| 1 | /*
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| 2 | * Copyright (c) 1992, 1993
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| 3 | * The Regents of the University of California. All rights reserved.
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| 4 | *
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| 5 | * Redistribution and use in source and binary forms, with or without
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| 6 | * modification, are permitted provided that the following conditions
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| 7 | * are met:
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| 8 | * 1. Redistributions of source code must retain the above copyright
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| 9 | * notice, this list of conditions and the following disclaimer.
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| 10 | * 2. Redistributions in binary form must reproduce the above copyright
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| 11 | * notice, this list of conditions and the following disclaimer in the
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| 12 | * documentation and/or other materials provided with the distribution.
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| 13 | * 3. All advertising materials mentioning features or use of this software
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| 14 | * must display the following acknowledgement:
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| 15 | * This product includes software developed by the University of
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| 16 | * California, Berkeley and its contributors.
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| 17 | * 4. Neither the name of the University nor the names of its contributors
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| 18 | * may be used to endorse or promote products derived from this software
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| 19 | * without specific prior written permission.
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| 20 | *
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| 21 | * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
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| 22 | * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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| 23 | * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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| 24 | * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
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| 25 | * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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| 26 | * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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| 27 | * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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| 28 | * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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| 29 | * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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| 30 | * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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| 31 | * SUCH DAMAGE.
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| 32 | */
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| 33 |
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| 34 | #ifndef lint
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| 35 | static char sccsid[] = "@(#)log.c 8.2 (Berkeley) 11/30/93";
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| 36 | #endif /* not lint */
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| 37 | #include <sys/cdefs.h>
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| 38 | __FBSDID("$FreeBSD: src/lib/msun/bsdsrc/b_log.c,v 1.7 2004/12/16 20:40:37 das Exp $");
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| 39 |
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| 40 | #include <math.h>
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| 41 | #include <errno.h>
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| 42 |
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| 43 | #include "mathimpl.h"
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| 44 |
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| 45 | /* Table-driven natural logarithm.
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| 46 | *
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| 47 | * This code was derived, with minor modifications, from:
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| 48 | * Peter Tang, "Table-Driven Implementation of the
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| 49 | * Logarithm in IEEE Floating-Point arithmetic." ACM Trans.
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| 50 | * Math Software, vol 16. no 4, pp 378-400, Dec 1990).
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| 51 | *
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| 52 | * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256,
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| 53 | * where F = j/128 for j an integer in [0, 128].
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| 54 | *
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| 55 | * log(2^m) = log2_hi*m + log2_tail*m
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| 56 | * since m is an integer, the dominant term is exact.
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| 57 | * m has at most 10 digits (for subnormal numbers),
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| 58 | * and log2_hi has 11 trailing zero bits.
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| 59 | *
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| 60 | * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
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| 61 | * logF_hi[] + 512 is exact.
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| 62 | *
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| 63 | * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ...
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| 64 | * the leading term is calculated to extra precision in two
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| 65 | * parts, the larger of which adds exactly to the dominant
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| 66 | * m and F terms.
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| 67 | * There are two cases:
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| 68 | * 1. when m, j are non-zero (m | j), use absolute
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| 69 | * precision for the leading term.
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| 70 | * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
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| 71 | * In this case, use a relative precision of 24 bits.
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| 72 | * (This is done differently in the original paper)
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| 73 | *
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| 74 | * Special cases:
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| 75 | * 0 return signalling -Inf
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| 76 | * neg return signalling NaN
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| 77 | * +Inf return +Inf
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| 78 | */
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| 79 |
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| 80 | #define endian (((*(int *) &one)) ? 1 : 0)
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| 81 | #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000
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| 82 |
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| 83 | #define N 128
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| 84 |
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| 85 | /* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
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| 86 | * Used for generation of extend precision logarithms.
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| 87 | * The constant 35184372088832 is 2^45, so the divide is exact.
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| 88 | * It ensures correct reading of logF_head, even for inaccurate
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| 89 | * decimal-to-binary conversion routines. (Everybody gets the
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| 90 | * right answer for integers less than 2^53.)
