Additional unit tests that I wrote and that check how close
exp (x) * exp (- x)
gets to 1.0 revealed that the gnulib implementation produces gross
rounding errors here. Where other implementations produce an error
of 2 ulp, with the gnulib expl we get:
- On FreeBSD: 104174 ulp
- On NetBSD and OpenBSD: 105970 ulp
- On HP-UX 11, Solaris 9/SPARC: 6e19 ulp
For example, where the correct values would be
x = 0.81790740799252277699530770566250073
y = exp(x) = 2.26575357544642719034140016719771465068
z = exp(-x) = 0.441354263251230865347549687292992855741
on Solaris 9 the actual values are
x = 0.81790740799252277699530770566250073
y = exp(x) = 2.2657535754464205164805133140966934
z = exp(-x) = 0.44135426325122969079664571540476187
y*z = 0.99999999999999439322013502361228368
err = -5.6067798649763877163169205963298112e-15 = -29112065479413222962.5 ulp
This means, only 15 correct digits!
Possibly the glibc based implementation could be adapted by carefully
thinking about how to compute which constants, depending on LDBL_MANT_DIG
and other machine parameters. But I found it easier to reimplement it
from scratch. A Chebychev polynomial is not even needed: The power series
is good enough. (Of course, reducing the exp function to an odd function
like sinh or tanh reduces by 2 the number of necessary multiplications.)
This implementation produces an error of 4 ulp in the exp (x) * exp (- x)
test. (Except on FreeBSD 6.4/x86, where the compiler truncates all
'long double' literals to 53 bits. One could work around it by converting
all 'long double' literals into references to 'unsigned int[3]' static
arrays, but I'm too lazy to do that.)
2012-03-05 Bruno Haible <br...@clisp.org>
expl: Fix precision of computed result.
* lib/expl.c: Completely rewritten.
* modules/expl (Depends-on): Add isnanl, roundl, ldexpl. Remove floorl.
(Maintainer): Add me.
* m4/expl.m4 (gl_FUNC_EXPL): Update computation of EXPL_LIBM.
================================= lib/expl.c =================================
/* Exponential function.
Copyright (C) 2011-2012 Free Software Foundation, Inc.
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 3 of the License, or
(at your option) any later version.
This program 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 General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>. */
#include <config.h>
/* Specification. */
#include <math.h>
#if HAVE_SAME_LONG_DOUBLE_AS_DOUBLE
long double
expl (long double x)
{
return exp (x);
}
#else
# include <float.h>
/* A value slightly larger than log(2). */
#define LOG2_PLUS_EPSILON 0.6931471805599454L
/* Best possible approximation of log(2) as a 'long double'. */
#define LOG2 0.693147180559945309417232121458176568075L
/* Best possible approximation of 1/log(2) as a 'long double'. */
#define LOG2_INVERSE 1.44269504088896340735992468100189213743L
/* Best possible approximation of log(2)/256 as a 'long double'. */
#define LOG2_BY_256 0.00270760617406228636491106297444600221904L
/* Best possible approximation of 256/log(2) as a 'long double'. */
#define LOG2_BY_256_INVERSE 369.329930467574632284140718336484387181L
/* The upper 32 bits of log(2)/256. */
#define LOG2_BY_256_HI_PART 0.0027076061733168899081647396087646484375L
/* log(2)/256 - LOG2_HI_PART. */
#define LOG2_BY_256_LO_PART \
0.000000000000745396456746323365681353781544922399845L
long double
expl (long double x)
{
if (isnanl (x))
return x;
if (x >= (long double) LDBL_MAX_EXP * LOG2_PLUS_EPSILON)
/* x > LDBL_MAX_EXP * log(2)
hence exp(x) > 2^LDBL_MAX_EXP, overflows to Infinity. */
return HUGE_VALL;
if (x <= (long double) (LDBL_MIN_EXP - 1 - LDBL_MANT_DIG) * LOG2_PLUS_EPSILON)
/* x < (LDBL_MIN_EXP - 1 - LDBL_MANT_DIG) * log(2)
hence exp(x) < 2^(LDBL_MIN_EXP-1-LDBL_MANT_DIG),
underflows to zero. */
return 0.0L;
/* Decompose x into
x = n * log(2) + m * log(2)/256 + y
where
n is an integer,
m is an integer, -128 <= m <= 128,
y is a number, |y| <= log(2)/512 + epsilon = 0.00135...
Then
exp(x) = 2^n * exp(m * log(2)/256) * exp(y)
The first factor is an ldexpl() call.
The second factor is a table lookup.
