0b42281ee9
to doubles as bits. fdlibm-1.1 had similar aliasing bugs, but these were fixed by NetBSD or Cygnus before a modified version of fdlibm was imported in 1994. TRUNC() is only used by tgamma() and some implementation-detail functions. The aliasing bugs were detected by compiling with gcc -O2 but don't seem to have broken tgamma() on i386's or amd64's. They broke my modified version of tgamma(). Moved the definition of TRUNC() to mathimpl.h so that it can be fixed in one place, although the general version is even slower than necessary because it has to operate on pointers to volatiles to handle its arg sometimes being volatile. Inefficiency of the fdlibm macros slows down libm generally, and tgamma() is a relatively unimportant part of libm. The macros act as if on 32-bit words in memory, so they are hard to optimize to direct actions on 64-bit double registers for (non-i386) machines where this is possible. The optimization is too hard for gcc on amd64's, and declaring variables as volatile makes it impossible.
474 lines
14 KiB
C
474 lines
14 KiB
C
/*
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* Copyright (c) 1992, 1993
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* The Regents of the University of California. All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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* 1. Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in the
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* documentation and/or other materials provided with the distribution.
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* 3. All advertising materials mentioning features or use of this software
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* must display the following acknowledgement:
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* This product includes software developed by the University of
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* California, Berkeley and its contributors.
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* 4. Neither the name of the University nor the names of its contributors
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* may be used to endorse or promote products derived from this software
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* without specific prior written permission.
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*
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* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
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* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
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* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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* SUCH DAMAGE.
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*/
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#ifndef lint
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static char sccsid[] = "@(#)log.c 8.2 (Berkeley) 11/30/93";
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#endif /* not lint */
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#include <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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#include <math.h>
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#include <errno.h>
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#include "mathimpl.h"
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/* Table-driven natural logarithm.
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*
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* This code was derived, with minor modifications, from:
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* Peter Tang, "Table-Driven Implementation of the
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* Logarithm in IEEE Floating-Point arithmetic." ACM Trans.
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* Math Software, vol 16. no 4, pp 378-400, Dec 1990).
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*
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* Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256,
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* where F = j/128 for j an integer in [0, 128].
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*
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* log(2^m) = log2_hi*m + log2_tail*m
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* since m is an integer, the dominant term is exact.
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* m has at most 10 digits (for subnormal numbers),
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* and log2_hi has 11 trailing zero bits.
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*
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* log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
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* logF_hi[] + 512 is exact.
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*
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* log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ...
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* the leading term is calculated to extra precision in two
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* parts, the larger of which adds exactly to the dominant
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* m and F terms.
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* There are two cases:
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* 1. when m, j are non-zero (m | j), use absolute
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* precision for the leading term.
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* 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
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* In this case, use a relative precision of 24 bits.
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* (This is done differently in the original paper)
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*
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* Special cases:
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* 0 return signalling -Inf
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* neg return signalling NaN
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* +Inf return +Inf
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*/
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#define N 128
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/* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
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* Used for generation of extend precision logarithms.
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* The constant 35184372088832 is 2^45, so the divide is exact.
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* It ensures correct reading of logF_head, even for inaccurate
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* decimal-to-binary conversion routines. (Everybody gets the
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* right answer for integers less than 2^53.)
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* Values for log(F) were generated using error < 10^-57 absolute
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* with the bc -l package.
