| /* logll.c |
| * |
| * Natural logarithm for 128-bit long double precision. |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * long double x, y, logq(); |
| * |
| * y = logq( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns the base e (2.718...) logarithm of x. |
| * |
| * The argument is separated into its exponent and fractional |
| * parts. Use of a lookup table increases the speed of the routine. |
| * The program uses logarithms tabulated at intervals of 1/128 to |
| * cover the domain from approximately 0.7 to 1.4. |
| * |
| * On the interval [-1/128, +1/128] the logarithm of 1+x is approximated by |
| * log(1+x) = x - 0.5 x^2 + x^3 P(x) . |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0.875, 1.125 100000 1.2e-34 4.1e-35 |
| * IEEE 0.125, 8 100000 1.2e-34 4.1e-35 |
| * |
| * |
| * WARNING: |
| * |
| * This program uses integer operations on bit fields of floating-point |
| * numbers. It does not work with data structures other than the |
| * structure assumed. |
| * |
| */ |
| |
| /* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov> |
| |
| This library is free software; you can redistribute it and/or |
| modify it under the terms of the GNU Lesser General Public |
| License as published by the Free Software Foundation; either |
| version 2.1 of the License, or (at your option) any later version. |
| |
| This library is distributed in the hope that it will be useful, |
| but WITHOUT ANY WARRANTY; without even the implied warranty of |
| MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
| Lesser General Public License for more details. |
| |
| You should have received a copy of the GNU Lesser General Public |
| License along with this library; if not, see |
| <http://www.gnu.org/licenses/>. */ |
| |
| #include "quadmath-imp.h" |
| |
| /* log(1+x) = x - .5 x^2 + x^3 l(x) |
| -.0078125 <= x <= +.0078125 |
| peak relative error 1.2e-37 */ |
| static const __float128 |
| l3 = 3.333333333333333333333333333333336096926E-1Q, |
| l4 = -2.499999999999999999999999999486853077002E-1Q, |
| l5 = 1.999999999999999999999999998515277861905E-1Q, |
| l6 = -1.666666666666666666666798448356171665678E-1Q, |
| l7 = 1.428571428571428571428808945895490721564E-1Q, |
| l8 = -1.249999999999999987884655626377588149000E-1Q, |
| l9 = 1.111111111111111093947834982832456459186E-1Q, |
| l10 = -1.000000000000532974938900317952530453248E-1Q, |
| l11 = 9.090909090915566247008015301349979892689E-2Q, |
| l12 = -8.333333211818065121250921925397567745734E-2Q, |
| l13 = 7.692307559897661630807048686258659316091E-2Q, |
| l14 = -7.144242754190814657241902218399056829264E-2Q, |
| l15 = 6.668057591071739754844678883223432347481E-2Q; |
| |
| /* Lookup table of ln(t) - (t-1) |
| t = 0.5 + (k+26)/128) |
| k = 0, ..., 91 */ |
| static const __float128 logtbl[92] = { |
| -5.5345593589352099112142921677820359632418E-2Q, |
| -5.2108257402767124761784665198737642086148E-2Q, |
| -4.8991686870576856279407775480686721935120E-2Q, |
| -4.5993270766361228596215288742353061431071E-2Q, |
| -4.3110481649613269682442058976885699556950E-2Q, |
| -4.0340872319076331310838085093194799765520E-2Q, |
| -3.7682072451780927439219005993827431503510E-2Q, |
| -3.5131785416234343803903228503274262719586E-2Q, |
| -3.2687785249045246292687241862699949178831E-2Q, |
| -3.0347913785027239068190798397055267411813E-2Q, |
