Mercurial > hg > CbC > CbC_gcc
comparison libquadmath/math/logq.c @ 68:561a7518be6b
update gcc-4.6
author | Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp> |
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date | Sun, 21 Aug 2011 07:07:55 +0900 |
parents | |
children | 04ced10e8804 |
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67:f6334be47118 | 68:561a7518be6b |
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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 "quadmath-imp.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 __float128 | |
67 l3 = 3.333333333333333333333333333333336096926E-1Q, | |
68 l4 = -2.499999999999999999999999999486853077002E-1Q, | |
69 l5 = 1.999999999999999999999999998515277861905E-1Q, | |
70 l6 = -1.666666666666666666666798448356171665678E-1Q, | |
71 l7 = 1.428571428571428571428808945895490721564E-1Q, | |
72 l8 = -1.249999999999999987884655626377588149000E-1Q, | |
73 l9 = 1.111111111111111093947834982832456459186E-1Q, | |
74 l10 = -1.000000000000532974938900317952530453248E-1Q, | |
75 l11 = 9.090909090915566247008015301349979892689E-2Q, | |
76 l12 = -8.333333211818065121250921925397567745734E-2Q, | |
77 l13 = 7.692307559897661630807048686258659316091E-2Q, | |
78 l14 = -7.144242754190814657241902218399056829264E-2Q, | |
79 l15 = 6.668057591071739754844678883223432347481E-2Q; | |
80 | |
81 /* Lookup table of ln(t) - (t-1) | |
82 t = 0.5 + (k+26)/128) | |
83 k = 0, ..., 91 */ | |
84 static const __float128 logtbl[92] = { | |
85 -5.5345593589352099112142921677820359632418E-2Q, | |
86 -5.2108257402767124761784665198737642086148E-2Q, | |
87 -4.8991686870576856279407775480686721935120E-2Q, | |
88 -4.5993270766361228596215288742353061431071E-2Q, | |
89 -4.3110481649613269682442058976885699556950E-2Q, | |
90 -4.0340872319076331310838085093194799765520E-2Q, | |
91 -3.7682072451780927439219005993827431503510E-2Q, | |
92 -3.5131785416234343803903228503274262719586E-2Q, | |
93 -3.2687785249045246292687241862699949178831E-2Q, | |
94 -3.0347913785027239068190798397055267411813E-2Q, | |
95 -2.8110077931525797884641940838507561326298E-2Q, | |
96 -2.5972247078357715036426583294246819637618E-2Q, | |
97 -2.3932450635346084858612873953407168217307E-2Q, | |
98 -2.1988775689981395152022535153795155900240E-2Q, | |
99 -2.0139364778244501615441044267387667496733E-2Q, | |
100 -1.8382413762093794819267536615342902718324E-2Q, | |
101 -1.6716169807550022358923589720001638093023E-2Q, | |
102 -1.5138929457710992616226033183958974965355E-2Q, | |
103 -1.3649036795397472900424896523305726435029E-2Q, | |
104 -1.2244881690473465543308397998034325468152E-2Q, | |
105 -1.0924898127200937840689817557742469105693E-2Q, | |
106 -9.6875626072830301572839422532631079809328E-3Q, | |
107 -8.5313926245226231463436209313499745894157E-3Q, | |
108 -7.4549452072765973384933565912143044991706E-3Q, | |
109 -6.4568155251217050991200599386801665681310E-3Q, | |
110 -5.5356355563671005131126851708522185605193E-3Q, | |
111 -4.6900728132525199028885749289712348829878E-3Q, | |
112 -3.9188291218610470766469347968659624282519E-3Q, | |
113 -3.2206394539524058873423550293617843896540E-3Q, | |
