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/* $OpenBSD: b_log__D.c,v 1.4 2009/10/27 23:59:29 deraadt Exp $ */ |
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/* |
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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. 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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#include "math.h" |
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#include "math_private.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 const double A1 = .08333333333333178827; |
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static const double A2 = .01250000000377174923; |
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static const double A3 = .002232139987919447809; |
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static const double A4 = .0004348877777076145742; |
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static const double logF_head[N+1] = { |
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0., |
87 |
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.007782140442060381246, |
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.015504186535963526694, |
89 |
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.023167059281547608406, |
90 |
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.030771658666765233647, |
91 |
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.038318864302141264488, |
92 |
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.045809536031242714670, |
93 |
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.053244514518837604555, |
94 |
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.060624621816486978786, |
95 |
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.067950661908525944454, |
96 |
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.075223421237524235039, |
97 |
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.082443669210988446138, |
98 |
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.089612158689760690322, |
99 |
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.096729626458454731618, |
100 |
|
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.103796793681567578460, |
101 |
|
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.110814366340264314203, |
102 |
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.117783035656430001836, |
103 |
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.124703478501032805070, |
104 |
|
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.131576357788617315236, |
105 |
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.138402322859292326029, |
106 |
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.145182009844575077295, |
107 |
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.151916042025732167530, |
108 |
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.158605030176659056451, |
109 |
|
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.165249572895390883786, |
110 |
|
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.171850256926518341060, |
111 |
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.178407657472689606947, |
112 |
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.184922338493834104156, |
113 |
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.191394852999565046047, |
114 |
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.197825743329758552135, |
115 |
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.204215541428766300668, |
116 |
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.210564769107350002741, |
117 |
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.216873938300523150246, |
118 |
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.223143551314024080056, |
119 |
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.229374101064877322642, |
120 |
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.235566071312860003672, |
121 |
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.241719936886966024758, |
122 |
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.247836163904594286577, |
123 |
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.253915209980732470285, |
124 |
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.259957524436686071567, |
125 |
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.265963548496984003577, |
126 |
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.271933715484010463114, |
127 |
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.277868451003087102435, |
128 |
|
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.283768173130738432519, |
129 |
|
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.289633292582948342896, |
130 |
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.295464212893421063199, |
131 |
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.301261330578199704177, |
132 |
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.307025035294827830512, |
133 |
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.312755710004239517729, |
134 |
|
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.318453731118097493890, |
135 |
|
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.324119468654316733591, |
136 |
|
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.329753286372579168528, |
137 |
|
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.335355541920762334484, |
138 |
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.340926586970454081892, |
139 |
|
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.346466767346100823488, |
140 |
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.351976423156884266063, |
141 |
|