The third factor is computed
- either as sinh(y) + cosh(y)
where sinh(y) is computed through the power series:
sinh(y) = y + y^3/3! + y^5/5! + ...
and cosh(y) is computed as hypot(1, sinh(y)),
- or as exp(2*z) = (1 + tanh(z))^2 / (1 - tanh(z)^2)
where z = y/2
and tanh(z) is computed through its power series:
tanh(z) = z
- 1/3 * z^3
+ 2/15 * z^5
- 17/315 * z^7
+ 62/2835 * z^9
- 1382/155925 * z^11
+ 21844/6081075 * z^13
- 929569/638512875 * z^15
+ ...
Since |z| <= log(2)/1024 < 0.0007, the relative error of the z^13 term
is < 0.0007^12 < 2^-120 <= 2^-LDBL_MANT_DIG, therefore we can truncate
the series after the z^11 term.
Given the usual bounds LDBL_MAX_EXP <= 16384, LDBL_MIN_EXP >= -16381,
LDBL_MANT_DIG <= 120, we can estimate x: -11440 <= x <= 11357.
This means, when dividing x by log(2), where we want x mod log(2)
to be precise to LDBL_MANT_DIG bits, we have to use an approximation
to log(2) that has 14+LDBL_MANT_DIG bits. */
{
long double nm = roundl (x * LOG2_BY_256_INVERSE); /* = 256 * n + m */
/* n has at most 15 bits, nm therefore has at most 23 bits, therefore
n * LOG2_HI_PART is computed exactly, and n * LOG2_LO_PART is computed
with an absolute error < 2^15 * 2e-10 * 2^-LDBL_MANT_DIG. */
long double y_tmp = x - nm * LOG2_BY_256_HI_PART;
long double y = y_tmp - nm * LOG2_BY_256_LO_PART;
long double z = 0.5L * y;
/* Coefficients of the power series for tanh(z). */
#define TANH_COEFF_1 1.0L
#define TANH_COEFF_3 -0.333333333333333333333333333333333333334L
#define TANH_COEFF_5 0.133333333333333333333333333333333333334L
#define TANH_COEFF_7 -0.053968253968253968253968253968253968254L
#define TANH_COEFF_9 0.0218694885361552028218694885361552028218L
#define TANH_COEFF_11 -0.00886323552990219656886323552990219656886L
#define TANH_COEFF_13 0.00359212803657248101692546136990581435026L
#define TANH_COEFF_15 -0.00145583438705131826824948518070211191904L
long double z2 = z * z;
long double tanh_z =
(((((TANH_COEFF_11
* z2 + TANH_COEFF_9)
* z2 + TANH_COEFF_7)
* z2 + TANH_COEFF_5)
* z2 + TANH_COEFF_3)
* z2 + TANH_COEFF_1)
* z;
long double exp_y =
((1.0L + tanh_z) * (1.0L + tanh_z)) / (1.0L - tanh_z * tanh_z);
int n = (int) roundl (nm * (1.0L / 256.0L));
int m = (int) nm - 256 * n;
/* expl_table[i] = exp((i - 128) * log(2)/256).
Computed in GNU clisp through
(progn
(setf (long-float-digits) 128)
(setq a 0L0)
(setf (long-float-digits) 256)
(dotimes (i 257)
(format t " ~D,~%"
(float (exp (* (/ (- i 128) 256) (log 2L0))) a)))) */
static const long double expl_table[257] =
{
0.707106781186547524400844362104849039284L,
0.709023942160207598920563322257676190836L,
0.710946301084582779904674297352120049962L,
0.71287387205274715340350157671438300618L,
0.714806669195985005617532889137569953044L,
0.71674470668389442125974978427737336719L,
0.71868799872449116280161304224785251353L,
0.720636559564312831364255957304947586072L,
0.72259040348852331001850312073583545284L,
0.724549544821017490259402705487111270714L,
0.726513997924526282423036245842287293786L,
0.728483777200721910815451524818606761737L,
0.730458897090323494325651445155310766577L,
0.732439372073202913296664682112279175616L,
0.734425216668490963430822513132890712652L,
0.736416445434683797507470506133110286942L,
0.738413072969749655693453740187024961962L,
0.740415113911235885228829945155951253966L,
0.742422582936376250272386395864403155277L,
0.744435494762198532693663597314273242753L,
0.746453864145632424600321765743336770838L,
0.748477705883617713391824861712720862423L,
0.750507034813212760132561481529764324813L,
0.752541865811703272039672277899716132493L,
0.75458221379671136988300977551659676571L,
0.756628093726304951096818488157633113612L,
0.75867952059910734940489114658718937343L,
0.760736509454407291763130627098242426467L,
0.762799075372269153425626844758470477304L,
0.76486723347364351194254345936342587308L,
0.766940998920478000900300751753859329456L,
0.769020386915828464216738479594307884331L,