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| 91 | * Values for log(F) were generated using error < 10^-57 absolute
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| 92 | * with the bc -l package.
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| 93 | */
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| 94 | static double A1 = .08333333333333178827;
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| 95 | static double A2 = .01250000000377174923;
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| 96 | static double A3 = .002232139987919447809;
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| 97 | static double A4 = .0004348877777076145742;
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| 98 |
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| 99 | static double logF_head[N+1] = {
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| 100 | 0.,
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| 101 | .007782140442060381246,
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| 102 | .015504186535963526694,
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| 103 | .023167059281547608406,
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| 104 | .030771658666765233647,
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| 105 | .038318864302141264488,
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| 106 | .045809536031242714670,
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| 107 | .053244514518837604555,
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| 108 | .060624621816486978786,
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| 109 | .067950661908525944454,
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| 110 | .075223421237524235039,
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| 111 | .082443669210988446138,
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| 112 | .089612158689760690322,
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| 113 | .096729626458454731618,
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| 114 | .103796793681567578460,
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| 115 | .110814366340264314203,
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| 116 | .117783035656430001836,
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| 117 | .124703478501032805070,
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| 118 | .131576357788617315236,
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| 119 | .138402322859292326029,
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| 120 | .145182009844575077295,
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| 121 | .151916042025732167530,
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| 122 | .158605030176659056451,
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| 123 | .165249572895390883786,
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| 124 | .171850256926518341060,
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| 125 | .178407657472689606947,
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| 126 | .184922338493834104156,
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| 127 | .191394852999565046047,
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| 128 | .197825743329758552135,
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| 129 | .204215541428766300668,
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| 130 | .210564769107350002741,
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| 131 | .216873938300523150246,
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| 132 | .223143551314024080056,
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| 133 | .229374101064877322642,
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| 134 | .235566071312860003672,
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| 135 | .241719936886966024758,
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| 136 | .247836163904594286577,
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| 137 | .253915209980732470285,
|
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| 138 | .259957524436686071567,
|
|---|
| 139 | .265963548496984003577,
|
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| 140 | .271933715484010463114,
|
|---|
| 141 | .277868451003087102435,
|
|---|
| 142 | .283768173130738432519,
|
|---|
| 143 | .289633292582948342896,
|
|---|
| 144 | .295464212893421063199,
|
|---|
| 145 | .301261330578199704177,
|
|---|
| 146 | .307025035294827830512,
|
|---|
