3 * Natural logarithm for 128-bit long double precision.
9 * long double x, y, logl();
17 * Returns the base e (2.718...) logarithm of x.
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.
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) .
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
39 * This program uses integer operations on bit fields of floating-point
40 * numbers. It does not work with data structures other than the
45 /* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov>
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.
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.
57 You should have received a copy of the GNU Lesser General Public
58 License along with this library; if not, see
59 <https://www.gnu.org/licenses/>. */
62 #include <math_private.h>
63 #include <libm-alias-finite.h>
65 /* log(1+x) = x - .5 x^2 + x^3 l(x)
66 -.0078125 <= x <= +.0078125
67 peak relative error 1.2e-37 */
68 static const long double
69 l3
= 3.333333333333333333333333333333336096926E-1L,
70 l4
= -2.499999999999999999999999999486853077002E-1L,
71 l5
= 1.999999999999999999999999998515277861905E-1L,
72 l6
= -1.666666666666666666666798448356171665678E-1L,
73 l7
= 1.428571428571428571428808945895490721564E-1L,
74 l8
= -1.249999999999999987884655626377588149000E-1L,
75 l9
= 1.111111111111111093947834982832456459186E-1L,
76 l10
= -1.000000000000532974938900317952530453248E-1L,
77 l11
= 9.090909090915566247008015301349979892689E-2L,
78 l12
= -8.333333211818065121250921925397567745734E-2L,
79 l13
= 7.692307559897661630807048686258659316091E-2L,
80 l14
= -7.144242754190814657241902218399056829264E-2L,
81 l15
= 6.668057591071739754844678883223432347481E-2L;
83 /* Lookup table of ln(t) - (t-1)
86 static const long double logtbl
[92] = {
87 -5.5345593589352099112142921677820359632418E-2L,
88 -5.2108257402767124761784665198737642086148E-2L,
89 -4.8991686870576856279407775480686721935120E-2L,
90 -4.5993270766361228596215288742353061431071E-2L,
91 -4.3110481649613269682442058976885699556950E-2L,
92 -4.0340872319076331310838085093194799765520E-2L,
93 -3.7682072451780927439219005993827431503510E-2L,
94 -3.5131785416234343803903228503274262719586E-2L,
95 -3.2687785249045246292687241862699949178831E-2L,
96 -3.0347913785027239068190798397055267411813E-2L,
97 -2.8110077931525797884641940838507561326298E-2L,
98 -2.5972247078357715036426583294246819637618E-2L,
99 -2.3932450635346084858612873953407168217307E-2L,
100 -2.1988775689981395152022535153795155900240E-2L,
101 -2.0139364778244501615441044267387667496733E-2L,
102 -1.8382413762093794819267536615342902718324E-2L,
103 -1.6716169807550022358923589720001638093023E-2L,
104 -1.5138929457710992616226033183958974965355E-2L,
105 -1.3649036795397472900424896523305726435029E-2L,
106 -1.2244881690473465543308397998034325468152E-2L,
107 -1.0924898127200937840689817557742469105693E-2L,
108 -9.6875626072830301572839422532631079809328E-3L,
109 -8.5313926245226231463436209313499745894157E-3L,
110 -7.4549452072765973384933565912143044991706E-3L,
111 -6.4568155251217050991200599386801665681310E-3L,
112 -5.5356355563671005131126851708522185605193E-3L,
113 -4.6900728132525199028885749289712348829878E-3L,
114 -3.9188291218610470766469347968659624282519E-3L,
115 -3.2206394539524058873423550293617843896540E-3L,
116 -2.5942708080877805657374888909297113032132E-3L,
117 -2.0385211375711716729239156839929281289086E-3L,
118 -1.5522183228760777967376942769773768850872E-3L,
119 -1.1342191863606077520036253234446621373191E-3L,
120 -7.8340854719967065861624024730268350459991E-4L,
121 -4.9869831458030115699628274852562992756174E-4L,
122 -2.7902661731604211834685052867305795169688E-4L,
123 -1.2335696813916860754951146082826952093496E-4L,
124 -3.0677461025892873184042490943581654591817E-5L,
125 #define ZERO logtbl[38]
126 0.0000000000000000000000000000000000000000E0L
,
127 -3.0359557945051052537099938863236321874198E-5L,
128 -1.2081346403474584914595395755316412213151E-4L,
129 -2.7044071846562177120083903771008342059094E-4L,
130 -4.7834133324631162897179240322783590830326E-4L,
131 -7.4363569786340080624467487620270965403695E-4L,
132 -1.0654639687057968333207323853366578860679E-3L,
