472 lines
14 KiB
C
472 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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/* @(#)log.c 8.2 (Berkeley) 11/30/93 */
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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,
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-.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,
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.00000000000039057462209830700,
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-.00000000000026570679203560751,
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.00000000000037135141919592021,
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-.00000000000017166921336082431,
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-.00000000000028658285157914353,
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-.00000000000023812542263446809,
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.00000000000006576659768580062,
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-.00000000000028210143846181267,
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.00000000000010701931762114254,
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.00000000000018119346366441110,
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.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,
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-.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,
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-.00000000000010960825046059278,
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.00000000000002311949383800537,
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-.00000000000058469058005299247,
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-.00000000000002103748251144494,
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-.00000000000023323182945587408,
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-.00000000000042333694288141916,
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-.00000000000043933937969737844,
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.00000000000041341647073835565,
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.00000000000006841763641591466,
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.00000000000047585534004430641,
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.00000000000083679678674757695,
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-.00000000000085763734646658640,
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.00000000000021913281229340092,
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-.00000000000062242842536431148,
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-.00000000000010983594325438430,
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.00000000000065310431377633651,
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-.00000000000047580199021710769,
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-.00000000000037854251265457040,
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.00000000000040939233218678664,
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.00000000000087424383914858291,
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.00000000000025218188456842882,
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-.00000000000003608131360422557,
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-.00000000000050518555924280902,
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.00000000000078699403323355317,
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-.00000000000067020876961949060,
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.00000000000016108575753932458,
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.00000000000058527188436251509,
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-.00000000000035246757297904791,
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-.00000000000018372084495629058,
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.00000000000088606689813494916,
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.00000000000066486268071468700,
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.00000000000063831615170646519,
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.00000000000025144230728376072,
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-.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);
|
|
if (m == -1022) {
|
|
j = logb(g), m += j;
|
|
g = ldexp(g, -j);
|
|
}
|
|
j = N*(g-1) + .5;
|
|
F = (1.0/N) * j + 1;
|
|
f = g - F;
|
|
|
|
g = 1/(2*F+f);
|
|
u = 2*f*g;
|
|
v = u*u;
|
|
q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
|
|
if (m | j)
|
|
u1 = u + 513, u1 -= 513;
|
|
else
|
|
u1 = u, TRUNC(u1);
|
|
u2 = (2.0*(f - F*u1) - u1*f) * g;
|
|
|
|
u1 += m*logF_head[N] + logF_head[j];
|
|
|
|
u2 += logF_tail[j]; u2 += q;
|
|
u2 += logF_tail[N]*m;
|
|
r.a = u1 + u2; /* Only difference is here */
|
|
TRUNC(r.a);
|
|
r.b = (u1 - r.a) + u2;
|
|
return (r);
|
|
}
|