| 147 | .312755710004239517729,
|
|---|
| 148 | .318453731118097493890,
|
|---|
| 149 | .324119468654316733591,
|
|---|
| 150 | .329753286372579168528,
|
|---|
| 151 | .335355541920762334484,
|
|---|
| 152 | .340926586970454081892,
|
|---|
| 153 | .346466767346100823488,
|
|---|
| 154 | .351976423156884266063,
|
|---|
| 155 | .357455888922231679316,
|
|---|
| 156 | .362905493689140712376,
|
|---|
| 157 | .368325561158599157352,
|
|---|
| 158 | .373716409793814818840,
|
|---|
| 159 | .379078352934811846353,
|
|---|
| 160 | .384411698910298582632,
|
|---|
| 161 | .389716751140440464951,
|
|---|
| 162 | .394993808240542421117,
|
|---|
| 163 | .400243164127459749579,
|
|---|
| 164 | .405465108107819105498,
|
|---|
| 165 | .410659924985338875558,
|
|---|
| 166 | .415827895143593195825,
|
|---|
| 167 | .420969294644237379543,
|
|---|
| 168 | .426084395310681429691,
|
|---|
| 169 | .431173464818130014464,
|
|---|
| 170 | .436236766774527495726,
|
|---|
| 171 | .441274560805140936281,
|
|---|
| 172 | .446287102628048160113,
|
|---|
| 173 | .451274644139630254358,
|
|---|
| 174 | .456237433481874177232,
|
|---|
| 175 | .461175715122408291790,
|
|---|
| 176 | .466089729924533457960,
|
|---|
| 177 | .470979715219073113985,
|
|---|
| 178 | .475845904869856894947,
|
|---|
| 179 | .480688529345570714212,
|
|---|
| 180 | .485507815781602403149,
|
|---|
| 181 | .490303988045525329653,
|
|---|
| 182 | .495077266798034543171,
|
|---|
| 183 | .499827869556611403822,
|
|---|
| 184 | .504556010751912253908,
|
|---|
| 185 | .509261901790523552335,
|
|---|
| 186 | .513945751101346104405,
|
|---|
| 187 | .518607764208354637958,
|
|---|
| 188 | .523248143765158602036,
|
|---|
| 189 | .527867089620485785417,
|
|---|
| 190 | .532464798869114019908,
|
|---|
| 191 | .537041465897345915436,
|
|---|
| 192 | .541597282432121573947,
|
|---|
| 193 | .546132437597407260909,
|
|---|
| 194 | .550647117952394182793,
|
|---|
| 195 | .555141507540611200965,
|
|---|
| 196 | .559615787935399566777,
|
|---|
| 197 | .564070138285387656651,
|
|---|
| 198 | .568504735352689749561,
|
|---|
| 199 | .572919753562018740922,
|
|---|
| 200 | .577315365035246941260,
|
|---|
| 201 | .581691739635061821900,
|
|---|
| 202 | .586049045003164792433,
|
|---|
| 203 | .590387446602107957005,
|
|---|
| 204 | .594707107746216934174,
|
|---|
| 205 | .599008189645246602594,
|
|---|
| 206 | .603290851438941899687,
|
|---|
| 207 | .607555250224322662688,
|
|---|
| 208 | .611801541106615331955,
|
|---|
| 209 | .616029877215623855590,
|
|---|
| 210 | .620240409751204424537,
|
|---|
| 211 | .624433288012369303032,
|
|---|
| 212 | .628608659422752680256,
|
|---|
| 213 | .632766669570628437213,
|
|---|
| 214 | .636907462236194987781,
|
|---|
| 215 | .641031179420679109171,
|
|---|
| 216 | .645137961373620782978,
|
|---|
| 217 | .649227946625615004450,
|
|---|
| 218 | .653301272011958644725,
|
|---|
| 219 | .657358072709030238911,
|
|---|
| 220 | .661398482245203922502,
|
|---|
| 221 | .665422632544505177065,
|
|---|
| 222 | .669430653942981734871,
|
|---|
| 223 | .673422675212350441142,
|
|---|
| 224 | .677398823590920073911,
|
|---|
| 225 | .681359224807238206267,
|
|---|
| 226 | .685304003098281100392,
|
|---|
| 227 | .689233281238557538017,
|
|---|
| 228 | .693147180560117703862
|
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| 229 | };
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| 230 |
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| 231 | static double logF_tail[N+1] = {
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| 232 | 0.,
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| 233 | -.00000000000000543229938420049,
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| 234 | .00000000000000172745674997061,
|
|---|
| 235 | -.00000000000001323017818229233,
|
|---|
| 236 | -.00000000000001154527628289872,
|
|---|
| 237 | -.00000000000000466529469958300,
|
|---|
| 238 | .00000000000005148849572685810,
|
|---|
| 239 | -.00000000000002532168943117445,
|
|---|
| 240 | -.00000000000005213620639136504,
|
|---|
| 241 | -.00000000000001819506003016881,
|
|---|
| 242 | .00000000000006329065958724544,
|
|---|
| 243 | .00000000000008614512936087814,
|
|---|
| 244 | -.00000000000007355770219435028,
|
|---|
| 245 | .00000000000009638067658552277,
|
|---|
| 246 | .00000000000007598636597194141,
|
|---|
| 247 | .00000000000002579999128306990,
|
|---|
| 248 | -.00000000000004654729747598444,
|
|---|
| 249 | -.00000000000007556920687451336,
|
|---|
| 250 | .00000000000010195735223708472,
|
