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1 | /* logll.c |
2 | * | |
3 | * Natural logarithm for 128-bit long double precision. | |
4 | * | |
5 | * | |
6 | * | |
7 | * SYNOPSIS: | |
8 | * | |
9 | * long double x, y, logl(); | |
10 | * | |
11 | * y = logl( x ); | |
12 | * | |
13 | * | |
14 | * | |
15 | * DESCRIPTION: | |
16 | * | |
17 | * Returns the base e (2.718...) logarithm of x. | |
18 | * | |
19 | * The argument is separated into its exponent and fractional | |
20 | * parts. Use of a lookup table increases the speed of the routine. | |
21 | * The program uses logarithms tabulated at intervals of 1/128 to | |
22 | * cover the domain from approximately 0.7 to 1.4. | |
23 | * | |
24 | * On the interval [-1/128, +1/128] the logarithm of 1+x is approximated by | |
25 | * log(1+x) = x - 0.5 x^2 + x^3 P(x) . | |
26 | * | |
27 | * | |
28 | * | |
29 | * ACCURACY: | |
30 | * | |
31 | * Relative error: | |
32 | * arithmetic domain # trials peak rms | |
33 | * IEEE 0.875, 1.125 100000 1.2e-34 4.1e-35 | |
34 | * IEEE 0.125, 8 100000 1.2e-34 4.1e-35 | |
35 | * | |
36 | * | |
37 | * WARNING: | |
38 | * | |
39 | * This program uses integer operations on bit fields of floating-point | |
40 | * numbers. It does not work with data structures other than the | |
41 | * structure assumed. | |
42 | * | |
43 | */ | |
44 | ||
45 | /* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov> | |
46 | ||
47 | This library is free software; you can redistribute it and/or | |
48 | modify it under the terms of the GNU Lesser General Public | |
49 | License as published by the Free Software Foundation; either | |
50 | version 2.1 of the License, or (at your option) any later version. | |
51 | ||
52 | This library is distributed in the hope that it will be useful, | |
53 | but WITHOUT ANY WARRANTY; without even the implied warranty of | |
54 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
55 | Lesser General Public License for more details. | |
56 | ||
57 | You should have received a copy of the GNU Lesser General Public | |
58 | License along with this library; if not, write to the Free Software | |
59 | Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ | |
60 | ||
61 | #include "math_private.h" | |
62 | ||
63 | /* log(1+x) = x - .5 x^2 + x^3 l(x) | |
64 | -.0078125 <= x <= +.0078125 | |
65 | peak relative error 1.2e-37 */ | |
66 | static const long double | |
67 | l3 = 3.333333333333333333333333333333336096926E-1L, | |
68 | l4 = -2.499999999999999999999999999486853077002E-1L, | |
69 | l5 = 1.999999999999999999999999998515277861905E-1L, | |
70 | l6 = -1.666666666666666666666798448356171665678E-1L, | |
71 | l7 = 1.428571428571428571428808945895490721564E-1L, | |
72 | l8 = -1.249999999999999987884655626377588149000E-1L, | |
73 | l9 = 1.111111111111111093947834982832456459186E-1L, | |
74 | l10 = -1.000000000000532974938900317952530453248E-1L, | |
75 | l11 = 9.090909090915566247008015301349979892689E-2L, | |
76 | l12 = -8.333333211818065121250921925397567745734E-2L, | |
77 | l13 = 7.692307559897661630807048686258659316091E-2L, | |
78 | l14 = -7.144242754190814657241902218399056829264E-2L, | |
79 | l15 = 6.668057591071739754844678883223432347481E-2L; | |
80 | ||
81 | /* Lookup table of ln(t) - (t-1) | |
82 | t = 0.5 + (k+26)/128) | |
83 | k = 0, ..., 91 */ | |
84 | static const long double logtbl[92] = { | |
85 | -5.5345593589352099112142921677820359632418E-2L, | |
86 | -5.2108257402767124761784665198737642086148E-2L, | |
87 | -4.8991686870576856279407775480686721935120E-2L, | |
88 | -4.5993270766361228596215288742353061431071E-2L, | |
89 | -4.3110481649613269682442058976885699556950E-2L, | |
90 | -4.0340872319076331310838085093194799765520E-2L, | |
91 | -3.7682072451780927439219005993827431503510E-2L, | |
92 | -3.5131785416234343803903228503274262719586E-2L, | |
93 | -3.2687785249045246292687241862699949178831E-2L, | |
94 | -3.0347913785027239068190798397055267411813E-2L, | |
95 | -2.8110077931525797884641940838507561326298E-2L, | |