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.357455888922231679316, |
142 |
|
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.362905493689140712376, |
143 |
|
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.368325561158599157352, |
144 |
|
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.373716409793814818840, |
145 |
|
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.379078352934811846353, |
146 |
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.384411698910298582632, |
147 |
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.389716751140440464951, |
148 |
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.394993808240542421117, |
149 |
|
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.400243164127459749579, |
150 |
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.405465108107819105498, |
151 |
|
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.410659924985338875558, |
152 |
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.415827895143593195825, |
153 |
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.420969294644237379543, |
154 |
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.426084395310681429691, |
155 |
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.431173464818130014464, |
156 |
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.436236766774527495726, |
157 |
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.441274560805140936281, |
158 |
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.446287102628048160113, |
159 |
|
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.451274644139630254358, |
160 |
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.456237433481874177232, |
161 |
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.461175715122408291790, |
162 |
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.466089729924533457960, |
163 |
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.470979715219073113985, |
164 |
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.475845904869856894947, |
165 |
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.480688529345570714212, |
166 |
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.485507815781602403149, |
167 |
|
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.490303988045525329653, |
168 |
|
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.495077266798034543171, |
169 |
|
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.499827869556611403822, |
170 |
|
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.504556010751912253908, |
171 |
|
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.509261901790523552335, |
172 |
|
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.513945751101346104405, |
173 |
|
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.518607764208354637958, |
174 |
|
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.523248143765158602036, |
175 |
|
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.527867089620485785417, |
176 |
|
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.532464798869114019908, |
177 |
|
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.537041465897345915436, |
178 |
|
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.541597282432121573947, |
179 |
|
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.546132437597407260909, |
180 |
|
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.550647117952394182793, |
181 |
|
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.555141507540611200965, |
182 |
|
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.559615787935399566777, |
183 |
|
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.564070138285387656651, |
184 |
|
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.568504735352689749561, |
185 |
|
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.572919753562018740922, |
186 |
|
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.577315365035246941260, |
187 |
|
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.581691739635061821900, |
188 |
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.586049045003164792433, |
189 |
|
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.590387446602107957005, |
190 |
|
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.594707107746216934174, |
191 |
|
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.599008189645246602594, |
192 |
|
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.603290851438941899687, |
193 |
|
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.607555250224322662688, |
194 |
|
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.611801541106615331955, |
195 |
|
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.616029877215623855590, |
196 |
|
|
.620240409751204424537, |
197 |
|
|
.624433288012369303032, |
198 |
|
|
.628608659422752680256, |
199 |
|
|
.632766669570628437213, |
200 |
|
|
.636907462236194987781, |
201 |
|
|
.641031179420679109171, |
202 |
|
|
.645137961373620782978, |
203 |
|
|
.649227946625615004450, |
204 |
|
|
.653301272011958644725, |
205 |
|
|
.657358072709030238911, |
206 |
|
|
.661398482245203922502, |
207 |
|
|
.665422632544505177065, |
208 |
|
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.669430653942981734871, |
209 |
|
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.673422675212350441142, |
210 |
|
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.677398823590920073911, |
211 |
|
|
.681359224807238206267, |
212 |
|
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.685304003098281100392, |
213 |
|
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.689233281238557538017, |
214 |
|
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.693147180560117703862 |