|---|
| 251 | -.00000000000017319034406422306,
|
|---|
| 252 | -.00000000000007718001336828098,
|
|---|
| 253 | .00000000000010980754099855238,
|
|---|
| 254 | -.00000000000002047235780046195,
|
|---|
| 255 | -.00000000000008372091099235912,
|
|---|
| 256 | .00000000000014088127937111135,
|
|---|
| 257 | .00000000000012869017157588257,
|
|---|
| 258 | .00000000000017788850778198106,
|
|---|
| 259 | .00000000000006440856150696891,
|
|---|
| 260 | .00000000000016132822667240822,
|
|---|
| 261 | -.00000000000007540916511956188,
|
|---|
| 262 | -.00000000000000036507188831790,
|
|---|
| 263 | .00000000000009120937249914984,
|
|---|
| 264 | .00000000000018567570959796010,
|
|---|
| 265 | -.00000000000003149265065191483,
|
|---|
| 266 | -.00000000000009309459495196889,
|
|---|
| 267 | .00000000000017914338601329117,
|
|---|
| 268 | -.00000000000001302979717330866,
|
|---|
| 269 | .00000000000023097385217586939,
|
|---|
| 270 | .00000000000023999540484211737,
|
|---|
| 271 | .00000000000015393776174455408,
|
|---|
| 272 | -.00000000000036870428315837678,
|
|---|
| 273 | .00000000000036920375082080089,
|
|---|
| 274 | -.00000000000009383417223663699,
|
|---|
| 275 | .00000000000009433398189512690,
|
|---|
| 276 | .00000000000041481318704258568,
|
|---|
| 277 | -.00000000000003792316480209314,
|
|---|
| 278 | .00000000000008403156304792424,
|
|---|
| 279 | -.00000000000034262934348285429,
|
|---|
| 280 | .00000000000043712191957429145,
|
|---|
| 281 | -.00000000000010475750058776541,
|
|---|
| 282 | -.00000000000011118671389559323,
|
|---|
| 283 | .00000000000037549577257259853,
|
|---|
| 284 | .00000000000013912841212197565,
|
|---|
| 285 | .00000000000010775743037572640,
|
|---|
| 286 | .00000000000029391859187648000,
|
|---|
| 287 | -.00000000000042790509060060774,
|
|---|
| 288 | .00000000000022774076114039555,
|
|---|
| 289 | .00000000000010849569622967912,
|
|---|
| 290 | -.00000000000023073801945705758,
|
|---|
| 291 | .00000000000015761203773969435,
|
|---|
| 292 | .00000000000003345710269544082,
|
|---|
| 293 | -.00000000000041525158063436123,
|
|---|
| 294 | .00000000000032655698896907146,
|
|---|
| 295 | -.00000000000044704265010452446,
|
|---|
| 296 | .00000000000034527647952039772,
|
|---|
| 297 | -.00000000000007048962392109746,
|
|---|
| 298 | .00000000000011776978751369214,
|
|---|
| 299 | -.00000000000010774341461609578,
|
|---|
| 300 | .00000000000021863343293215910,
|
|---|
| 301 | .00000000000024132639491333131,
|
|---|
| 302 | .00000000000039057462209830700,
|
|---|
| 303 | -.00000000000026570679203560751,
|
|---|
| 304 | .00000000000037135141919592021,
|
|---|
| 305 | -.00000000000017166921336082431,
|
|---|
| 306 | -.00000000000028658285157914353,
|
|---|
| 307 | -.00000000000023812542263446809,
|
|---|
| 308 | .00000000000006576659768580062,
|
|---|
| 309 | -.00000000000028210143846181267,
|
|---|
| 310 | .00000000000010701931762114254,
|
|---|
| 311 | .00000000000018119346366441110,
|
|---|
| 312 | .00000000000009840465278232627,
|
|---|
| 313 | -.00000000000033149150282752542,
|
|---|
| 314 | -.00000000000018302857356041668,
|
|---|
| 315 | -.00000000000016207400156744949,
|
|---|
| 316 | .00000000000048303314949553201,
|
|---|
| 317 | -.00000000000071560553172382115,
|
|---|
| 318 | .00000000000088821239518571855,
|
|---|
| 319 | -.00000000000030900580513238244,
|
|---|
| 320 | -.00000000000061076551972851496,
|
|---|
| 321 | .00000000000035659969663347830,
|
|---|
| 322 | .00000000000035782396591276383,
|
|---|
| 323 | -.00000000000046226087001544578,
|
|---|
| 324 | .00000000000062279762917225156,
|
|---|
| 325 | .00000000000072838947272065741,
|
|---|
| 326 | .00000000000026809646615211673,
|
|---|
| 327 | -.00000000000010960825046059278,
|
|---|
| 328 | .00000000000002311949383800537,
|
|---|
| 329 | -.00000000000058469058005299247,
|
|---|
| 330 | -.00000000000002103748251144494,
|
|---|
| 331 | -.00000000000023323182945587408,
|
|---|
| 332 | -.00000000000042333694288141916,
|
|---|
| 333 | -.00000000000043933937969737844,
|
|---|
| 334 | .00000000000041341647073835565,
|
|---|
| 335 | .00000000000006841763641591466,
|
|---|
| 336 | .00000000000047585534004430641,
|
|---|
| 337 | .00000000000083679678674757695,
|
|---|
| 338 | -.00000000000085763734646658640,
|
|---|
| 339 | .00000000000021913281229340092,
|
|---|
| 340 | -.00000000000062242842536431148,
|
|---|
| 341 | -.00000000000010983594325438430,
|
|---|
| 342 | .00000000000065310431377633651,
|
|---|
| 343 | -.00000000000047580199021710769,
|
|---|