0.771105412703970411806145931045367420652L,
0.773196091570510777431255778146135325272L,
0.77529243884249997956151370535341912283L,
0.777394469888544286059157168801667390437L,
0.779502200118918483516864044737428940745L,
0.781615644985678852072965367573877941354L,
0.783734819982776446532455855478222575498L,
0.78585974064617068462428149076570281356L,
0.787990422553943243227635080090952504452L,
0.790126881326412263402248482007960521995L,
0.79226913262624686505993407346567890838L,
0.794417192158581972116898048814333564685L,
0.796571075671133448968624321559534367934L,
0.798730798954313549131410147104316569576L,
0.800896377841346676896923120795476813684L,
0.803067828208385462848443946517563571584L,
0.805245165974627154089760333678700291728L,
0.807428407102430320039984581575729114268L,
0.809617567597431874649880866726368203972L,
0.81181266350866441589760797777344082227L,
0.814013710928673883424109261007007338614L,
0.816220725993637535170713864466769240053L,
0.818433724883482243883852017078007231025L,
0.82065272382200311435413206848451310067L,
0.822877739076982422259378362362911222833L,
0.825108786960308875483586738272485101678L,
0.827345883828097198786118571797909120834L,
0.829589046080808042697824787210781231927L,
0.831838290163368217523168228488195222638L,
0.834093632565291253329796170708536192903L,
0.836355089820798286809404612069230711295L,
0.83862267850893927589613232455870870518L,
0.84089641525371454303112547623321489504L,
0.84317631672419664796432298771385230143L,
0.84546239963465259098692866759361830709L,
0.84775468074466634749045860363936420312L,
0.850053176859261734750681286748751167545L,
0.852357904829025611837203530384718316326L,
0.854668881550231413551897437515331498025L,
0.856986123964963019301812477839166009452L,
0.859309649061238957814672188228156252257L,
0.861639473873136948607517116872358729753L,
0.863975615480918781121524414614366207052L,
0.866318091011155532438509953514163469652L,
0.868666917636853124497101040936083380124L,
0.871022112577578221729056715595464682243L,
0.873383693099584470038708278290226842228L,
0.875751676515939078050995142767930296012L,
0.878126080186649741556080309687656610647L,
0.880506921518791912081045787323636256171L,
0.882894217966636410521691124969260937028L,
0.885287987031777386769987907431242017412L,
0.88768824626326062627527960009966160388L,
0.89009501325771220447985955243623523504L,
0.892508305659467490072110281986409916153L,
0.8949281411607004980029443898876582985L,
0.897354537501553593213851621063890907178L,
0.899787512470267546027427696662514569756L,
0.902227083903311940153838631655504844215L,
0.904673269685515934269259325789226871994L,
0.907126087750199378124917300181170171233L,
0.909585556079304284147971563828178746372L,
0.91205169270352665549806275316460097744L,
0.914524515702448671545983912696158354092L,
0.91700404320467123174354159479414442804L,
0.919490293387946858856304371174663918816L,
0.921983284479312962533570386670938449637L,
0.92448303475522546419252726694739603678L,
0.92698956254169278419622653516884831976L,
0.929502886214410192307650717745572682403L,
0.932023024198894522404814545597236289343L,
0.934549994970619252444512104439799143264L,
0.93708381705514995066499947497722326722L,
0.93962450902828008902058735120448448827L,
0.942172089516167224843810351983745154882L,
0.944726577195469551733539267378681531548L,
0.947287990793482820670109326713462307376L,
0.949856349088277632361251759806996099924L,
0.952431670908837101825337466217860725517L,
0.955013975135194896221170529572799135168L,
0.957603280698573646936305635147915443924L,
0.960199606581523736948607188887070611744L,
0.962802971818062464478519115091191368377L,
0.965413395493813583952272948264534783197L,
0.968030896746147225299027952283345762418L,
0.970655494764320192607710617437589705184L,
0.973287208789616643172102023321302921373L,
0.97592605811548914795551023340047499377L,
