From 2d0ecbd37030e185556536ea00b3bca28d079a77 Mon Sep 17 00:00:00 2001 From: Bruno Haible Date: Tue, 6 Mar 2012 03:01:51 +0100 Subject: [PATCH] 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. --- ChangeLog | 8 + lib/expl.c | 489 +++++++++++++++++++++++++++++++++++++++++++++-------------- m4/expl.m4 | 23 ++- modules/expl | 6 +- 4 files changed, 411 insertions(+), 115 deletions(-) diff --git a/ChangeLog b/ChangeLog index 740e11315..a18d03370 100644 --- a/ChangeLog +++ b/ChangeLog @@ -1,5 +1,13 @@ 2012-03-05 Bruno Haible + 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. + +2012-03-05 Bruno Haible + cbrt* tests: More tests. * tests/test-cbrt.h: New file. * tests/test-cbrt.c: Include and test-cbrt.h. diff --git a/lib/expl.c b/lib/expl.c index 59cd6c378..28c12b1b8 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 -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, + 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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 diff --git a/m4/expl.m4 b/m4/expl.m4 index 8bf7378d3..a94b2b18f 100644 --- a/m4/expl.m4 +++ b/m4/expl.m4 @@ -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_DEFUN([gl_FUNC_EXPL], 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]) diff --git a/modules/expl b/modules/expl index 7df4fb653..210a25ab0 100644 --- a/modules/expl +++ b/modules/expl @@ -10,7 +10,9 @@ math 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 @@ License: LGPL Maintainer: -Paolo Bonzini +Bruno Haible -- 2.11.0