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177
libm/bsdsrc/b_exp.c Normal file
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/*
* Copyright (c) 1985, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* This product includes software developed by the University of
* California, Berkeley and its contributors.
* 4. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#ifndef lint
static char sccsid[] = "@(#)exp.c 8.1 (Berkeley) 6/4/93";
#endif /* not lint */
#include <sys/cdefs.h>
/* __FBSDID("$FreeBSD: src/lib/msun/bsdsrc/b_exp.c,v 1.7 2004/12/16 20:40:37 das Exp $"); */
/* EXP(X)
* RETURN THE EXPONENTIAL OF X
* DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
* CODED IN C BY K.C. NG, 1/19/85;
* REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
*
* Required system supported functions:
* scalb(x,n)
* copysign(x,y)
* finite(x)
*
* Method:
* 1. Argument Reduction: given the input x, find r and integer k such
* that
* x = k*ln2 + r, |r| <= 0.5*ln2 .
* r will be represented as r := z+c for better accuracy.
*
* 2. Compute exp(r) by
*
* exp(r) = 1 + r + r*R1/(2-R1),
* where
* R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
*
* 3. exp(x) = 2^k * exp(r) .
*
* Special cases:
* exp(INF) is INF, exp(NaN) is NaN;
* exp(-INF)= 0;
* for finite argument, only exp(0)=1 is exact.
*
* Accuracy:
* exp(x) returns the exponential of x nearly rounded. In a test run
* with 1,156,000 random arguments on a VAX, the maximum observed
* error was 0.869 ulps (units in the last place).
*/
#include "mathimpl.h"
const static double p1 = 0x1.555555555553ep-3;
const static double p2 = -0x1.6c16c16bebd93p-9;
const static double p3 = 0x1.1566aaf25de2cp-14;
const static double p4 = -0x1.bbd41c5d26bf1p-20;
const static double p5 = 0x1.6376972bea4d0p-25;
const static double ln2hi = 0x1.62e42fee00000p-1;
const static double ln2lo = 0x1.a39ef35793c76p-33;
const static double lnhuge = 0x1.6602b15b7ecf2p9;
const static double lntiny = -0x1.77af8ebeae354p9;
const static double invln2 = 0x1.71547652b82fep0;
#if 0
double exp(x)
double x;
{
double z,hi,lo,c;
int k;
#if !defined(vax)&&!defined(tahoe)
if(x!=x) return(x); /* x is NaN */
#endif /* !defined(vax)&&!defined(tahoe) */
if( x <= lnhuge ) {
if( x >= lntiny ) {
/* argument reduction : x --> x - k*ln2 */
k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */
/* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
hi=x-k*ln2hi;
x=hi-(lo=k*ln2lo);
/* return 2^k*[1+x+x*c/(2+c)] */
z=x*x;
c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
}
/* end of x > lntiny */
else
/* exp(-big#) underflows to zero */
if(finite(x)) return(scalb(1.0,-5000));
/* exp(-INF) is zero */
else return(0.0);
}
/* end of x < lnhuge */
else
/* exp(INF) is INF, exp(+big#) overflows to INF */
return( finite(x) ? scalb(1.0,5000) : x);
}
#endif
/* returns exp(r = x + c) for |c| < |x| with no overlap. */
double __exp__D(x, c)
double x, c;
{
double z,hi,lo;
int k;
if (x != x) /* x is NaN */
return(x);
if ( x <= lnhuge ) {
if ( x >= lntiny ) {
/* argument reduction : x --> x - k*ln2 */
z = invln2*x;
k = z + copysign(.5, x);
/* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
hi=(x-k*ln2hi); /* Exact. */
x= hi - (lo = k*ln2lo-c);
/* return 2^k*[1+x+x*c/(2+c)] */
z=x*x;
c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
c = (x*c)/(2.0-c);
return scalb(1.+(hi-(lo - c)), k);
}
/* end of x > lntiny */
else
/* exp(-big#) underflows to zero */
if(finite(x)) return(scalb(1.0,-5000));
/* exp(-INF) is zero */
else return(0.0);
}
/* end of x < lnhuge */
else
/* exp(INF) is INF, exp(+big#) overflows to INF */
return( finite(x) ? scalb(1.0,5000) : x);
}

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/*
* Copyright (c) 1992, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* This product includes software developed by the University of
* California, Berkeley and its contributors.