96 | -2.5972247078357715036426583294246819637618E-2L, | |
97 | -2.3932450635346084858612873953407168217307E-2L, | |
98 | -2.1988775689981395152022535153795155900240E-2L, | |
99 | -2.0139364778244501615441044267387667496733E-2L, | |
100 | -1.8382413762093794819267536615342902718324E-2L, | |
101 | -1.6716169807550022358923589720001638093023E-2L, | |
102 | -1.5138929457710992616226033183958974965355E-2L, | |
103 | -1.3649036795397472900424896523305726435029E-2L, | |
104 | -1.2244881690473465543308397998034325468152E-2L, | |
105 | -1.0924898127200937840689817557742469105693E-2L, | |
106 | -9.6875626072830301572839422532631079809328E-3L, | |
107 | -8.5313926245226231463436209313499745894157E-3L, | |
108 | -7.4549452072765973384933565912143044991706E-3L, | |
109 | -6.4568155251217050991200599386801665681310E-3L, | |
110 | -5.5356355563671005131126851708522185605193E-3L, | |
111 | -4.6900728132525199028885749289712348829878E-3L, | |
112 | -3.9188291218610470766469347968659624282519E-3L, | |
113 | -3.2206394539524058873423550293617843896540E-3L, | |
114 | -2.5942708080877805657374888909297113032132E-3L, | |
115 | -2.0385211375711716729239156839929281289086E-3L, | |
116 | -1.5522183228760777967376942769773768850872E-3L, | |
117 | -1.1342191863606077520036253234446621373191E-3L, | |
118 | -7.8340854719967065861624024730268350459991E-4L, | |
119 | -4.9869831458030115699628274852562992756174E-4L, | |
120 | -2.7902661731604211834685052867305795169688E-4L, | |
121 | -1.2335696813916860754951146082826952093496E-4L, | |
122 | -3.0677461025892873184042490943581654591817E-5L, | |
123 | #define ZERO logtbl[38] | |
124 | 0.0000000000000000000000000000000000000000E0L, | |
125 | -3.0359557945051052537099938863236321874198E-5L, | |
126 | -1.2081346403474584914595395755316412213151E-4L, | |
127 | -2.7044071846562177120083903771008342059094E-4L, | |
128 | -4.7834133324631162897179240322783590830326E-4L, | |
129 | -7.4363569786340080624467487620270965403695E-4L, | |
130 | -1.0654639687057968333207323853366578860679E-3L, | |
131 | -1.4429854811877171341298062134712230604279E-3L, | |
132 | -1.8753781835651574193938679595797367137975E-3L, | |
133 | -2.3618380914922506054347222273705859653658E-3L, | |
134 | -2.9015787624124743013946600163375853631299E-3L, | |
135 | -3.4938307889254087318399313316921940859043E-3L, | |
136 | -4.1378413103128673800485306215154712148146E-3L, | |
137 | -4.8328735414488877044289435125365629849599E-3L, | |
138 | -5.5782063183564351739381962360253116934243E-3L, | |
139 | -6.3731336597098858051938306767880719015261E-3L, | |
140 | -7.2169643436165454612058905294782949315193E-3L, | |
141 | -8.1090214990427641365934846191367315083867E-3L, | |
142 | -9.0486422112807274112838713105168375482480E-3L, | |
143 | -1.0035177140880864314674126398350812606841E-2L, | |
144 | -1.1067990155502102718064936259435676477423E-2L, | |
145 | -1.2146457974158024928196575103115488672416E-2L, | |
146 | -1.3269969823361415906628825374158424754308E-2L, | |
147 | -1.4437927104692837124388550722759686270765E-2L, | |
148 | -1.5649743073340777659901053944852735064621E-2L, | |
149 | -1.6904842527181702880599758489058031645317E-2L, | |
150 | -1.8202661505988007336096407340750378994209E-2L, | |
151 | -1.9542647000370545390701192438691126552961E-2L, | |
152 | -2.0924256670080119637427928803038530924742E-2L, | |
153 | -2.2346958571309108496179613803760727786257E-2L, | |
154 | -2.3810230892650362330447187267648486279460E-2L, | |
155 | -2.5313561699385640380910474255652501521033E-2L, | |
156 | -2.6856448685790244233704909690165496625399E-2L, | |
157 | -2.8438398935154170008519274953860128449036E-2L, | |
158 | -3.0058928687233090922411781058956589863039E-2L, | |
159 | -3.1717563112854831855692484086486099896614E-2L, | |
160 | -3.3413836095418743219397234253475252001090E-2L, | |
161 | -3.5147290019036555862676702093393332533702E-2L, | |
162 | -3.6917475563073933027920505457688955423688E-2L, | |
163 | -3.8723951502862058660874073462456610731178E-2L, | |
