1/* 2 * Copyright (c) 1992, 1993 3 * The Regents of the University of California. All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 1. Redistributions of source code must retain the above copyright 9 * notice, this list of conditions and the following disclaimer. 10 * 2. Redistributions in binary form must reproduce the above copyright 11 * notice, this list of conditions and the following disclaimer in the 12 * documentation and/or other materials provided with the distribution. 13 * 3. All advertising materials mentioning features or use of this software 14 * must display the following acknowledgement: 15 * This product includes software developed by the University of 16 * California, Berkeley and its contributors. 17 * 4. Neither the name of the University nor the names of its contributors 18 * may be used to endorse or promote products derived from this software 19 * without specific prior written permission. 20 * 21 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 22 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 23 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 24 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 25 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 26 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 27 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 28 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 29 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 31 * SUCH DAMAGE. 32 */ 33 34/* @(#)log.c 8.2 (Berkeley) 11/30/93 */ 35#include <sys/cdefs.h> 36__FBSDID("$FreeBSD: stable/10/lib/msun/bsdsrc/b_log.c 326747 2017-12-10 14:49:06Z dim $"); 37 38#include <math.h> 39 40#include "mathimpl.h" 41 42/* Table-driven natural logarithm. 43 * 44 * This code was derived, with minor modifications, from: 45 * Peter Tang, "Table-Driven Implementation of the 46 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans. 47 * Math Software, vol 16. no 4, pp 378-400, Dec 1990). 48 * 49 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256, 50 * where F = j/128 for j an integer in [0, 128]. 51 * 52 * log(2^m) = log2_hi*m + log2_tail*m 53 * since m is an integer, the dominant term is exact. 54 * m has at most 10 digits (for subnormal numbers), 55 * and log2_hi has 11 trailing zero bits. 56 * 57 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h 58 * logF_hi[] + 512 is exact. 59 * 60 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ... 61 * the leading term is calculated to extra precision in two 62 * parts, the larger of which adds exactly to the dominant 63 * m and F terms. 64 * There are two cases: 65 * 1. when m, j are non-zero (m | j), use absolute 66 * precision for the leading term. 67 * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1). 68 * In this case, use a relative precision of 24 bits. 69 * (This is done differently in the original paper) 70 * 71 * Special cases: 72 * 0 return signalling -Inf 73 * neg return signalling NaN 74 * +Inf return +Inf 75*/ 76 77#define N 128 78 79/* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. 80 * Used for generation of extend precision logarithms. 81 * The constant 35184372088832 is 2^45, so the divide is exact. 82 * It ensures correct reading of logF_head, even for inaccurate 83 * decimal-to-binary conversion routines. (Everybody gets the 84 * right answer for integers less than 2^53.) 85 * Values for log(F) were generated using error < 10^-57 absolute 86 * with the bc -l package. 87*/ 88static double A1 = .08333333333333178827; 89static double A2 = .01250000000377174923; 90static double A3 = .002232139987919447809; 91static double A4 = .0004348877777076145742; 92 93static double logF_head[N+1] = { 94 0., 95 .007782140442060381246, 96 .015504186535963526694, 97 .023167059281547608406, 98 .030771658666765233647, 99 .038318864302141264488, 100 .045809536031242714670, 101 .053244514518837604555, 102 .060624621816486978786, 103 .067950661908525944454, 104 .075223421237524235039, 105 .082443669210988446138, 106 .089612158689760690322, 107 .096729626458454731618, 108 .103796793681567578460, 109 .110814366340264314203, 110 .117783035656430001836, 111 .124703478501032805070, 112 .131576357788617315236, 113 .138402322859292326029, 114 .145182009844575077295, 115 .151916042025732167530, 116 .158605030176659056451, 117 .165249572895390883786, 118 .171850256926518341060, 119 .178407657472689606947, 120 .184922338493834104156, 121 .191394852999565046047, 122 .197825743329758552135, 123 .204215541428766300668, 124 .210564769107350002741, 125 .216873938300523150246, 126 .223143551314024080056, 127 .229374101064877322642, 128 .235566071312860003672, 129 .241719936886966024758, 130 .247836163904594286577, 131 .253915209980732470285, 132 .259957524436686071567, 133 .265963548496984003577, 134 .271933715484010463114, 135 .277868451003087102435, 136 .283768173130738432519, 137 .289633292582948342896, 138 .295464212893421063199, 139 .301261330578199704177, 140 .307025035294827830512, 141 .312755710004239517729, 142 .318453731118097493890, 143 .324119468654316733591, 144 .329753286372579168528, 145 .335355541920762334484, 146 .340926586970454081892, 147 .346466767346100823488, 148 .351976423156884266063, 149 .357455888922231679316, 150 .362905493689140712376, 151 .368325561158599157352, 152 .373716409793814818840, 153 .379078352934811846353, 154 .384411698910298582632, 155 .389716751140440464951, 156 .394993808240542421117, 157 .400243164127459749579, 158 .405465108107819105498, 159 .410659924985338875558, 160 .415827895143593195825, 161 .420969294644237379543, 162 .426084395310681429691, 163 .431173464818130014464, 164 .436236766774527495726, 165 .441274560805140936281, 166 .446287102628048160113, 167 .451274644139630254358, 168 .456237433481874177232, 169 .461175715122408291790, 170 .466089729924533457960, 171 .470979715219073113985, 172 .475845904869856894947, 173 .480688529345570714212, 174 .485507815781602403149, 175 .490303988045525329653, 176 .495077266798034543171, 177 .499827869556611403822, 178 .504556010751912253908, 179 .509261901790523552335, 180 .513945751101346104405, 181 .518607764208354637958, 182 .523248143765158602036, 183 .527867089620485785417, 184 .532464798869114019908, 185 .537041465897345915436, 186 .541597282432121573947, 187 .546132437597407260909, 188 .550647117952394182793, 189 .555141507540611200965, 190 .559615787935399566777, 191 .564070138285387656651, 192 .568504735352689749561, 193 .572919753562018740922, 194 .577315365035246941260, 195 .581691739635061821900, 196 .586049045003164792433, 197 .590387446602107957005, 198 .594707107746216934174, 199 .599008189645246602594, 200 .603290851438941899687, 201 .607555250224322662688, 202 .611801541106615331955, 203 .616029877215623855590, 204 .620240409751204424537, 205 .624433288012369303032, 206 .628608659422752680256, 207 .632766669570628437213, 208 .636907462236194987781, 209 .641031179420679109171, 210 .645137961373620782978, 211 .649227946625615004450, 212 .653301272011958644725, 213 .657358072709030238911, 214 .661398482245203922502, 215 .665422632544505177065, 216 .669430653942981734871, 217 .673422675212350441142, 218 .677398823590920073911, 219 .681359224807238206267, 220 .685304003098281100392, 221 .689233281238557538017, 222 .693147180560117703862 223}; 224 225static double logF_tail[N+1] = { 226 0., 227 -.00000000000000543229938420049, 228 .00000000000000172745674997061, 229 -.00000000000001323017818229233, 230 -.00000000000001154527628289872, 231 -.00000000000000466529469958300, 232 .00000000000005148849572685810, 233 -.00000000000002532168943117445, 234 -.00000000000005213620639136504, 235 -.00000000000001819506003016881, 236 .00000000000006329065958724544, 237 .00000000000008614512936087814, 238 -.00000000000007355770219435028, 239 .00000000000009638067658552277, 240 .00000000000007598636597194141, 241 .00000000000002579999128306990, 242 -.00000000000004654729747598444, 243 -.00000000000007556920687451336, 244 .00000000000010195735223708472, 245 -.00000000000017319034406422306, 246 -.00000000000007718001336828098, 247 .00000000000010980754099855238, 248 -.00000000000002047235780046195, 249 -.00000000000008372091099235912, 250 .00000000000014088127937111135, 251 .00000000000012869017157588257, 252 .00000000000017788850778198106, 253 .00000000000006440856150696891, 254 .00000000000016132822667240822, 255 -.00000000000007540916511956188, 256 -.00000000000000036507188831790, 257 .00000000000009120937249914984, 258 .00000000000018567570959796010, 259 -.00000000000003149265065191483, 260 -.00000000000009309459495196889, 261 .00000000000017914338601329117, 262 -.00000000000001302979717330866, 263 .00000000000023097385217586939, 264 .00000000000023999540484211737, 265 .00000000000015393776174455408, 266 -.00000000000036870428315837678, 267 .00000000000036920375082080089, 268 -.00000000000009383417223663699, 269 .00000000000009433398189512690, 270 .00000000000041481318704258568, 271 -.00000000000003792316480209314, 272 .00000000000008403156304792424, 273 -.00000000000034262934348285429, 274 .00000000000043712191957429145, 275 -.00000000000010475750058776541, 276 -.00000000000011118671389559323, 277 .00000000000037549577257259853, 278 .00000000000013912841212197565, 279 .00000000000010775743037572640, 280 .00000000000029391859187648000, 281 -.00000000000042790509060060774, 282 .00000000000022774076114039555, 283 .00000000000010849569622967912, 284 -.00000000000023073801945705758, 285 .00000000000015761203773969435, 286 .00000000000003345710269544082, 287 -.00000000000041525158063436123, 288 .00000000000032655698896907146, 289 -.00000000000044704265010452446, 290 .00000000000034527647952039772, 291 -.00000000000007048962392109746, 292 .00000000000011776978751369214, 293 -.00000000000010774341461609578, 294 .00000000000021863343293215910, 295 .00000000000024132639491333131, 296 .00000000000039057462209830700, 297 -.00000000000026570679203560751, 298 .00000000000037135141919592021, 299 -.00000000000017166921336082431, 300 -.00000000000028658285157914353, 301 -.00000000000023812542263446809, 302 .00000000000006576659768580062, 303 -.00000000000028210143846181267, 304 .00000000000010701931762114254, 305 .00000000000018119346366441110, 306 .00000000000009840465278232627, 307 -.00000000000033149150282752542, 308 -.00000000000018302857356041668, 309 -.00000000000016207400156744949, 310 .00000000000048303314949553201, 311 -.00000000000071560553172382115, 312 .00000000000088821239518571855, 313 -.00000000000030900580513238244, 314 -.00000000000061076551972851496, 315 .00000000000035659969663347830, 316 .00000000000035782396591276383, 317 -.00000000000046226087001544578, 318 .00000000000062279762917225156, 319 .00000000000072838947272065741, 320 .00000000000026809646615211673, 321 -.00000000000010960825046059278, 322 .00000000000002311949383800537, 323 -.00000000000058469058005299247, 324 -.00000000000002103748251144494, 325 -.00000000000023323182945587408, 326 -.00000000000042333694288141916, 327 -.00000000000043933937969737844, 328 .00000000000041341647073835565, 329 .00000000000006841763641591466, 330 .00000000000047585534004430641, 331 .00000000000083679678674757695, 332 -.00000000000085763734646658640, 333 .00000000000021913281229340092, 334 -.00000000000062242842536431148, 335 -.00000000000010983594325438430, 336 .00000000000065310431377633651, 337 -.00000000000047580199021710769, 338 -.00000000000037854251265457040, 339 .00000000000040939233218678664, 340 .00000000000087424383914858291, 341 .00000000000025218188456842882, 342 -.00000000000003608131360422557, 343 -.00000000000050518555924280902, 344 .00000000000078699403323355317, 345 -.00000000000067020876961949060, 346 .00000000000016108575753932458, 347 .00000000000058527188436251509, 348 -.00000000000035246757297904791, 349 -.00000000000018372084495629058, 350 .00000000000088606689813494916, 351 .00000000000066486268071468700, 352 .00000000000063831615170646519, 353 .00000000000025144230728376072, 354 -.00000000000017239444525614834 355}; 356 357#if 0 358double 359#ifdef _ANSI_SOURCE 360log(double x) 361#else 362log(x) double x; 363#endif 364{ 365 int m, j; 366 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0; 367 volatile double u1; 368 369 /* Catch special cases */ 370 if (x <= 0) 371 if (x == zero) /* log(0) = -Inf */ 372 return (-one/zero); 373 else /* log(neg) = NaN */ 374 return (zero/zero); 375 else if (!finite(x)) 376 return (x+x); /* x = NaN, Inf */ 377 378 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 379 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 380 381 m = logb(x); 382 g = ldexp(x, -m); 383 if (m == -1022) { 384 j = logb(g), m += j; 385 g = ldexp(g, -j); 386 } 387 j = N*(g-1) + .5; 388 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */ 389 f = g - F; 390 391 /* Approximate expansion for log(1+f/F) ~= u + q */ 392 g = 1/(2*F+f); 393 u = 2*f*g; 394 v = u*u; 395 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 396 397 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8, 398 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits. 399 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750 400 */ 401 if (m | j) 402 u1 = u + 513, u1 -= 513; 403 404 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero; 405 * u1 = u to 24 bits. 406 */ 407 else 408 u1 = u, TRUNC(u1); 409 u2 = (2.0*(f - F*u1) - u1*f) * g; 410 /* u1 + u2 = 2f/(2F+f) to extra precision. */ 411 412 /* log(x) = log(2^m*F*(1+f/F)) = */ 413 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */ 414 /* (exact) + (tiny) */ 415 416 u1 += m*logF_head[N] + logF_head[j]; /* exact */ 417 u2 = (u2 + logF_tail[j]) + q; /* tiny */ 418 u2 += logF_tail[N]*m; 419 return (u1 + u2); 420} 421#endif 422 423/* 424 * Extra precision variant, returning struct {double a, b;}; 425 * log(x) = a+b to 63 bits, with a rounded to 26 bits. 426 */ 427struct Double 428#ifdef _ANSI_SOURCE 429__log__D(double x) 430#else 431__log__D(x) double x; 432#endif 433{ 434 int m, j; 435 double F, f, g, q, u, v, u2; 436 volatile double u1; 437 struct Double r; 438 439 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 440 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 441 442 m = logb(x); 443 g = ldexp(x, -m); 444 if (m == -1022) { 445 j = logb(g), m += j; 446 g = ldexp(g, -j); 447 } 448 j = N*(g-1) + .5; 449 F = (1.0/N) * j + 1; 450 f = g - F; 451 452 g = 1/(2*F+f); 453 u = 2*f*g; 454 v = u*u; 455 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 456 if (m | j) 457 u1 = u + 513, u1 -= 513; 458 else 459 u1 = u, TRUNC(u1); 460 u2 = (2.0*(f - F*u1) - u1*f) * g; 461 462 u1 += m*logF_head[N] + logF_head[j]; 463 464 u2 += logF_tail[j]; u2 += q; 465 u2 += logF_tail[N]*m; 466 r.a = u1 + u2; /* Only difference is here */ 467 TRUNC(r.a); 468 r.b = (u1 - r.a) + u2; 469 return (r); 470} 471