* 4. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#ifndef lint
static char sccsid[] = "@(#)log.c 8.2 (Berkeley) 11/30/93";
#endif /* not lint */
#include <sys/cdefs.h>
/* __FBSDID("$FreeBSD: src/lib/msun/bsdsrc/b_log.c,v 1.8 2005/09/19 11:28:19 bde Exp $"); */
#include <math.h>
#include <errno.h>
#include "mathimpl.h"
/* Table-driven natural logarithm.
*
* This code was derived, with minor modifications, from:
* Peter Tang, "Table-Driven Implementation of the
* Logarithm in IEEE Floating-Point arithmetic." ACM Trans.
* Math Software, vol 16. no 4, pp 378-400, Dec 1990).
*
* Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256,
* where F = j/128 for j an integer in [0, 128].
*
* log(2^m) = log2_hi*m + log2_tail*m
* since m is an integer, the dominant term is exact.
* m has at most 10 digits (for subnormal numbers),
* and log2_hi has 11 trailing zero bits.
*
* log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
* logF_hi[] + 512 is exact.
*
* log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ...
* the leading term is calculated to extra precision in two
* parts, the larger of which adds exactly to the dominant
* m and F terms.
* There are two cases:
* 1. when m, j are non-zero (m | j), use absolute
* precision for the leading term.
* 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
* In this case, use a relative precision of 24 bits.
* (This is done differently in the original paper)
*
* Special cases:
* 0 return signalling -Inf
* neg return signalling NaN
* +Inf return +Inf
*/
#define N 128
/* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
* Used for generation of extend precision logarithms.
* The constant 35184372088832 is 2^45, so the divide is exact.
* It ensures correct reading of logF_head, even for inaccurate
* decimal-to-binary conversion routines. (Everybody gets the
* right answer for integers less than 2^53.)
* Values for log(F) were generated using error < 10^-57 absolute
* with the bc -l package.
*/
static double A1 = .08333333333333178827;
static double A2 = .01250000000377174923;
static double A3 = .002232139987919447809;
static double A4 = .0004348877777076145742;
static double logF_head[N+1] = {
0.,
.007782140442060381246,
.015504186535963526694,
.023167059281547608406,
.030771658666765233647,
.038318864302141264488,
.045809536031242714670,
.053244514518837604555,
.060624621816486978786,
.067950661908525944454,
.075223421237524235039,
.082443669210988446138,
.089612158689760690322,
.096729626458454731618,
.103796793681567578460,
.110814366340264314203,
.117783035656430001836,
.124703478501032805070,
.131576357788617315236,
.138402322859292326029,
.145182009844575077295,
.151916042025732167530,
.158605030176659056451,
.165249572895390883786,
.171850256926518341060,
.178407657472689606947,
.184922338493834104156,
.191394852999565046047,
.197825743329758552135,
.204215541428766300668,
.210564769107350002741,
.216873938300523150246,
.223143551314024080056,
.229374101064877322642,
.235566071312860003672,
.241719936886966024758,
.247836163904594286577,
.253915209980732470285,
.259957524436686071567,
.265963548496984003577,
.271933715484010463114,
.277868451003087102435,
.283768173130738432519,
.289633292582948342896,
.295464212893421063199,
.301261330578199704177,
.307025035294827830512,
.312755710004239517729,
.318453731118097493890,
.324119468654316733591,
.329753286372579168528,
.335355541920762334484,
.340926586970454081892,
.346466767346100823488,
.351976423156884266063,
.357455888922231679316,
.362905493689140712376,
.368325561158599157352,
.373716409793814818840,
.379078352934811846353,
.384411698910298582632,
.389716751140440464951,
.394993808240542421117,
.400243164127459749579,
.405465108107819105498,
.410659924985338875558,
.415827895143593195825,
.420969294644237379543,
.426084395310681429691,
.431173464818130014464,
.436236766774527495726,
.441274560805140936281,
.446287102628048160113,
.451274644139630254358,
.456237433481874177232,
.461175715122408291790,
.466089729924533457960,
.470979715219073113985,
.475845904869856894947,
.480688529345570714212,
.485507815781602403149,
.490303988045525329653,
.495077266798034543171,
.499827869556611403822,