| 344 | -.00000000000037854251265457040,
|
|---|
| 345 | .00000000000040939233218678664,
|
|---|
| 346 | .00000000000087424383914858291,
|
|---|
| 347 | .00000000000025218188456842882,
|
|---|
| 348 | -.00000000000003608131360422557,
|
|---|
| 349 | -.00000000000050518555924280902,
|
|---|
| 350 | .00000000000078699403323355317,
|
|---|
| 351 | -.00000000000067020876961949060,
|
|---|
| 352 | .00000000000016108575753932458,
|
|---|
| 353 | .00000000000058527188436251509,
|
|---|
| 354 | -.00000000000035246757297904791,
|
|---|
| 355 | -.00000000000018372084495629058,
|
|---|
| 356 | .00000000000088606689813494916,
|
|---|
| 357 | .00000000000066486268071468700,
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| 358 | .00000000000063831615170646519,
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| 359 | .00000000000025144230728376072,
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| 360 | -.00000000000017239444525614834
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| 361 | };
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| 362 |
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| 363 | #if 0
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| 364 | double
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| 365 | #ifdef _ANSI_SOURCE
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| 366 | log(double x)
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| 367 | #else
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| 368 | log(x) double x;
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| 369 | #endif
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| 370 | {
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| 371 | int m, j;
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| 372 | double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0;
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| 373 | volatile double u1;
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| 374 |
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| 375 | /* Catch special cases */
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| 376 | if (x <= 0)
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| 377 | if (x == zero) /* log(0) = -Inf */
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| 378 | return (-one/zero);
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| 379 | else /* log(neg) = NaN */
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| 380 | return (zero/zero);
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| 381 | else if (!finite(x))
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| 382 | return (x+x); /* x = NaN, Inf */
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| 383 |
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| 384 | /* Argument reduction: 1 <= g < 2; x/2^m = g; */
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| 385 | /* y = F*(1 + f/F) for |f| <= 2^-8 */
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| 386 |
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| 387 | m = logb(x);
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| 388 | g = ldexp(x, -m);
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| 389 | if (m == -1022) {
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| 390 | j = logb(g), m += j;
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| 391 | g = ldexp(g, -j);
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| 392 | }
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| 393 | j = N*(g-1) + .5;
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| 394 | F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */
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| 395 | f = g - F;
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| 396 |
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| 397 | /* Approximate expansion for log(1+f/F) ~= u + q */
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| 398 | g = 1/(2*F+f);
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| 399 | u = 2*f*g;
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| 400 | v = u*u;
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| 401 | q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
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| 402 |
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| 403 | /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8,
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| 404 | * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits.