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*/
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static double A1 = .08333333333333178827;
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static double A2 = .01250000000377174923;
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static double A3 = .002232139987919447809;
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static double A4 = .0004348877777076145742;
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static double logF_head[N+1] = {
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0.,
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.007782140442060381246,
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.015504186535963526694,
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.023167059281547608406,
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.030771658666765233647,
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.038318864302141264488,
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.045809536031242714670,
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.053244514518837604555,
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.060624621816486978786,
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.067950661908525944454,
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.075223421237524235039,
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.082443669210988446138,
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.089612158689760690322,
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.096729626458454731618,
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.103796793681567578460,
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.110814366340264314203,
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.117783035656430001836,
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.124703478501032805070,
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.131576357788617315236,
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.138402322859292326029,
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.145182009844575077295,
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.151916042025732167530,
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.158605030176659056451,
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.165249572895390883786,
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.171850256926518341060,
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.178407657472689606947,
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.184922338493834104156,
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.191394852999565046047,
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.197825743329758552135,
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.204215541428766300668,
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.210564769107350002741,
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.216873938300523150246,
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.223143551314024080056,
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.229374101064877322642,
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.235566071312860003672,
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.241719936886966024758,
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.247836163904594286577,
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.253915209980732470285,
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.259957524436686071567,
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.265963548496984003577,
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.271933715484010463114,
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.277868451003087102435,
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.283768173130738432519,
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.289633292582948342896,
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.295464212893421063199,
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.301261330578199704177,
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.307025035294827830512,
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.312755710004239517729,
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.318453731118097493890,
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.324119468654316733591,
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.329753286372579168528,
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.335355541920762334484,
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.340926586970454081892,
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.346466767346100823488,
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.351976423156884266063,
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.357455888922231679316,
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.362905493689140712376,
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.368325561158599157352,
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.373716409793814818840,
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.379078352934811846353,
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.384411698910298582632,
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.389716751140440464951,
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.394993808240542421117,
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.400243164127459749579,
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.405465108107819105498,
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.410659924985338875558,
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.415827895143593195825,
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.420969294644237379543,
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.426084395310681429691,
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.431173464818130014464,
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.436236766774527495726,
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.441274560805140936281,
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.446287102628048160113,
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.451274644139630254358,
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.456237433481874177232,
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.461175715122408291790,
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.466089729924533457960,
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.470979715219073113985,
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.475845904869856894947,
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.480688529345570714212,
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.485507815781602403149,
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.490303988045525329653,
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.495077266798034543171,
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.499827869556611403822,
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.504556010751912253908,
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.509261901790523552335,
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.513945751101346104405,
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.518607764208354637958,
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.523248143765158602036,
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.527867089620485785417,
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.532464798869114019908,
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.537041465897345915436,
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.541597282432121573947,
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.546132437597407260909,
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.550647117952394182793,
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.555141507540611200965,
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.559615787935399566777,
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.564070138285387656651,
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.568504735352689749561,
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.572919753562018740922,
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.577315365035246941260,
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.581691739635061821900,
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.586049045003164792433,
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.590387446602107957005,
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.594707107746216934174,
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.599008189645246602594,
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.603290851438941899687,
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.607555250224322662688,
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.611801541106615331955,
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.616029877215623855590,
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.620240409751204424537,
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.624433288012369303032,