.504556010751912253908,
.509261901790523552335,
.513945751101346104405,
.518607764208354637958,
.523248143765158602036,
.527867089620485785417,
.532464798869114019908,
.537041465897345915436,
.541597282432121573947,
.546132437597407260909,
.550647117952394182793,
.555141507540611200965,
.559615787935399566777,
.564070138285387656651,
.568504735352689749561,
.572919753562018740922,
.577315365035246941260,
.581691739635061821900,
.586049045003164792433,
.590387446602107957005,
.594707107746216934174,
.599008189645246602594,
.603290851438941899687,
.607555250224322662688,
.611801541106615331955,
.616029877215623855590,
.620240409751204424537,
.624433288012369303032,
.628608659422752680256,
.632766669570628437213,
.636907462236194987781,
.641031179420679109171,
.645137961373620782978,
.649227946625615004450,
.653301272011958644725,
.657358072709030238911,
.661398482245203922502,
.665422632544505177065,
.669430653942981734871,
.673422675212350441142,
.677398823590920073911,
.681359224807238206267,
.685304003098281100392,
.689233281238557538017,
.693147180560117703862
};
static double logF_tail[N+1] = {
0.,
-.00000000000000543229938420049,
.00000000000000172745674997061,
-.00000000000001323017818229233,
-.00000000000001154527628289872,
-.00000000000000466529469958300,
.00000000000005148849572685810,
-.00000000000002532168943117445,
-.00000000000005213620639136504,
-.00000000000001819506003016881,
.00000000000006329065958724544,
.00000000000008614512936087814,
-.00000000000007355770219435028,
.00000000000009638067658552277,
.00000000000007598636597194141,
.00000000000002579999128306990,
-.00000000000004654729747598444,
-.00000000000007556920687451336,
.00000000000010195735223708472,
-.00000000000017319034406422306,
-.00000000000007718001336828098,
.00000000000010980754099855238,
-.00000000000002047235780046195,
-.00000000000008372091099235912,
.00000000000014088127937111135,
.00000000000012869017157588257,
.00000000000017788850778198106,
.00000000000006440856150696891,
.00000000000016132822667240822,
-.00000000000007540916511956188,
-.00000000000000036507188831790,
.00000000000009120937249914984,
.00000000000018567570959796010,
-.00000000000003149265065191483,
-.00000000000009309459495196889,
.00000000000017914338601329117,
-.00000000000001302979717330866,
.00000000000023097385217586939,
.00000000000023999540484211737,
.00000000000015393776174455408,
-.00000000000036870428315837678,
.00000000000036920375082080089,
-.00000000000009383417223663699,
.00000000000009433398189512690,
.00000000000041481318704258568,
-.00000000000003792316480209314,
.00000000000008403156304792424,
-.00000000000034262934348285429,
.00000000000043712191957429145,
-.00000000000010475750058776541,
-.00000000000011118671389559323,
.00000000000037549577257259853,
.00000000000013912841212197565,
.00000000000010775743037572640,
.00000000000029391859187648000,
-.00000000000042790509060060774,
.00000000000022774076114039555,
.00000000000010849569622967912,
-.00000000000023073801945705758,
.00000000000015761203773969435,
.00000000000003345710269544082,
-.00000000000041525158063436123,
.00000000000032655698896907146,
-.00000000000044704265010452446,
.00000000000034527647952039772,
-.00000000000007048962392109746,
.00000000000011776978751369214,
-.00000000000010774341461609578,
.00000000000021863343293215910,
.00000000000024132639491333131,
.00000000000039057462209830700,
-.00000000000026570679203560751,
.00000000000037135141919592021,
-.00000000000017166921336082431,
-.00000000000028658285157914353,
-.00000000000023812542263446809,
.00000000000006576659768580062,
-.00000000000028210143846181267,
.00000000000010701931762114254,
.00000000000018119346366441110,
.00000000000009840465278232627,
-.00000000000033149150282752542,
-.00000000000018302857356041668,
-.00000000000016207400156744949,
.00000000000048303314949553201,
-.00000000000071560553172382115,
.00000000000088821239518571855,
-.00000000000030900580513238244,
-.00000000000061076551972851496,
.00000000000035659969663347830,
.00000000000035782396591276383,
-.00000000000046226087001544578,
.00000000000062279762917225156,
.00000000000072838947272065741,
.00000000000026809646615211673,
-.00000000000010960825046059278,
