, 275 -.00000000000034262934348285429, 276 .00000000000043712191957429145, 277 -.00000000000010475750058776541, 278 -.00000000000011118671389559323, 279 .00000000000037549577257259853, 280 .00000000000013912841212197565, 281 .00000000000010775743037572640, 282 .00000000000029391859187648000, 283 -.00000000000042790509060060774, 284 .00000000000022774076114039555, 285 .00000000000010849569622967912, 286 -.00000000000023073801945705758, 287 .00000000000015761203773969435, 288 .00000000000003345710269544082, 289 -.00000000000041525158063436123, 290 .00000000000032655698896907146, 291 -.00000000000044704265010452446, 292 .00000000000034527647952039772, 293 -.00000000000007048962392109746, 294 .00000000000011776978751369214, 295 -.00000000000010774341461609578, 296 .00000000000021863343293215910, 297 .00000000000024132639491333131, 298 .00000000000039057462209830700, 299 -.00000000000026570679203560751, 300 .00000000000037135141919592021, 301 -.00000000000017166921336082431, 302 -.00000000000028658285157914353, 303 -.00000000000023812542263446809, 304 .00000000000006576659768580062, 305 -.00000000000028210143846181267, 306 .00000000000010701931762114254, 307 .00000000000018119346366441110, 308 .00000000000009840465278232627, 309 -.00000000000033149150282752542, 310 -.00000000000018302857356041668, 311 -.00000000000016207400156744949, 312 .00000000000048303314949553201, 313 -.00000000000071560553172382115, 314 .00000000000088821239518571855, 315 -.00000000000030900580513238244, 316 -.00000000000061076551972851496, 317 .00000000000035659969663347830, 318 .00000000000035782396591276383, 319 -.00000000000046226087001544578, 320 .00000000000062279762917225156, 321 .00000000000072838947272065741, 322 .00000000000026809646615211673, 323 -.00000000000010960825046059278, 324 .00000000000002311949383800537, 325 -.00000000000058469058005299247, 326 -.00000000000002103748251144494, 327 -.00000000000023323182945587408, 328 -.00000000000042333694288141916, 329 -.00000000000043933937969737844, 330 .00000000000041341647073835565, 331 .00000000000006841763641591466, 332 .00000000000047585534004430641, 333 .00000000000083679678674757695, 334 -.00000000000085763734646658640, 335 .00000000000021913281229340092, 336 -.00000000000062242842536431148, 337 -.00000000000010983594325438430, 338 .00000000000065310431377633651, 339 -.00000000000047580199021710769, 340 -.00000000000037854251265457040, 341 .00000000000040939233218678664, 342 .00000000000087424383914858291, 343 .00000000000025218188456842882, 344 -.00000000000003608131360422557, 345 -.00000000000050518555924280902, 346 .00000000000078699403323355317, 347 -.00000000000067020876961949060, 348 .00000000000016108575753932458, 349 .00000000000058527188436251509, 350 -.00000000000035246757297904791, 351 -.00000000000018372084495629058, 352 .00000000000088606689813494916, 353 .00000000000066486268071468700, 354 .00000000000063831615170646519, 355 .00000000000025144230728376072, 356 -.00000000000017239444525614834 357 }; 358 359 #if 0 360 double 361 #ifdef _ANSI_SOURCE 362 log(double x) 363 #else 364 log(x) double x; 365 #endif 366 { 367 int m, j; 368 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0; 369 volatile double u1; 370 371 /* Catch special cases */ 372 if (x <= 0) 373 if (x == zero) /* log(0) = -Inf */ 374 return (-one/zero); 375 else /* log(neg) = NaN */ 376 return (zero/zero); 377 else if (!finite(x)) 378 return (x+x); /* x = NaN, Inf */ 379 380 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 381 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 382 383 m = logb(x); 384 g = ldexp(x, -m); 385 if (m == -1022) { 386 j = logb(g), m += j; 387 g = ldexp(g, -j); 388 } 389 j = N*(g-1) + .5; 390 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */ 391 f = g - F; 392 393 /* Approximate expansion for log(1+f/F) ~= u + q */ 394 g = 1/(2*F+f); 395 u = 2*f*g; 396 v = u*u; 397 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 398 399 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8, 400 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits. 401 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750 402 */ 403 if (m | j) 404 u1 = u + 513, u1 -= 513; 405 406 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero; 407 * u1 = u to 24 bits. 408 */ 409 else 410 u1 = u, TRUNC(u1); 411 u2 = (2.0*(f - F*u1) - u1*f) * g; 412 /* u1 + u2 = 2f/(2F+f) to extra precision. */ 413 414 /* log(x) = log(2^m*F*(1+f/F)) = */ 415 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */ 416 /* (exact) + (tiny) */ 417 418 u1 += m*logF_head[N] + logF_head[j]; /* exact */ 419 u2 = (u2 + logF_tail[j]) + q; /* tiny */ 420 u2 += logF_tail[N]*m; 421 return (u1 + u2); 422 } 423 #endif 424 425 /* 426 * Extra precision variant, returning struct {double a, b;}; 427 * log(x) = a+b to 63 bits, with a rounded to 26 bits. 428 */ 429 struct Double 430 #ifdef _ANSI_SOURCE 431 __log__D(double x) 432 #else 433 __log__D(x) double x; 434 #endif 435 { 436 int m, j; 437 double F, f, g, q, u, v, u2; 438 volatile double u1; 439 struct Double r; 440 441 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 442 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 443 444 m = logb(x); 445 g = ldexp(x, -m); 446 if (m == -1022) { 447 j = logb(g), m += j; 448 g = ldexp(g, -j); 449 } 450 j = N*(g-1) + .5; 451 F = (1.0/N) * j + 1; 452 f = g - F; 453 454 g = 1/(2*F+f); 455 u = 2*f*g; 456 v = u*u; 457 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 458 if (m | j) 459 u1 = u + 513, u1 -= 513; 460 else 461 u1 = u, TRUNC(u1); 462 u2 = (2.0*(f - F*u1) - u1*f) * g; 463 464 u1 += m*logF_head[N] + logF_head[j]; 465 466 u2 += logF_tail[j]; u2 += q; 467 u2 += logF_tail[N]*m; 468 r.a = u1 + u2; /* Only difference is here */ 469 TRUNC(r.a); 470 r.b = (u1 - r.a) + u2; 471 return (r); 472 } 473