215 |
|
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}; |
216 |
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|
217 |
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static const double logF_tail[N+1] = { |
218 |
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0., |
219 |
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-.00000000000000543229938420049, |
220 |
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.00000000000000172745674997061, |
221 |
|
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-.00000000000001323017818229233, |
222 |
|
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-.00000000000001154527628289872, |
223 |
|
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-.00000000000000466529469958300, |
224 |
|
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.00000000000005148849572685810, |
225 |
|
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-.00000000000002532168943117445, |
226 |
|
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-.00000000000005213620639136504, |
227 |
|
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-.00000000000001819506003016881, |
228 |
|
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.00000000000006329065958724544, |
229 |
|
|
.00000000000008614512936087814, |
230 |
|
|
-.00000000000007355770219435028, |
231 |
|
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.00000000000009638067658552277, |
232 |
|
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.00000000000007598636597194141, |
233 |
|
|
.00000000000002579999128306990, |
234 |
|
|
-.00000000000004654729747598444, |
235 |
|
|
-.00000000000007556920687451336, |
236 |
|
|
.00000000000010195735223708472, |
237 |
|
|
-.00000000000017319034406422306, |
238 |
|
|
-.00000000000007718001336828098, |
239 |
|
|
.00000000000010980754099855238, |
240 |
|
|
-.00000000000002047235780046195, |
241 |
|
|
-.00000000000008372091099235912, |
242 |
|
|
.00000000000014088127937111135, |
243 |
|
|
.00000000000012869017157588257, |
244 |
|
|
.00000000000017788850778198106, |
245 |
|
|
.00000000000006440856150696891, |
246 |
|
|
.00000000000016132822667240822, |
247 |
|
|
-.00000000000007540916511956188, |
248 |
|
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-.00000000000000036507188831790, |
249 |
|
|
.00000000000009120937249914984, |
250 |
|
|
.00000000000018567570959796010, |
251 |
|
|
-.00000000000003149265065191483, |
252 |
|
|
-.00000000000009309459495196889, |
253 |
|
|
.00000000000017914338601329117, |
254 |
|
|
-.00000000000001302979717330866, |
255 |
|
|
.00000000000023097385217586939, |
256 |
|
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.00000000000023999540484211737, |
257 |
|
|
.00000000000015393776174455408, |
258 |
|
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-.00000000000036870428315837678, |
259 |
|
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.00000000000036920375082080089, |
260 |
|
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-.00000000000009383417223663699, |
261 |
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.00000000000009433398189512690, |
262 |
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.00000000000041481318704258568, |
263 |
|
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-.00000000000003792316480209314, |
264 |
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.00000000000008403156304792424, |
265 |
|
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-.00000000000034262934348285429, |
266 |
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.00000000000043712191957429145, |
267 |
|
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-.00000000000010475750058776541, |
268 |
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-.00000000000011118671389559323, |
269 |
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.00000000000037549577257259853, |
270 |
|
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.00000000000013912841212197565, |
271 |
|
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.00000000000010775743037572640, |
272 |
|
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.00000000000029391859187648000, |
273 |
|
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-.00000000000042790509060060774, |
274 |
|
|
.00000000000022774076114039555, |
275 |
|
|
.00000000000010849569622967912, |
276 |
|
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-.00000000000023073801945705758, |
277 |
|
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.00000000000015761203773969435, |
278 |
|
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.00000000000003345710269544082, |
279 |
|
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-.00000000000041525158063436123, |
280 |
|
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.00000000000032655698896907146, |
281 |
|
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-.00000000000044704265010452446, |
282 |
|
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.00000000000034527647952039772, |
283 |
|
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-.00000000000007048962392109746, |
284 |
|
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.00000000000011776978751369214, |
285 |
|
|
-.00000000000010774341461609578, |
286 |
|
|
.00000000000021863343293215910, |
287 |
|
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.00000000000024132639491333131, |
288 |
|
|
.00000000000039057462209830700, |
289 |
|
|
-.00000000000026570679203560751, |
290 |
|
|
.00000000000037135141919592021, |
291 |
|
|
-.00000000000017166921336082431, |
292 |
|
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-.00000000000028658285157914353, |
293 |
|