0.978572062087700134509161125813435745597L,
0.981225240104463713381244885057070325016L,
0.983885611616587889056366801238014683926L,
0.98655319612761715646797006813220671315L,
0.989228013193975484129124959065583667775L,
0.99191008242510968492991311132615581644L,
0.994599423483633175652477686222166314457L,
0.997296056085470126257659913847922601123L,
1.0L,
1.00271127505020248543074558845036204047L,
1.0054299011128028213513839559347998147L,
1.008155898118417515783094890817201039276L,
1.01088928605170046002040979056186052439L,
1.013630084951489438840258929063939929597L,
1.01637831491095303794049311378629406276L,
1.0191339960777379496848780958207928794L,
1.02189714865411667823448013478329943978L,
1.02466779289713564514828907627081492763L,
1.0274459491187636965388611939222137815L,
1.030231637686041012871707902453904567093L,
1.033024879021228422500108283970460918086L,
1.035825693601957120029983209018081371844L,
1.03863410196137879061243669795463973258L,
1.04145012468831614126454607901189312648L,
1.044273782427413840321966478739929008784L,
1.04710509587928986612990725022711224056L,
1.04994408580068726608203812651590790906L,
1.05279077300462632711989120298074630319L,
1.05564517836055715880834132515293865216L,
1.058507322794512690105772109683716645074L,
1.061377227289262080950567678003883726294L,
1.06425491288446454978861125700158022068L,
1.06714040067682361816952112099280916261L,
1.0700337118202417735424119367576235685L,
1.072934867525975551385035450873827585343L,
1.075843889062791037803228648476057074063L,
1.07876079775711979374068003743848295849L,
1.081685614993215201942115594422531125643L,
1.08461836221330923781610517190661434161L,
1.087559060917769665346797830944039707867L,
1.09050773266525765920701065576070797899L,
1.09346439907288585422822014625044716208L,
1.096429081816376823386138295859248481766L,
1.09940180263022198546369696823882990404L,
1.10238258330784094355641420942564685751L,
1.10537144570174125558827469625695031104L,
1.108368411723678638009423649426619850137L,
1.111373503344817603850149254228916637444L,
1.1143867425958925363088129569196030678L,
1.11740815156736919905457996308578026665L,
1.12043775240960668442900387986631301277L,
1.123475567333019800733729739775321431954L,
1.12652161860824189979479864378703477763L,
1.129575928566288145997264988840249825907L,
1.13263851959871922798707372367762308438L,
1.13570941415780551424039033067611701343L,
1.13878863475669165370383028384151125472L,
1.14187620396956162271229760828788093894L,
1.14497214443180421939441388822291589579L,
1.14807647884017900677879966269734268003L,
1.15118922995298270581775963520198253612L,
1.154310420590216039548221528724806960684L,
1.157440073633751029613085766293796821106L,
1.16057821202749874636945947257609098625L,
1.16372485877757751381357359909218531234L,
1.166880036952481570555516298414089287834L,
1.170043769683250188080259035792738573L,
1.17321608016363724753480435451324538889L,
1.176396991650281276284645728483848641054L,
1.17958652746287594548610056676944051898L,
1.182784710984341029924457204693850757966L,
1.18599156566099383137126564953421556374L,
1.18920711500272106671749997056047591529L,
1.19243138258315122214272755814543101148L,
1.195664392039827374583837049865451975705L,
1.19890616707438048177030255797630020695L,
1.202156731452703142096396957497765876003L,
1.205416109005123825604211432558411335666L,
1.208684323626581577354792255889216998484L,
1.21196139927680119446816891773249304545L,
1.215247359980468878116520251338798457624L,
1.218542229827408361758207148117394510724L,
1.221846032972757516903891841911570785836L,
1.225158793637145437709464594384845353707L,
1.22848053610687000569400895779278184036L,
1.2318112847340759358845566532127948166L,
1.235151063936933305692912507415415760294L,
1.238499898199816567833368865859612431545L,
1.24185781207348404859367746872659560551L,
1.24522483017525793277520496748615267417L,
1.24860097718920473662176609730249554519L,