| -2.8110077931525797884641940838507561326298E-2Q, |
| -2.5972247078357715036426583294246819637618E-2Q, |
| -2.3932450635346084858612873953407168217307E-2Q, |
| -2.1988775689981395152022535153795155900240E-2Q, |
| -2.0139364778244501615441044267387667496733E-2Q, |
| -1.8382413762093794819267536615342902718324E-2Q, |
| -1.6716169807550022358923589720001638093023E-2Q, |
| -1.5138929457710992616226033183958974965355E-2Q, |
| -1.3649036795397472900424896523305726435029E-2Q, |
| -1.2244881690473465543308397998034325468152E-2Q, |
| -1.0924898127200937840689817557742469105693E-2Q, |
| -9.6875626072830301572839422532631079809328E-3Q, |
| -8.5313926245226231463436209313499745894157E-3Q, |
| -7.4549452072765973384933565912143044991706E-3Q, |
| -6.4568155251217050991200599386801665681310E-3Q, |
| -5.5356355563671005131126851708522185605193E-3Q, |
| -4.6900728132525199028885749289712348829878E-3Q, |
| -3.9188291218610470766469347968659624282519E-3Q, |
| -3.2206394539524058873423550293617843896540E-3Q, |
| -2.5942708080877805657374888909297113032132E-3Q, |
| -2.0385211375711716729239156839929281289086E-3Q, |
| -1.5522183228760777967376942769773768850872E-3Q, |
| -1.1342191863606077520036253234446621373191E-3Q, |
| -7.8340854719967065861624024730268350459991E-4Q, |
| -4.9869831458030115699628274852562992756174E-4Q, |
| -2.7902661731604211834685052867305795169688E-4Q, |
| -1.2335696813916860754951146082826952093496E-4Q, |
| -3.0677461025892873184042490943581654591817E-5Q, |
| #define ZERO logtbl[38] |
| 0.0000000000000000000000000000000000000000E0Q, |
| -3.0359557945051052537099938863236321874198E-5Q, |
| -1.2081346403474584914595395755316412213151E-4Q, |
| -2.7044071846562177120083903771008342059094E-4Q, |
| -4.7834133324631162897179240322783590830326E-4Q, |
| -7.4363569786340080624467487620270965403695E-4Q, |
| -1.0654639687057968333207323853366578860679E-3Q, |
| -1.4429854811877171341298062134712230604279E-3Q, |
| -1.8753781835651574193938679595797367137975E-3Q, |
| -2.3618380914922506054347222273705859653658E-3Q, |
| -2.9015787624124743013946600163375853631299E-3Q, |
| -3.4938307889254087318399313316921940859043E-3Q, |
| -4.1378413103128673800485306215154712148146E-3Q, |
| -4.8328735414488877044289435125365629849599E-3Q, |
| -5.5782063183564351739381962360253116934243E-3Q, |
| -6.3731336597098858051938306767880719015261E-3Q, |
| -7.2169643436165454612058905294782949315193E-3Q, |
| -8.1090214990427641365934846191367315083867E-3Q, |
| -9.0486422112807274112838713105168375482480E-3Q, |
| -1.0035177140880864314674126398350812606841E-2Q, |
| -1.1067990155502102718064936259435676477423E-2Q, |
| -1.2146457974158024928196575103115488672416E-2Q, |
| -1.3269969823361415906628825374158424754308E-2Q, |
| -1.4437927104692837124388550722759686270765E-2Q, |
| -1.5649743073340777659901053944852735064621E-2Q, |
| -1.6904842527181702880599758489058031645317E-2Q, |
| -1.8202661505988007336096407340750378994209E-2Q, |
| -1.9542647000370545390701192438691126552961E-2Q, |
| -2.0924256670080119637427928803038530924742E-2Q, |
| -2.2346958571309108496179613803760727786257E-2Q, |
| -2.3810230892650362330447187267648486279460E-2Q, |
| -2.5313561699385640380910474255652501521033E-2Q, |
| -2.6856448685790244233704909690165496625399E-2Q, |
| -2.8438398935154170008519274953860128449036E-2Q, |
| -3.0058928687233090922411781058956589863039E-2Q, |