164 | -4.0566284516358241168330505467000838017425E-2L, | |
165 | -4.2444048996543693813649967076598766917965E-2L, | |
166 | -4.4356826869355401653098777649745233339196E-2L, | |
167 | -4.6304207416957323121106944474331029996141E-2L, | |
168 | -4.8285787106164123613318093945035804818364E-2L, | |
169 | -5.0301169421838218987124461766244507342648E-2L, | |
170 | -5.2349964705088137924875459464622098310997E-2L, | |
171 | -5.4431789996103111613753440311680967840214E-2L, | |
172 | -5.6546268881465384189752786409400404404794E-2L, | |
173 | -5.8693031345788023909329239565012647817664E-2L, | |
174 | -6.0871713627532018185577188079210189048340E-2L, | |
175 | -6.3081958078862169742820420185833800925568E-2L, | |
176 | -6.5323413029406789694910800219643791556918E-2L, | |
177 | -6.7595732653791419081537811574227049288168E-2L | |
178 | }; | |
179 | ||
180 | /* ln(2) = ln2a + ln2b with extended precision. */ | |
181 | static const long double | |
182 | ln2a = 6.93145751953125e-1L, | |
183 | ln2b = 1.4286068203094172321214581765680755001344E-6L; | |
184 | ||
185 | long double | |
186 | __ieee754_logl(long double x) | |
187 | { | |
188 | long double z, y, w; | |
189 | ieee854_long_double_shape_type u, t; | |
190 | unsigned int m; | |
191 | int k, e; | |
192 | ||
193 | u.value = x; | |
194 | m = u.parts32.w0; | |
195 | ||
196 | /* Check for IEEE special cases. */ | |
197 | k = m & 0x7fffffff; | |
198 | /* log(0) = -infinity. */ | |
199 | if ((k | u.parts32.w1 | (u.parts32.w2 & 0x7fffffff) | u.parts32.w3) == 0) | |
200 | { | |
201 | return -0.5L / ZERO; | |
202 | } | |
203 | /* log ( x < 0 ) = NaN */ | |
204 | if (m & 0x80000000) | |
205 | { | |
206 | return (x - x) / ZERO; | |
207 | } | |
208 | /* log (infinity or NaN) */ | |
209 | if (k >= 0x7ff00000) | |
210 | { | |
211 | return x + x; | |
212 | } | |
213 | ||
214 | /* On this interval the table is not used due to cancellation error. */ | |
215 | if ((x <= 1.0078125L) && (x >= 0.9921875L)) | |
216 | { | |
217 | z = x - 1.0L; | |
218 | k = 64; | |
219 | t.value = 1.0L; | |
220 | e = 0; | |
221 | } | |
222 | else | |
223 | { | |
224 | /* Extract exponent and reduce domain to 0.703125 <= u < 1.40625 */ | |
225 | unsigned int w0; | |
226 | e = (int) (m >> 20) - (int) 0x3fe; | |
227 | m &= 0xfffff; | |
228 | w0 = m | 0x3fe00000; | |
229 | m |= 0x100000; | |
230 | /* Find lookup table index k from high order bits of the significand. */ | |
231 | if (m < 0x168000) | |
232 | { | |
233 | k = (m - 0xff000) >> 13; | |
234 | /* t is the argument 0.5 + (k+26)/128 | |
235 | of the nearest item to u in the lookup table. */ | |
236 | t.parts32.w0 = 0x3ff00000 + (k << 13); | |
237 | t.parts32.w1 = 0; | |
238 | t.parts32.w2 = 0; | |
239 | t.parts32.w3 = 0; | |
240 | w0 += 0x100000; | |
241 | e -= 1; | |
242 | k += 64; | |
243 | } | |
244 | else | |
245 | { | |
246 | k = (m - 0xfe000) >> 14; | |
247 | t.parts32.w0 = 0x3fe00000 + (k << 14); | |
248 | t.parts32.w1 = 0; | |
249 | t.parts32.w2 = 0; | |
250 | t.parts32.w3 = 0; | |
251 | } | |
252 | u.value = __scalbnl (u.value, ((int) ((w0 - u.parts32.w0) * 2)) >> 21); | |
253 | /* log(u) = log( t u/t ) = log(t) + log(u/t) | |
254 | log(t) is tabulated in the lookup table. | |
255 | Express log(u/t) = log(1+z), where z = u/t - 1 = (u-t)/t. | |
256 | cf. Cody & Waite. */ | |
257 | z = (u.value - t.value) / t.value; | |
258 | } | |
259 | /* Series expansion of log(1+z). */ | |
260 | w = z * z; | |
261 | y = ((((((((((((l15 * z | |
262 | + l14) * z | |
263 | + l13) * z | |
264 | + l12) * z | |
265 | + l11) * z | |
266 | + l10) * z | |
267 | + l9) * z | |
268 | + l8) * z | |
269 | + l7) * z | |
270 | + l6) * z | |
271 | + l5) * z | |
272 | + l4) * z | |
273 | + l3) * z * w; | |
274 | y -= 0.5 * w; | |
275 | y += e * ln2b; /* Base 2 exponent offset times ln(2). */ | |
276 | y += z; | |
277 | y += logtbl[k-26]; /* log(t) - (t-1) */ | |
278 | y += (t.value - 1.0L); | |
279 | y += e * ln2a; | |
280 | return y; | |
281 | } |