.00000000000002311949383800537,
-.00000000000058469058005299247,
-.00000000000002103748251144494,
-.00000000000023323182945587408,
-.00000000000042333694288141916,
-.00000000000043933937969737844,
.00000000000041341647073835565,
.00000000000006841763641591466,
.00000000000047585534004430641,
.00000000000083679678674757695,
-.00000000000085763734646658640,
.00000000000021913281229340092,
-.00000000000062242842536431148,
-.00000000000010983594325438430,
.00000000000065310431377633651,
-.00000000000047580199021710769,
-.00000000000037854251265457040,
.00000000000040939233218678664,
.00000000000087424383914858291,
.00000000000025218188456842882,
-.00000000000003608131360422557,
-.00000000000050518555924280902,
.00000000000078699403323355317,
-.00000000000067020876961949060,
.00000000000016108575753932458,
.00000000000058527188436251509,
-.00000000000035246757297904791,
-.00000000000018372084495629058,
.00000000000088606689813494916,
.00000000000066486268071468700,
.00000000000063831615170646519,
.00000000000025144230728376072,
-.00000000000017239444525614834
};
#if 0
double
#ifdef _ANSI_SOURCE
log(double x)
#else
log(x) double x;
#endif
{
int m, j;
double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0;
volatile double u1;
/* Catch special cases */
if (x <= 0)
if (x == zero) /* log(0) = -Inf */
return (-one/zero);
else /* log(neg) = NaN */
return (zero/zero);
else if (!finite(x))
return (x+x); /* x = NaN, Inf */
/* Argument reduction: 1 <= g < 2; x/2^m = g; */
/* y = F*(1 + f/F) for |f| <= 2^-8 */
m = logb(x);
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*128 is an integer in [128, 512] */
f = g - F;
/* Approximate expansion for log(1+f/F) ~= u + q */
g = 1/(2*F+f);
u = 2*f*g;
v = u*u;
q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
/* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8,
* u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits.
* It also adds exactly to |m*log2_hi + log_F_head[j] | < 750
*/
if (m | j)
u1 = u + 513, u1 -= 513;
/* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero;
* u1 = u to 24 bits.
*/
else
u1 = u, TRUNC(u1);
u2 = (2.0*(f - F*u1) - u1*f) * g;
/* u1 + u2 = 2f/(2F+f) to extra precision. */
/* log(x) = log(2^m*F*(1+f/F)) = */
/* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */
/* (exact) + (tiny) */
u1 += m*logF_head[N] + logF_head[j]; /* exact */
u2 = (u2 + logF_tail[j]) + q; /* tiny */
u2 += logF_tail[N]*m;
return (u1 + u2);
}
#endif
/*
* Extra precision variant, returning struct {double a, b;};
* log(x) = a+b to 63 bits, with a rounded to 26 bits.
*/
struct Double
#ifdef _ANSI_SOURCE
__log__D(double x)
#else
__log__D(x) double x;
#endif
{
int m, j;
double F, f, g, q, u, v, u2;
volatile double u1;
struct Double r;
/* Argument reduction: 1 <= g < 2; x/2^m = g; */
/* y = F*(1 + f/F) for |f| <= 2^-8 */
m = logb(x);
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);
}

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/*-
* Copyright (c) 1992, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* This product includes software developed by the University of
* California, Berkeley and its contributors.
* 4. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#ifndef lint
static char sccsid[] = "@(#)gamma.c 8.1 (Berkeley) 6/4/93";
#endif /* not lint */
#include <sys/cdefs.h>
/* __FBSDID("$FreeBSD: src/lib/msun/bsdsrc/b_tgamma.c,v 1.7 2005/09/19 11:28:19 bde Exp $"); */
/*
* This code by P. McIlroy, Oct 1992;
*
* The financial support of UUNET Communications Services is greatfully
* acknowledged.
*/
//#include <math.h>
#include "../include/math.h"
#include "mathimpl.h"
#include <errno.h>
/* METHOD:
* x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
* At negative integers, return +Inf, and set errno.
*
* x < 6.5:
* Use argument reduction G(x+1) = xG(x) to reach the
* range [1.066124,2.066124]. Use a rational
* approximation centered at the minimum (x0+1) to
* ensure monotonicity.
*
* x >= 6.5: Use the asymptotic approximation (Stirling's formula)
* adjusted for equal-ripples:
*
* log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
*
* Keep extra precision in multiplying (x-.5)(log(x)-1), to
* avoid premature round-off.