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.628608659422752680256,
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.632766669570628437213,
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.636907462236194987781,
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.641031179420679109171,
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.645137961373620782978,
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.649227946625615004450,
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.653301272011958644725,
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.657358072709030238911,
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.661398482245203922502,
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.665422632544505177065,
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.669430653942981734871,
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.673422675212350441142,
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.677398823590920073911,
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.681359224807238206267,
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.685304003098281100392,
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.689233281238557538017,
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.693147180560117703862
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};
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static double logF_tail[N+1] = {
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0.,
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-.00000000000000543229938420049,
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.00000000000000172745674997061,
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-.00000000000001323017818229233,
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-.00000000000001154527628289872,
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-.00000000000000466529469958300,
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.00000000000005148849572685810,
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-.00000000000002532168943117445,
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-.00000000000005213620639136504,
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-.00000000000001819506003016881,
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.00000000000006329065958724544,
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.00000000000008614512936087814,
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-.00000000000007355770219435028,
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.00000000000009638067658552277,
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.00000000000007598636597194141,
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.00000000000002579999128306990,
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-.00000000000004654729747598444,
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-.00000000000007556920687451336,
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.00000000000010195735223708472,
|
|
-.00000000000017319034406422306,
|
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-.00000000000007718001336828098,
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.00000000000010980754099855238,
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-.00000000000002047235780046195,
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-.00000000000008372091099235912,
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.00000000000014088127937111135,
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.00000000000012869017157588257,
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.00000000000017788850778198106,
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.00000000000006440856150696891,
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.00000000000016132822667240822,
|
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-.00000000000007540916511956188,
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-.00000000000000036507188831790,
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.00000000000009120937249914984,
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.00000000000018567570959796010,
|
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-.00000000000003149265065191483,
|
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-.00000000000009309459495196889,
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.00000000000017914338601329117,
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-.00000000000001302979717330866,
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.00000000000023097385217586939,
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.00000000000023999540484211737,
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.00000000000015393776174455408,
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-.00000000000036870428315837678,
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.00000000000036920375082080089,
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-.00000000000009383417223663699,
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.00000000000009433398189512690,
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.00000000000041481318704258568,
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-.00000000000003792316480209314,
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.00000000000008403156304792424,
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-.00000000000034262934348285429,
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.00000000000043712191957429145,
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-.00000000000010475750058776541,
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-.00000000000011118671389559323,
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.00000000000037549577257259853,
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.00000000000013912841212197565,
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.00000000000010775743037572640,
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.00000000000029391859187648000,
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-.00000000000042790509060060774,
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.00000000000022774076114039555,
|
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.00000000000010849569622967912,
|
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-.00000000000023073801945705758,
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.00000000000015761203773969435,
|
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.00000000000003345710269544082,
|
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-.00000000000041525158063436123,
|
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.00000000000032655698896907146,
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-.00000000000044704265010452446,
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.00000000000034527647952039772,
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-.00000000000007048962392109746,
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.00000000000011776978751369214,
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-.00000000000010774341461609578,
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.00000000000021863343293215910,
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.00000000000024132639491333131,
|
|
.00000000000039057462209830700,
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-.00000000000026570679203560751,
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|
.00000000000037135141919592021,
|
|
-.00000000000017166921336082431,
|
|
-.00000000000028658285157914353,
|
|
-.00000000000023812542263446809,
|
|
.00000000000006576659768580062,
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|
-.00000000000028210143846181267,
|
|
.00000000000010701931762114254,
|
|
.00000000000018119346366441110,
|
|
.00000000000009840465278232627,
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-.00000000000033149150282752542,
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-.00000000000018302857356041668,
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-.00000000000016207400156744949,
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.00000000000048303314949553201,
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-.00000000000071560553172382115,
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|
.00000000000088821239518571855,
|
|
-.00000000000030900580513238244,
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-.00000000000061076551972851496,
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|
.00000000000035659969663347830,
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.00000000000035782396591276383,
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-.00000000000046226087001544578,
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.00000000000062279762917225156,
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|
.00000000000072838947272065741,
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|
.00000000000026809646615211673,
|
|