| -3.1717563112854831855692484086486099896614E-2Q, |
| -3.3413836095418743219397234253475252001090E-2Q, |
| -3.5147290019036555862676702093393332533702E-2Q, |
| -3.6917475563073933027920505457688955423688E-2Q, |
| -3.8723951502862058660874073462456610731178E-2Q, |
| -4.0566284516358241168330505467000838017425E-2Q, |
| -4.2444048996543693813649967076598766917965E-2Q, |
| -4.4356826869355401653098777649745233339196E-2Q, |
| -4.6304207416957323121106944474331029996141E-2Q, |
| -4.8285787106164123613318093945035804818364E-2Q, |
| -5.0301169421838218987124461766244507342648E-2Q, |
| -5.2349964705088137924875459464622098310997E-2Q, |
| -5.4431789996103111613753440311680967840214E-2Q, |
| -5.6546268881465384189752786409400404404794E-2Q, |
| -5.8693031345788023909329239565012647817664E-2Q, |
| -6.0871713627532018185577188079210189048340E-2Q, |
| -6.3081958078862169742820420185833800925568E-2Q, |
| -6.5323413029406789694910800219643791556918E-2Q, |
| -6.7595732653791419081537811574227049288168E-2Q |
| }; |
| |
| /* ln(2) = ln2a + ln2b with extended precision. */ |
| static const __float128 |
| ln2a = 6.93145751953125e-1Q, |
| ln2b = 1.4286068203094172321214581765680755001344E-6Q; |
| |
| __float128 |
| logq(__float128 x) |
| { |
| __float128 z, y, w; |
| ieee854_float128 u, t; |
| unsigned int m; |
| int k, e; |
| |
| u.value = x; |
| m = u.words32.w0; |
| |
| /* Check for IEEE special cases. */ |
| k = m & 0x7fffffff; |
| /* log(0) = -infinity. */ |
| if ((k | u.words32.w1 | u.words32.w2 | u.words32.w3) == 0) |
| { |
| return -0.5Q / ZERO; |
| } |
| /* log ( x < 0 ) = NaN */ |
| if (m & 0x80000000) |
| { |
| return (x - x) / ZERO; |
| } |
| /* log (infinity or NaN) */ |
| if (k >= 0x7fff0000) |
| { |
| return x + x; |
| } |
| |
| /* Extract exponent and reduce domain to 0.703125 <= u < 1.40625 */ |
| u.value = frexpq (x, &e); |
| m = u.words32.w0 & 0xffff; |
| m |= 0x10000; |
| /* Find lookup table index k from high order bits of the significand. */ |
| if (m < 0x16800) |
| { |
| k = (m - 0xff00) >> 9; |
| /* t is the argument 0.5 + (k+26)/128 |
| of the nearest item to u in the lookup table. */ |
| t.words32.w0 = 0x3fff0000 + (k << 9); |
| t.words32.w1 = 0; |
| t.words32.w2 = 0; |
| t.words32.w3 = 0; |
| u.words32.w0 += 0x10000; |
| e -= 1; |
| k += 64; |
| } |
| else |
| { |
| k = (m - 0xfe00) >> 10; |
| t.words32.w0 = 0x3ffe0000 + (k << 10); |
| t.words32.w1 = 0; |
| t.words32.w2 = 0; |
| t.words32.w3 = 0; |
| } |
| /* On this interval the table is not used due to cancellation error. */ |
| if ((x <= 1.0078125Q) && (x >= 0.9921875Q)) |
| { |
| if (x == 1) |
| return 0; |
| z = x - 1; |
| k = 64; |
| t.value = 1; |
| e = 0; |
| } |
| else |
| { |
| /* log(u) = log( t u/t ) = log(t) + log(u/t) |
| log(t) is tabulated in the lookup table. |
| Express log(u/t) = log(1+z), where z = u/t - 1 = (u-t)/t. |
| cf. Cody & Waite. */ |
| z = (u.value - t.value) / t.value; |
| } |
| /* Series expansion of log(1+z). */ |
| w = z * z; |
| y = ((((((((((((l15 * z |
| + l14) * z |
| + l13) * z |
| + l12) * z |
| + l11) * z |
| + l10) * z |
| + l9) * z |
| + l8) * z |
| + l7) * z |
| + l6) * z |
| + l5) * z |
| + l4) * z |
| + l3) * z * w; |
| y -= 0.5 * w; |
| y += e * ln2b; /* Base 2 exponent offset times ln(2). */ |
| y += z; |
| y += logtbl[k-26]; /* log(t) - (t-1) */ |
| y += (t.value - 1); |
| y += e * ln2a; |
| return y; |
| } |