1.25198627786631627006020603178920359732L,
1.255380757024691089579390657442301194595L,
1.25878443954971644307786044181516261876L,
1.26219735039425070801401025851841645967L,
1.265619514578806324196273999873453036296L,
1.26905095719173322255441908103233800472L,
1.27249170338940275123669204418460217677L,
1.27594177839639210038120243475928938891L,
1.27940120750566922691358797002785254596L,
1.28287001607877828072666978102151405111L,
1.286348229546025533601482208069738348355L,
1.28983587340666581223274729549155218968L,
1.293332973229089436725559789048704304684L,
1.296839554651009665933754117792451159835L,
1.30035564337965065101414056707091779129L,
1.30388126519193589857452364895199736833L,
1.30741644593467724479715157747196172848L,
1.310961211524764341922991786330755849366L,
1.314515587949354658485983613383997794965L,
1.318079601266063994690185647066116617664L,
1.32165327760315751432651181233060922616L,
1.32523664315974129462953709549872167411L,
1.32882972420595439547865089632866510792L,
1.33243254708316144935164337949073577407L,
1.33604513820414577344262790437186975929L,
1.33966752405330300536003066972435257602L,
1.34329973118683526382421714618163087542L,
1.346941786232945835788173713229537282075L,
1.35059371589203439140852219606013396004L,
1.35425554693689272829801474014070280434L,
1.357927306212901046494536695671766697446L,
1.36160902063822475558553593883194147464L,
1.36530071720401181543069836033754285543L,
1.36900242297459061192960113298219283217L,
1.37271416508766836928499785714471721579L,
1.37643597075453010021632280551868696026L,
1.380167867260238095581945274358283464697L,
1.383909881963831954872659527265192818L,
1.387662042298529159042861017950775988896L,
1.39142437577192618714983552956624344668L,
1.395196909966200178275574599249220994716L,
1.398979672538311140209528136715194969206L,
1.40277269122020470637471352433337881711L,
1.40657599381901544248361973255451684411L,
1.410389608217270704414375128268675481145L,
1.41421356237309504880168872420969807857L
};
return ldexpl (expl_table[128 + m] * exp_y, n);
}
}
#endif
==============================================================================
--- m4/expl.m4.orig Tue Mar 6 02:41:43 2012
+++ m4/expl.m4 Tue Mar 6 01:32:23 2012
@@ -1,4 +1,4 @@
-# expl.m4 serial 6
+# expl.m4 serial 7
dnl Copyright (C) 2010-2012 Free Software Foundation, Inc.
dnl This file is free software; the Free Software Foundation
dnl gives unlimited permission to copy and/or distribute it,
@@ -66,8 +66,25 @@
AC_REQUIRE([gl_FUNC_EXP])
EXPL_LIBM="$EXP_LIBM"
else
- AC_REQUIRE([gl_FUNC_FLOORL])
- EXPL_LIBM="$FLOORL_LIBM"
+ AC_REQUIRE([gl_FUNC_ISNANL])
+ AC_REQUIRE([gl_FUNC_ROUNDL])
+ AC_REQUIRE([gl_FUNC_LDEXPL])
+ EXPL_LIBM=
+ dnl Append $ISNANL_LIBM to EXPL_LIBM, avoiding gratuitous duplicates.
+ case " $EXPL_LIBM " in
+ *" $ISNANL_LIBM "*) ;;
+ *) EXPL_LIBM="$EXPL_LIBM $ISNANL_LIBM" ;;
+ esac
+ dnl Append $ROUNDL_LIBM to EXPL_LIBM, avoiding gratuitous duplicates.
+ case " $EXPL_LIBM " in
+ *" $ROUNDL_LIBM "*) ;;
+ *) EXPL_LIBM="$EXPL_LIBM $ROUNDL_LIBM" ;;
+ esac
+ dnl Append $LDEXPL_LIBM to EXPL_LIBM, avoiding gratuitous duplicates.
+ case " $EXPL_LIBM " in
+ *" $LDEXPL_LIBM "*) ;;
+ *) EXPL_LIBM="$EXPL_LIBM $LDEXPL_LIBM" ;;
+ esac
fi
fi
AC_SUBST([EXPL_LIBM])
--- modules/expl.orig Tue Mar 6 02:41:43 2012
+++ modules/expl Tue Mar 6 02:08:52 2012
@@ -10,7 +10,9 @@
extensions
exp [test $HAVE_EXPL = 0 && test $HAVE_SAME_LONG_DOUBLE_AS_DOUBLE
= 1]
float [test $HAVE_EXPL = 0 && test $HAVE_SAME_LONG_DOUBLE_AS_DOUBLE
= 0]
-floorl [test $HAVE_EXPL = 0 && test $HAVE_SAME_LONG_DOUBLE_AS_DOUBLE
= 0]
+isnanl [test $HAVE_EXPL = 0 && test $HAVE_SAME_LONG_DOUBLE_AS_DOUBLE
= 0]
+roundl [test $HAVE_EXPL = 0 && test $HAVE_SAME_LONG_DOUBLE_AS_DOUBLE
= 0]
+ldexpl [test $HAVE_EXPL = 0 && test $HAVE_SAME_LONG_DOUBLE_AS_DOUBLE
= 0]
configure.ac:
gl_FUNC_EXPL
@@ -31,4 +33,4 @@
LGPL
Maintainer:
-Paolo Bonzini
+Bruno Haible