*
* Special values:
* non-positive integer: Set overflow trap; return +Inf;
* x > 171.63: Set overflow trap; return +Inf;
* NaN: Set invalid trap; return NaN
*
* Accuracy: Gamma(x) is accurate to within
* x > 0: error provably < 0.9ulp.
* Maximum observed in 1,000,000 trials was .87ulp.
* x < 0:
* Maximum observed error < 4ulp in 1,000,000 trials.
*/
static double neg_gam(double);
static double small_gam(double);
static double smaller_gam(double);
static struct Double large_gam(double);
static struct Double ratfun_gam(double, double);
/*
* Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
* [1.066.., 2.066..] accurate to 4.25e-19.
*/
#define LEFT -.3955078125 /* left boundary for rat. approx */
#define x0 .461632144968362356785 /* xmin - 1 */
#define a0_hi 0.88560319441088874992
#define a0_lo -.00000000000000004996427036469019695
#define P0 6.21389571821820863029017800727e-01
#define P1 2.65757198651533466104979197553e-01
#define P2 5.53859446429917461063308081748e-03
#define P3 1.38456698304096573887145282811e-03
#define P4 2.40659950032711365819348969808e-03
#define Q0 1.45019531250000000000000000000e+00
#define Q1 1.06258521948016171343454061571e+00
#define Q2 -2.07474561943859936441469926649e-01
#define Q3 -1.46734131782005422506287573015e-01
#define Q4 3.07878176156175520361557573779e-02
#define Q5 5.12449347980666221336054633184e-03
#define Q6 -1.76012741431666995019222898833e-03
#define Q7 9.35021023573788935372153030556e-05
#define Q8 6.13275507472443958924745652239e-06
/*
* Constants for large x approximation (x in [6, Inf])
* (Accurate to 2.8*10^-19 absolute)
*/
#define lns2pi_hi 0.418945312500000
#define lns2pi_lo -.000006779295327258219670263595
#define Pa0 8.33333333333333148296162562474e-02
#define Pa1 -2.77777777774548123579378966497e-03
#define Pa2 7.93650778754435631476282786423e-04
#define Pa3 -5.95235082566672847950717262222e-04
#define Pa4 8.41428560346653702135821806252e-04
#define Pa5 -1.89773526463879200348872089421e-03
#define Pa6 5.69394463439411649408050664078e-03
#define Pa7 -1.44705562421428915453880392761e-02
static const double zero = 0., one = 1.0, tiny = 1e-300;
double
tgamma(x)
double x;
{
struct Double u;
if (x >= 6) {
if(x > 171.63)
return(one/zero);
u = large_gam(x);
return(__exp__D(u.a, u.b));
} else if (x >= 1.0 + LEFT + x0)
return (small_gam(x));
else if (x > 1.e-17)
return (smaller_gam(x));
else if (x > -1.e-17) {
if (x == 0.0)
return (one/x);
one+1e-20; /* Raise inexact flag. */
return (one/x);
} else if (!finite(x))
return (x*x); /* x = NaN, -Inf */
else
return (neg_gam(x));
}
/*
* Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
*/
static struct Double
large_gam(x)
double x;
{
double z, p;
struct Double t, u, v;
z = one/(x*x);
p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
p = p/x;
u = __log__D(x);
u.a -= one;
v.a = (x -= .5);
TRUNC(v.a);
v.b = x - v.a;
t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */
t.b = v.b*u.a + x*u.b;
/* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
t.b += lns2pi_lo; t.b += p;
u.a = lns2pi_hi + t.b; u.a += t.a;
u.b = t.a - u.a;
u.b += lns2pi_hi; u.b += t.b;
return (u);
}
/*
* Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
* It also has correct monotonicity.
*/
static double
small_gam(x)
double x;
{
double y, ym1, t;
struct Double yy, r;
y = x - one;
ym1 = y - one;
if (y <= 1.0 + (LEFT + x0)) {
yy = ratfun_gam(y - x0, 0);
return (yy.a + yy.b);
}
r.a = y;
TRUNC(r.a);
yy.a = r.a - one;
y = ym1;
yy.b = r.b = y - yy.a;
/* Argument reduction: G(x+1) = x*G(x) */
for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
t = r.a*yy.a;
r.b = r.a*yy.b + y*r.b;
r.a = t;
TRUNC(r.a);
r.b += (t - r.a);
}
/* Return r*tgamma(y). */
yy = ratfun_gam(y - x0, 0);
y = r.b*(yy.a + yy.b) + r.a*yy.b;
y += yy.a*r.a;
return (y);
}
/*
* Good on (0, 1+x0+LEFT]. Accurate to 1ulp.
