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.

1 files changed, 379 insertions, 110 deletions

diff --git a/lib/expl.c b/lib/expl.c
index 59cd6c378a..28c12b1b86 100644
--- a/lib/expl.c
+++ b/lib/expl.c
@@ -1,9 +1,5 @@
-/* Emulation for expl.
-   Contributed by Paolo Bonzini
-
-   Copyright 2002-2003, 2007, 2009-2012 Free Software Foundation, Inc.
-
-   This file is part of gnulib.
+/* 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
@@ -33,117 +29,390 @@ expl (long double x)
 
 #else
 
-/* Code based on glibc/sysdeps/ieee754/ldbl-128/e_expl.c.  */
-
 # include <float.h>
 
-static const long double C[] = {
-/* Chebyshev polynomial coefficients for (exp(x)-1)/x */
-# define P1 C[0]
-# define P2 C[1]
-# define P3 C[2]
-# define P4 C[3]
-# define P5 C[4]
-# define P6 C[5]
- 0.5L,
- 1.66666666666666666666666666666666683E-01L,
- 4.16666666666666666666654902320001674E-02L,
- 8.33333333333333333333314659767198461E-03L,
- 1.38888888889899438565058018857254025E-03L,
- 1.98412698413981650382436541785404286E-04L,
-
-/* Smallest integer x for which e^x overflows.  */
-# define himark C[6]
- 11356.523406294143949491931077970765L,
-
-/* Largest integer x for which e^x underflows.  */
-# define lomark C[7]
--11433.4627433362978788372438434526231L,
-
-/* very small number */
-# define TINY C[8]
- 1.0e-4900L,
-
-/* 2^16383 */
-# define TWO16383 C[9]
- 5.94865747678615882542879663314003565E+4931L};
+/* 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)
 {
-  /* Check for usual case.  */
-  if (x < himark && x > lomark)
-    {
-      int exponent;
-      long double t, x22;
-      int k = 1;
-      long double result = 1.0;
-
-      /* Compute an integer power of e with a granularity of 0.125.  */
-      exponent = (int) floorl (x * 8.0L);
-      x -= exponent / 8.0L;
-
-      if (x > 0.0625)
-        {
-          exponent++;
-          x -= 0.125L;
-        }
-
-      if (exponent < 0)
-        {
-          t = 0.8824969025845954028648921432290507362220L; /* e^-0.25 */
-          exponent = -exponent;
-        }
-      else
-        t = 1.1331484530668263168290072278117938725655L; /* e^0.25 */
-
-      while (exponent)
-        {
-          if (exponent & k)
-            {
-              result *= t;
-              exponent ^= k;
-            }
-          t *= t;
-          k <<= 1;
-        }
-
-      /* Approximate (e^x - 1)/x, using a seventh-degree polynomial,
-         with maximum error in [-2^-16-2^-53,2^-16+2^-53]
-         less than 4.8e-39.  */
-      x22 = x + x*x*(P1+x*(P2+x*(P3+x*(P4+x*(P5+x*P6)))));
-
-      return result + result * x22;
-    }
-  /* Exceptional cases:  */
-  else if (x < himark)
-    {
-      if (x + x == x)
-        /* e^-inf == 0, with no error.  */
-        return 0;
-      else
-        /* Underflow */
-        return TINY * TINY;
-    }
-  else
-    /* Return x, if x is a NaN or Inf; or overflow, otherwise.  */
-    return TWO16383*x;
-}
+  if (isnanl (x))
+    return x;
 
-#endif
+  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 0
-int
-main (void)
-{
-  printf ("%.16Lg\n", expl (1.0L));
-  printf ("%.16Lg\n", expl (-1.0L));
-  printf ("%.16Lg\n", expl (2.0L));
-  printf ("%.16Lg\n", expl (4.0L));
-  printf ("%.16Lg\n", expl (-2.0L));
-  printf ("%.16Lg\n", expl (-4.0L));
-  printf ("%.16Lg\n", expl (0.0625L));
-  printf ("%.16Lg\n", expl (0.3L));
-  printf ("%.16Lg\n", expl (0.6L));
+  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