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-.00000000000023812542263446809, |
294 |
|
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.00000000000006576659768580062, |
295 |
|
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-.00000000000028210143846181267, |
296 |
|
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.00000000000010701931762114254, |
297 |
|
|
.00000000000018119346366441110, |
298 |
|
|
.00000000000009840465278232627, |
299 |
|
|
-.00000000000033149150282752542, |
300 |
|
|
-.00000000000018302857356041668, |
301 |
|
|
-.00000000000016207400156744949, |
302 |
|
|
.00000000000048303314949553201, |
303 |
|
|
-.00000000000071560553172382115, |
304 |
|
|
.00000000000088821239518571855, |
305 |
|
|
-.00000000000030900580513238244, |
306 |
|
|
-.00000000000061076551972851496, |
307 |
|
|
.00000000000035659969663347830, |
308 |
|
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.00000000000035782396591276383, |
309 |
|
|
-.00000000000046226087001544578, |
310 |
|
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.00000000000062279762917225156, |
311 |
|
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.00000000000072838947272065741, |
312 |
|
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.00000000000026809646615211673, |
313 |
|
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-.00000000000010960825046059278, |
314 |
|
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.00000000000002311949383800537, |
315 |
|
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-.00000000000058469058005299247, |
316 |
|
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-.00000000000002103748251144494, |
317 |
|
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-.00000000000023323182945587408, |
318 |
|
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-.00000000000042333694288141916, |
319 |
|
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-.00000000000043933937969737844, |
320 |
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.00000000000041341647073835565, |
321 |
|
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.00000000000006841763641591466, |
322 |
|
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.00000000000047585534004430641, |
323 |
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.00000000000083679678674757695, |
324 |
|
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-.00000000000085763734646658640, |
325 |
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.00000000000021913281229340092, |
326 |
|
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-.00000000000062242842536431148, |
327 |
|
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-.00000000000010983594325438430, |
328 |
|
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.00000000000065310431377633651, |
329 |
|
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-.00000000000047580199021710769, |
330 |
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-.00000000000037854251265457040, |
331 |
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.00000000000040939233218678664, |
332 |
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.00000000000087424383914858291, |
333 |
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.00000000000025218188456842882, |
334 |
|
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-.00000000000003608131360422557, |
335 |
|
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-.00000000000050518555924280902, |
336 |
|
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.00000000000078699403323355317, |
337 |
|
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-.00000000000067020876961949060, |
338 |
|
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.00000000000016108575753932458, |
339 |
|
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.00000000000058527188436251509, |
340 |
|
|
-.00000000000035246757297904791, |
341 |
|
|
-.00000000000018372084495629058, |
342 |
|
|
.00000000000088606689813494916, |
343 |
|
|
.00000000000066486268071468700, |
344 |
|
|
.00000000000063831615170646519, |
345 |
|
|
.00000000000025144230728376072, |
346 |
|
|
-.00000000000017239444525614834 |
347 |
|
|
}; |
348 |
|
|
|
349 |
|
|
/* |
350 |
|
|
* Extra precision variant, returning struct {double a, b;}; |
351 |
|
|
* log(x) = a+b to 63 bits, with a rounded to 26 bits. |
352 |
|
|
*/ |
353 |
|
|
struct Double |
354 |
|
|
__log__D(double x) |
355 |
|
|
{ |
356 |
|
|
int m, j; |
357 |
|
|
double F, f, g, q, u, v, u2; |
358 |
|
30 |
volatile double u1; |
359 |
|
15 |
struct Double r; |
360 |
|
|
|
361 |
|
|
/* Argument reduction: 1 <= g < 2; x/2^m = g; */ |
362 |
|
|
/* y = F*(1 + f/F) for |f| <= 2^-8 */ |
363 |
|
|
|
364 |
|
15 |
m = logb(x); |
365 |
|
15 |
g = ldexp(x, -m); |
366 |
✗✓ |
15 |
if (m == -1022) { |
367 |
|
|
j = logb(g); |
368 |
|
|
m += j; |
369 |
|
|
g = ldexp(g, -j); |
370 |
|
|
} |
371 |
|
15 |
j = N*(g-1) + .5; |
372 |
|
15 |
F = (1.0/N) * j + 1; |
373 |
|
15 |
f = g - F; |
374 |
|
|
|
375 |
|
15 |
g = 1/(2*F+f); |
376 |
|
15 |
u = 2*f*g; |
377 |
|
15 |
v = u*u; |
378 |
|
15 |
q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); |
379 |
✓✗ |
15 |
if (m | j) { |
380 |
|
15 |
u1 = u + 513; |
381 |
|
15 |
u1 -= 513; |
382 |
|
15 |
} |
383 |
|
|
else { |
384 |
|
|
u1 = u; |
385 |
|
|
TRUNC(u1); |
386 |
|
|
} |
387 |
|
15 |
u2 = (2.0*(f - F*u1) - u1*f) * g; |
388 |
|
|
|
389 |
|
15 |
u1 += m*logF_head[N] + logF_head[j]; |
390 |
|
|
|
391 |
|
15 |
u2 += logF_tail[j]; u2 += q; |
392 |
|
15 |
u2 += logF_tail[N]*m; |
393 |
|
15 |
r.a = u1 + u2; /* Only difference is here */ |
394 |
|
15 |
TRUNC(r.a); |
395 |
|
15 |
r.b = (u1 - r.a) + u2; |
396 |
|
15 |
return (r); |
397 |
|
15 |
} |