-.00000000000010960825046059278,
|
|
.00000000000002311949383800537,
|
|
-.00000000000058469058005299247,
|
|
-.00000000000002103748251144494,
|
|
-.00000000000023323182945587408,
|
|
-.00000000000042333694288141916,
|
|
-.00000000000043933937969737844,
|
|
.00000000000041341647073835565,
|
|
.00000000000006841763641591466,
|
|
.00000000000047585534004430641,
|
|
.00000000000083679678674757695,
|
|
-.00000000000085763734646658640,
|
|
.00000000000021913281229340092,
|
|
-.00000000000062242842536431148,
|
|
-.00000000000010983594325438430,
|
|
.00000000000065310431377633651,
|
|
-.00000000000047580199021710769,
|
|
-.00000000000037854251265457040,
|
|
.00000000000040939233218678664,
|
|
.00000000000087424383914858291,
|
|
.00000000000025218188456842882,
|
|
-.00000000000003608131360422557,
|
|
-.00000000000050518555924280902,
|
|
.00000000000078699403323355317,
|
|
-.00000000000067020876961949060,
|
|
.00000000000016108575753932458,
|
|
.00000000000058527188436251509,
|
|
-.00000000000035246757297904791,
|
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-.00000000000018372084495629058,
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.00000000000088606689813494916,
|
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.00000000000066486268071468700,
|
|
.00000000000063831615170646519,
|
|
.00000000000025144230728376072,
|
|
-.00000000000017239444525614834
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};
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#if 0
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double
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#ifdef _ANSI_SOURCE
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log(double x)
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#else
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log(x) double x;
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#endif
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{
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int m, j;
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double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0;
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volatile double u1;
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/* Catch special cases */
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if (x <= 0)
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if (x == zero) /* log(0) = -Inf */
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return (-one/zero);
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else /* log(neg) = NaN */
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return (zero/zero);
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else if (!finite(x))
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return (x+x); /* x = NaN, Inf */
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/* Argument reduction: 1 <= g < 2; x/2^m = g; */
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/* y = F*(1 + f/F) for |f| <= 2^-8 */
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m = logb(x);
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g = ldexp(x, -m);
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if (m == -1022) {
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j = logb(g), m += j;
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g = ldexp(g, -j);
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}
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j = N*(g-1) + .5;
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F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */
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f = g - F;
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/* Approximate expansion for log(1+f/F) ~= u + q */
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g = 1/(2*F+f);
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u = 2*f*g;
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v = u*u;
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q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
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/* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8,
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* u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits.
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* It also adds exactly to |m*log2_hi + log_F_head[j] | < 750
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*/
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if (m | j)
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u1 = u + 513, u1 -= 513;
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/* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero;
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* u1 = u to 24 bits.
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*/
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else
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u1 = u, TRUNC(u1);
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u2 = (2.0*(f - F*u1) - u1*f) * g;
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/* u1 + u2 = 2f/(2F+f) to extra precision. */
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/* log(x) = log(2^m*F*(1+f/F)) = */
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/* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */
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/* (exact) + (tiny) */
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u1 += m*logF_head[N] + logF_head[j]; /* exact */
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u2 = (u2 + logF_tail[j]) + q; /* tiny */
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u2 += logF_tail[N]*m;
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return (u1 + u2);
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}
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#endif
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/*
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* Extra precision variant, returning struct {double a, b;};
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* log(x) = a+b to 63 bits, with a rounded to 26 bits.
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*/
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struct Double
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#ifdef _ANSI_SOURCE
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__log__D(double x)
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#else
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__log__D(x) double x;
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#endif
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{
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int m, j;
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double F, f, g, q, u, v, u2;
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volatile double u1;
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struct Double r;
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/* Argument reduction: 1 <= g < 2; x/2^m = g; */
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/* y = F*(1 + f/F) for |f| <= 2^-8 */
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m = logb(x);
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g = ldexp(x, -m);
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if (m == -1022) {
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j = logb(g), m += j;
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g = ldexp(g, -j);
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}
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j = N*(g-1) + .5;
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F = (1.0/N) * j + 1;
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f = g - F;
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g = 1/(2*F+f);
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u = 2*f*g;
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v = u*u;
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q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
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if (m | j)
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u1 = u + 513, u1 -= 513;
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else
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u1 = u, TRUNC(u1);
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u2 = (2.0*(f - F*u1) - u1*f) * g;
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u1 += m*logF_head[N] + logF_head[j];
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u2 += logF_tail[j]; u2 += q;
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u2 += logF_tail[N]*m;
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r.a = u1 + u2; /* Only difference is here */
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TRUNC(r.a);
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r.b = (u1 - r.a) + u2;
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return (r);
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}
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