*/
static double
smaller_gam(x)
double x;
{
double t, d;
struct Double r, xx;
if (x < x0 + LEFT) {
t = x, TRUNC(t);
d = (t+x)*(x-t);
t *= t;
xx.a = (t + x), TRUNC(xx.a);
xx.b = x - xx.a; xx.b += t; xx.b += d;
t = (one-x0); t += x;
d = (one-x0); d -= t; d += x;
x = xx.a + xx.b;
} else {
xx.a = x, TRUNC(xx.a);
xx.b = x - xx.a;
t = x - x0;
d = (-x0 -t); d += x;
}
r = ratfun_gam(t, d);
d = r.a/x, TRUNC(d);
r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
return (d + r.a/x);
}
/*
* returns (z+c)^2 * P(z)/Q(z) + a0
*/
static struct Double
ratfun_gam(z, c)
double z, c;
{
double p, q;
struct Double r, t;
q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
/* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
p = p/q;
t.a = z, TRUNC(t.a); /* t ~= z + c */
t.b = (z - t.a) + c;
t.b *= (t.a + z);
q = (t.a *= t.a); /* t = (z+c)^2 */
TRUNC(t.a);
t.b += (q - t.a);
r.a = p, TRUNC(r.a); /* r = P/Q */
r.b = p - r.a;
t.b = t.b*p + t.a*r.b + a0_lo;
t.a *= r.a; /* t = (z+c)^2*(P/Q) */
r.a = t.a + a0_hi, TRUNC(r.a);
r.b = ((a0_hi-r.a) + t.a) + t.b;
return (r); /* r = a0 + t */
}
static double
neg_gam(x)
double x;
{
int sgn = 1;
struct Double lg, lsine;
double y, z;
y = floor(x + .5);
if (y == x) /* Negative integer. */
return (one/zero);
z = fabs(x - y);
y = .5*ceil(x);
if (y == ceil(y))
sgn = -1;
if (z < .25)
z = sin(M_PI*z);
else
z = cos(M_PI*(0.5-z));
/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
if (x < -170) {
if (x < -190)
return ((double)sgn*tiny*tiny);
y = one - x; /* exact: 128 < |x| < 255 */
lg = large_gam(y);
lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */
lg.a -= lsine.a; /* exact (opposite signs) */
lg.b -= lsine.b;
y = -(lg.a + lg.b);
z = (y + lg.a) + lg.b;
y = __exp__D(y, z);
if (sgn < 0) y = -y;
return (y);
}
y = one-x;
if (one-y == x)
y = tgamma(y);
else /* 1-x is inexact */
y = -x*tgamma(-x);
if (sgn < 0) y = -y;
return (M_PI / (y*z));
}

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/*
* Copyright (c) 1988, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* This product includes software developed by the University of
* California, Berkeley and its contributors.
* 4. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*
* @(#)mathimpl.h 8.1 (Berkeley) 6/4/93
* $FreeBSD: src/lib/msun/bsdsrc/mathimpl.h,v 1.7 2005/11/18 05:03:12 bde Exp $
*/
#ifndef _MATHIMPL_H_
#define _MATHIMPL_H_
#include <sys/cdefs.h>
#include <math.h>
#include "../src/math_private.h"
/*
* TRUNC() is a macro that sets the trailing 27 bits in the mantissa of an
* IEEE double variable to zero. It must be expression-like for syntactic
* reasons, and we implement this expression using an inline function
* instead of a pure macro to avoid depending on the gcc feature of
* statement-expressions.
*/
#define TRUNC(d) (_b_trunc(&(d)))
static __inline void
_b_trunc(volatile double *_dp)
{
uint32_t _lw;
GET_LOW_WORD(_lw, *_dp);
SET_LOW_WORD(*_dp, _lw & 0xf8000000);
}
struct Double {
double a;
double b;
};
/*
* Functions internal to the math package, yet not static.
*/
double __exp__D(double, double);
struct Double __log__D(double);
#endif /* !_MATHIMPL_H_ */