114 -2.5942708080877805657374888909297113032132E-3Q, | |
115 -2.0385211375711716729239156839929281289086E-3Q, | |
116 -1.5522183228760777967376942769773768850872E-3Q, | |
117 -1.1342191863606077520036253234446621373191E-3Q, | |
118 -7.8340854719967065861624024730268350459991E-4Q, | |
119 -4.9869831458030115699628274852562992756174E-4Q, | |
120 -2.7902661731604211834685052867305795169688E-4Q, | |
121 -1.2335696813916860754951146082826952093496E-4Q, | |
122 -3.0677461025892873184042490943581654591817E-5Q, | |
123 #define ZERO logtbl[38] | |
124 0.0000000000000000000000000000000000000000E0Q, | |
125 -3.0359557945051052537099938863236321874198E-5Q, | |
126 -1.2081346403474584914595395755316412213151E-4Q, | |
127 -2.7044071846562177120083903771008342059094E-4Q, | |
128 -4.7834133324631162897179240322783590830326E-4Q, | |
129 -7.4363569786340080624467487620270965403695E-4Q, | |
130 -1.0654639687057968333207323853366578860679E-3Q, | |
131 -1.4429854811877171341298062134712230604279E-3Q, | |
132 -1.8753781835651574193938679595797367137975E-3Q, | |
133 -2.3618380914922506054347222273705859653658E-3Q, | |
134 -2.9015787624124743013946600163375853631299E-3Q, | |
135 -3.4938307889254087318399313316921940859043E-3Q, | |
136 -4.1378413103128673800485306215154712148146E-3Q, | |
137 -4.8328735414488877044289435125365629849599E-3Q, | |
138 -5.5782063183564351739381962360253116934243E-3Q, | |
139 -6.3731336597098858051938306767880719015261E-3Q, | |
140 -7.2169643436165454612058905294782949315193E-3Q, | |
141 -8.1090214990427641365934846191367315083867E-3Q, | |
142 -9.0486422112807274112838713105168375482480E-3Q, | |
143 -1.0035177140880864314674126398350812606841E-2Q, | |
144 -1.1067990155502102718064936259435676477423E-2Q, | |
145 -1.2146457974158024928196575103115488672416E-2Q, | |
146 -1.3269969823361415906628825374158424754308E-2Q, | |
147 -1.4437927104692837124388550722759686270765E-2Q, | |
148 -1.5649743073340777659901053944852735064621E-2Q, | |
149 -1.6904842527181702880599758489058031645317E-2Q, | |
150 -1.8202661505988007336096407340750378994209E-2Q, | |
151 -1.9542647000370545390701192438691126552961E-2Q, | |
152 -2.0924256670080119637427928803038530924742E-2Q, | |
153 -2.2346958571309108496179613803760727786257E-2Q, | |
154 -2.3810230892650362330447187267648486279460E-2Q, | |
155 -2.5313561699385640380910474255652501521033E-2Q, | |
156 -2.6856448685790244233704909690165496625399E-2Q, | |
157 -2.8438398935154170008519274953860128449036E-2Q, | |
158 -3.0058928687233090922411781058956589863039E-2Q, | |
159 -3.1717563112854831855692484086486099896614E-2Q, | |
160 -3.3413836095418743219397234253475252001090E-2Q, | |
161 -3.5147290019036555862676702093393332533702E-2Q, | |
162 -3.6917475563073933027920505457688955423688E-2Q, | |
163 -3.8723951502862058660874073462456610731178E-2Q, | |
164 -4.0566284516358241168330505467000838017425E-2Q, | |
165 -4.2444048996543693813649967076598766917965E-2Q, | |
166 -4.4356826869355401653098777649745233339196E-2Q, | |
167 -4.6304207416957323121106944474331029996141E-2Q, | |
168 -4.8285787106164123613318093945035804818364E-2Q, | |
169 -5.0301169421838218987124461766244507342648E-2Q, | |