133 -1.4429854811877171341298062134712230604279E-3L,
134 -1.8753781835651574193938679595797367137975E-3L,
135 -2.3618380914922506054347222273705859653658E-3L,
136 -2.9015787624124743013946600163375853631299E-3L,
137 -3.4938307889254087318399313316921940859043E-3L,
138 -4.1378413103128673800485306215154712148146E-3L,
139 -4.8328735414488877044289435125365629849599E-3L,
140 -5.5782063183564351739381962360253116934243E-3L,
141 -6.3731336597098858051938306767880719015261E-3L,
142 -7.2169643436165454612058905294782949315193E-3L,
143 -8.1090214990427641365934846191367315083867E-3L,
144 -9.0486422112807274112838713105168375482480E-3L,
145 -1.0035177140880864314674126398350812606841E-2L,
146 -1.1067990155502102718064936259435676477423E-2L,
147 -1.2146457974158024928196575103115488672416E-2L,
148 -1.3269969823361415906628825374158424754308E-2L,
149 -1.4437927104692837124388550722759686270765E-2L,
150 -1.5649743073340777659901053944852735064621E-2L,
151 -1.6904842527181702880599758489058031645317E-2L,
152 -1.8202661505988007336096407340750378994209E-2L,
153 -1.9542647000370545390701192438691126552961E-2L,
154 -2.0924256670080119637427928803038530924742E-2L,
155 -2.2346958571309108496179613803760727786257E-2L,
156 -2.3810230892650362330447187267648486279460E-2L,
157 -2.5313561699385640380910474255652501521033E-2L,
158 -2.6856448685790244233704909690165496625399E-2L,
159 -2.8438398935154170008519274953860128449036E-2L,
160 -3.0058928687233090922411781058956589863039E-2L,
161 -3.1717563112854831855692484086486099896614E-2L,
162 -3.3413836095418743219397234253475252001090E-2L,
163 -3.5147290019036555862676702093393332533702E-2L,
164 -3.6917475563073933027920505457688955423688E-2L,
165 -3.8723951502862058660874073462456610731178E-2L,
166 -4.0566284516358241168330505467000838017425E-2L,
167 -4.2444048996543693813649967076598766917965E-2L,
168 -4.4356826869355401653098777649745233339196E-2L,
169 -4.6304207416957323121106944474331029996141E-2L,
170 -4.8285787106164123613318093945035804818364E-2L,
171 -5.0301169421838218987124461766244507342648E-2L,
172 -5.2349964705088137924875459464622098310997E-2L,
173 -5.4431789996103111613753440311680967840214E-2L,
174 -5.6546268881465384189752786409400404404794E-2L,
175 -5.8693031345788023909329239565012647817664E-2L,
176 -6.0871713627532018185577188079210189048340E-2L,
177 -6.3081958078862169742820420185833800925568E-2L,
178 -6.5323413029406789694910800219643791556918E-2L,
179 -6.7595732653791419081537811574227049288168E-2L
182 /* ln(2) = ln2a + ln2b with extended precision. */
183 static const long double
184 ln2a
= 6.93145751953125e-1L,
185 ln2b
= 1.4286068203094172321214581765680755001344E-6L;
187 static const long double
188 ldbl_epsilon
= 0x1p
-106L;
191 __ieee754_logl(long double x
)
193 long double z
, y
, w
, t
;
200 EXTRACT_WORDS (hx
, lx
, xhi
);
203 /* Check for IEEE special cases. */
205 /* log(0) = -infinity. */
210 /* log ( x < 0 ) = NaN */
213 return (x
- x
) / ZERO
;
215 /* log (infinity or NaN) */
221 /* On this interval the table is not used due to cancellation error. */
222 if ((x
<= 1.0078125L) && (x
>= 0.9921875L))
233 /* Extract exponent and reduce domain to 0.703125 <= u < 1.40625 */
235 e
= (int) (m
>> 20) - (int) 0x3fe;
240 EXTRACT_WORDS (hx
, lx
, xhi
);
242 e
= (int) (m
>> 20) - (int) 0x3fe - 106;
247 /* Find lookup table index k from high order bits of the significand. */
250 k
= (m
- 0xff000) >> 13;
251 /* t is the argument 0.5 + (k+26)/128
252 of the nearest item to u in the lookup table. */
253 INSERT_WORDS (xhi
, 0x3ff00000 + (k
<< 13), 0);
261 k
= (m
- 0xfe000) >> 14;
262 INSERT_WORDS (xhi
, 0x3fe00000 + (k
<< 14), 0);
265 x
= __scalbnl (x
, ((int) ((w0
- hx
) * 2)) >> 21);
266 /* log(u) = log( t u/t ) = log(t) + log(u/t)
267 log(t) is tabulated in the lookup table.
268 Express log(u/t) = log(1+z), where z = u/t - 1 = (u-t)/t.
272 /* Series expansion of log(1+z). */
274 /* Avoid spurious underflows. */
275 if (__glibc_unlikely (fabsl (z
) <= ldbl_epsilon
))
279 y
= ((((((((((((l15
* z
294 y
+= e
* ln2b
; /* Base 2 exponent offset times ln(2). */
296 y
+= logtbl
[k
-26]; /* log(t) - (t-1) */
301 libm_alias_finite (__ieee754_logl
, __logl
)