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. Neither the name of the University nor the names of its contributors 14 * may be used to endorse or promote products derived from this software 15 * without specific prior written permission. 16 * 17 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 18 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 19 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 20 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 21 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 22 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 23 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 24 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 25 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 26 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 27 * SUCH DAMAGE. 28 */ 29 30 /* @(#)log.c 8.2 (Berkeley) 11/30/93 */ 31 #include "cdefs-compat.h" 32 //__FBSDID("$FreeBSD: src/lib/msun/bsdsrc/b_log.c,v 1.9 2008/02/22 02:26:51 das Exp $"); 33 34 #include <openlibm.h> 35 #include <errno.h> 36 37 #include "mathimpl.h" 38 39 /* Table-driven natural logarithm. 40 * 41 * This code was derived, with minor modifications, from: 42 * Peter Tang, "Table-Driven Implementation of the 43 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans. 44 * Math Software, vol 16. no 4, pp 378-400, Dec 1990). 45 * 46 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256, 47 * where F = j/128 for j an integer in [0, 128]. 48 * 49 * log(2^m) = log2_hi*m + log2_tail*m 50 * since m is an integer, the dominant term is exact. 51 * m has at most 10 digits (for subnormal numbers), 52 * and log2_hi has 11 trailing zero bits. 53 * 54 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h 55 * logF_hi[] + 512 is exact. 56 * 57 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ... 58 * the leading term is calculated to extra precision in two 59 * parts, the larger of which adds exactly to the dominant 60 * m and F terms. 61 * There are two cases: 62 * 1. when m, j are non-zero (m | j), use absolute 63 * precision for the leading term. 64 * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1). 65 * In this case, use a relative precision of 24 bits. 66 * (This is done differently in the original paper) 67 * 68 * Special cases: 69 * 0 return signalling -Inf 70 * neg return signalling NaN 71 * +Inf return +Inf 72 */ 73 74 #define N 128 75 76 /* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. 77 * Used for generation of extend precision logarithms. 78 * The constant 35184372088832 is 2^45, so the divide is exact. 79 * It ensures correct reading of logF_head, even for inaccurate 80 * decimal-to-binary conversion routines. (Everybody gets the 81 * right answer for integers less than 2^53.) 82 * Values for log(F) were generated using error < 10^-57 absolute 83 * with the bc -l package. 84 */ 85 static double A1 = .08333333333333178827; 86 static double A2 = .01250000000377174923; 87 static double A3 = .002232139987919447809; 88 static double A4 = .0004348877777076145742; 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 #if 0 355 DLLEXPORT double 356 #ifdef _ANSI_SOURCE 357 log(double x) 358 #else 359 log(x) double x; 360 #endif 361 { 362 int m, j; 363 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0; 364 volatile double u1; 365 366 /* Catch special cases */ 367 if (x <= 0) 368 if (x == zero) /* log(0) = -Inf */ 369 return (-one/zero); 370 else /* log(neg) = NaN */ 371 return (zero/zero); 372 else if (!finite(x)) 373 return (x+x); /* x = NaN, Inf */ 374 375 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 376 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 377 378 m = logb(x); 379 g = ldexp(x, -m); 380 if (m == -1022) { 381 j = logb(g), m += j; 382 g = ldexp(g, -j); 383 } 384 j = N*(g-1) + .5; 385 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */ 386 f = g - F; 387 388 /* Approximate expansion for log(1+f/F) ~= u + q */ 389 g = 1/(2*F+f); 390 u = 2*f*g; 391 v = u*u; 392 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 393 394 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8, 395 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits. 396 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750 397 */ 398 if (m | j) 399 u1 = u + 513, u1 -= 513; 400 401 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero; 402 * u1 = u to 24 bits. 403 */ 404 else 405 u1 = u, TRUNC(u1); 406 u2 = (2.0*(f - F*u1) - u1*f) * g; 407 /* u1 + u2 = 2f/(2F+f) to extra precision. */ 408 409 /* log(x) = log(2^m*F*(1+f/F)) = */ 410 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */ 411 /* (exact) + (tiny) */ 412 413 u1 += m*logF_head[N] + logF_head[j]; /* exact */ 414 u2 = (u2 + logF_tail[j]) + q; /* tiny */ 415 u2 += logF_tail[N]*m; 416 return (u1 + u2); 417 } 418 #endif 419 420 /* 421 * Extra precision variant, returning struct {double a, b;}; 422 * log(x) = a+b to 63 bits, with a rounded to 26 bits. 423 */ 424 struct Double 425 #ifdef _ANSI_SOURCE 426 __log__D(double x) 427 #else 428 __log__D(x) double x; 429 #endif 430 { 431 int m, j; 432 double F, f, g, q, u, v, u2; 433 volatile double u1; 434 struct Double r; 435 436 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 437 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 438 439 m = logb(x); 440 g = ldexp(x, -m); 441 if (m == -1022) { 442 j = logb(g), m += j; 443 g = ldexp(g, -j); 444 } 445 j = N*(g-1) + .5; 446 F = (1.0/N) * j + 1; 447 f = g - F; 448 449 g = 1/(2*F+f); 450 u = 2*f*g; 451 v = u*u; 452 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 453 if (m | j) 454 u1 = u + 513, u1 -= 513; 455 else 456 u1 = u, TRUNC(u1); 457 u2 = (2.0*(f - F*u1) - u1*f) * g; 458 459 u1 += m*logF_head[N] + logF_head[j]; 460 461 u2 += logF_tail[j]; u2 += q; 462 u2 += logF_tail[N]*m; 463 r.a = u1 + u2; /* Only difference is here */ 464 TRUNC(r.a); 465 r.b = (u1 - r.a) + u2; 466 return (r); 467 } 468