170 -5.2349964705088137924875459464622098310997E-2Q, | |
171 -5.4431789996103111613753440311680967840214E-2Q, | |
172 -5.6546268881465384189752786409400404404794E-2Q, | |
173 -5.8693031345788023909329239565012647817664E-2Q, | |
174 -6.0871713627532018185577188079210189048340E-2Q, | |
175 -6.3081958078862169742820420185833800925568E-2Q, | |
176 -6.5323413029406789694910800219643791556918E-2Q, | |
177 -6.7595732653791419081537811574227049288168E-2Q | |
178 }; | |
179 | |
180 /* ln(2) = ln2a + ln2b with extended precision. */ | |
181 static const __float128 | |
182 ln2a = 6.93145751953125e-1Q, | |
183 ln2b = 1.4286068203094172321214581765680755001344E-6Q; | |
184 | |
185 __float128 | |
186 logq (__float128 x) | |
187 { | |
188 __float128 z, y, w; | |
189 ieee854_float128 u, t; | |
190 unsigned int m; | |
191 int k, e; | |
192 | |
193 u.value = x; | |
194 m = u.words32.w0; | |
195 | |
196 /* Check for IEEE special cases. */ | |
197 k = m & 0x7fffffff; | |
198 /* log(0) = -infinity. */ | |
199 if ((k | u.words32.w1 | u.words32.w2 | u.words32.w3) == 0) | |
200 { | |
201 return -0.5Q / 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 >= 0x7fff0000) | |
210 { | |
211 return x + x; | |
212 } | |
213 | |
214 /* Extract exponent and reduce domain to 0.703125 <= u < 1.40625 */ | |
215 e = (int) (m >> 16) - (int) 0x3ffe; | |
216 m &= 0xffff; | |
217 u.words32.w0 = m | 0x3ffe0000; | |
218 m |= 0x10000; | |
219 /* Find lookup table index k from high order bits of the significand. */ | |
220 if (m < 0x16800) | |
221 { | |
222 k = (m - 0xff00) >> 9; | |
223 /* t is the argument 0.5 + (k+26)/128 | |
224 of the nearest item to u in the lookup table. */ | |
225 t.words32.w0 = 0x3fff0000 + (k << 9); | |
226 t.words32.w1 = 0; | |
227 t.words32.w2 = 0; | |
228 t.words32.w3 = 0; | |
229 u.words32.w0 += 0x10000; | |
230 e -= 1; | |
231 k += 64; | |
232 } | |
233 else | |
234 { | |
235 k = (m - 0xfe00) >> 10; | |
236 t.words32.w0 = 0x3ffe0000 + (k << 10); | |
237 t.words32.w1 = 0; | |
238 t.words32.w2 = 0; | |
239 t.words32.w3 = 0; | |
240 } | |
241 /* On this interval the table is not used due to cancellation error. */ | |
242 if ((x <= 1.0078125Q) && (x >= 0.9921875Q)) | |
243 { | |
244 z = x - 1.0Q; | |
245 k = 64; | |
246 t.value = 1.0Q; | |
247 e = 0; | |
248 } | |
249 else | |
250 { | |
251 /* log(u) = log( t u/t ) = log(t) + log(u/t) | |
252 log(t) is tabulated in the lookup table. | |
253 Express log(u/t) = log(1+z), where z = u/t - 1 = (u-t)/t. | |
254 cf. Cody & Waite. */ | |
255 z = (u.value - t.value) / t.value; | |
256 } | |
257 /* Series expansion of log(1+z). */ | |
258 w = z * z; | |
259 y = ((((((((((((l15 * z | |
260 + l14) * z | |
261 + l13) * z | |
262 + l12) * z | |
263 + l11) * z | |
264 + l10) * z | |
265 + l9) * z | |
266 + l8) * z | |
267 + l7) * z | |
268 + l6) * z | |
269 + l5) * z | |
270 + l4) * z | |
271 + l3) * z * w; | |
272 y -= 0.5 * w; | |
273 y += e * ln2b; /* Base 2 exponent offset times ln(2). */ | |
274 y += z; | |
275 y += logtbl[k-26]; /* log(t) - (t-1) */ | |
276 y += (t.value - 1.0Q); | |
277 y += e * ln2a; | |
278 return y; | |
279 } |