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| 405 | * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750
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| 406 | */
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| 407 | if (m | j)
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| 408 | u1 = u + 513, u1 -= 513;
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| 409 |
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| 410 | /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero;
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| 411 | * u1 = u to 24 bits.
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| 412 | */
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| 413 | else
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| 414 | u1 = u, TRUNC(u1);
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| 415 | u2 = (2.0*(f - F*u1) - u1*f) * g;
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| 416 | /* u1 + u2 = 2f/(2F+f) to extra precision. */
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| 417 |
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| 418 | /* log(x) = log(2^m*F*(1+f/F)) = */
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| 419 | /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */
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| 420 | /* (exact) + (tiny) */
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| 421 |
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| 422 | u1 += m*logF_head[N] + logF_head[j]; /* exact */
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| 423 | u2 = (u2 + logF_tail[j]) + q; /* tiny */
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| 424 | u2 += logF_tail[N]*m;
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| 425 | return (u1 + u2);
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| 426 | }
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| 427 | #endif
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| 428 |
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| 429 | /*
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| 430 | * Extra precision variant, returning struct {double a, b;};
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| 431 | * log(x) = a+b to 63 bits, with a rounded to 26 bits.
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| 432 | */
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| 433 | struct Double
|
|---|
| 434 | #ifdef _ANSI_SOURCE
|
|---|
| 435 | __log__D(double x)
|
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| 436 | #else
|
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| 437 | __log__D(x) double x;
|
|---|
| 438 | #endif
|
|---|
| 439 | {
|
|---|
| 440 | int m, j;
|
|---|
| 441 | double F, f, g, q, u, v, u2, one = 1.0;
|
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| 442 | volatile double u1;
|
|---|
| 443 | struct Double r;
|
|---|
| 444 |
|
|---|
| 445 | /* Argument reduction: 1 <= g < 2; x/2^m = g; */
|
|---|
| 446 | /* y = F*(1 + f/F) for |f| <= 2^-8 */
|
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| 447 |
|
|---|
| 448 | m = logb(x);
|
|---|
| 449 | g = ldexp(x, -m);
|
|---|
| 450 | if (m == -1022) {
|
|---|
| 451 | j = logb(g), m += j;
|
|---|
| 452 | g = ldexp(g, -j);
|
|---|
| 453 | }
|
|---|
| 454 | j = N*(g-1) + .5;
|
|---|
| 455 | F = (1.0/N) * j + 1;
|
|---|
| 456 | f = g - F;
|
|---|
| 457 |
|
|---|
| 458 | g = 1/(2*F+f);
|
|---|
| 459 | u = 2*f*g;
|
|---|
| 460 | v = u*u;
|
|---|
| 461 | q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
|
|---|
| 462 | if (m | j)
|
|---|
| 463 | u1 = u + 513, u1 -= 513;
|
|---|
| 464 | else
|
|---|
| 465 | u1 = u, TRUNC(u1);
|
|---|
| 466 | u2 = (2.0*(f - F*u1) - u1*f) * g;
|
|---|
| 467 |
|
|---|
| 468 | u1 += m*logF_head[N] + logF_head[j];
|
|---|
| 469 |
|
|---|
| 470 | u2 += logF_tail[j]; u2 += q;
|
|---|
| 471 | u2 += logF_tail[N]*m;
|
|---|
| 472 | r.a = u1 + u2; /* Only difference is here */
|
|---|
| 473 | TRUNC(r.a);
|
|---|
| 474 | r.b = (u1 - r.a) + u2;
|
|---|
| 475 | return (r);
|
|---|
| 476 | }
|
|---|
