2 * Copyright (c) 1992, 1993
3 * The Regents of the University of California. All rights reserved.
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6 * modification, are permitted provided that the following conditions
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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
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18 * may be used to endorse or promote products derived from this software
19 * without specific prior written permission.
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34 /* @(#)log.c 8.2 (Berkeley) 11/30/93 */
35 #include <sys/cdefs.h>
36 __FBSDID("$FreeBSD$");
42 /* Table-driven natural logarithm.
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).
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].
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.
57 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
58 * logF_hi[] + 512 is exact.
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
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)
72 * 0 return signalling -Inf
73 * neg return signalling NaN
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.
88 static double A1 = .08333333333333178827;
89 static double A2 = .01250000000377174923;
90 static double A3 = .002232139987919447809;
91 static double A4 = .0004348877777076145742;
93 static double logF_head[N+1] = {
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
225 static double logF_tail[N+1] = {
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
366 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0;
369 /* Catch special cases */
371 if (x == zero) /* log(0) = -Inf */
373 else /* log(neg) = NaN */
376 return (x+x); /* x = NaN, Inf */
378 /* Argument reduction: 1 <= g < 2; x/2^m = g; */
379 /* y = F*(1 + f/F) for |f| <= 2^-8 */
388 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */
391 /* Approximate expansion for log(1+f/F) ~= u + q */
395 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
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
402 u1 = u + 513, u1 -= 513;
404 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero;
409 u2 = (2.0*(f - F*u1) - u1*f) * g;
410 /* u1 + u2 = 2f/(2F+f) to extra precision. */
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) */
416 u1 += m*logF_head[N] + logF_head[j]; /* exact */
417 u2 = (u2 + logF_tail[j]) + q; /* tiny */
418 u2 += logF_tail[N]*m;
424 * Extra precision variant, returning struct {double a, b;};
425 * log(x) = a+b to 63 bits, with a rounded to 26 bits.
431 __log__D(x) double x;
435 double F, f, g, q, u, v, u2;
439 /* Argument reduction: 1 <= g < 2; x/2^m = g; */
440 /* y = F*(1 + f/F) for |f| <= 2^-8 */
455 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
457 u1 = u + 513, u1 -= 513;
460 u2 = (2.0*(f - F*u1) - u1*f) * g;
462 u1 += m*logF_head[N] + logF_head[j];
464 u2 += logF_tail[j]; u2 += q;
465 u2 += logF_tail[N]*m;
466 r.a = u1 + u2; /* Only difference is here */
468 r.b = (u1 - r.a) + u2;