Index: head/lib/msun/bsdsrc/b_exp.c =================================================================== --- head/lib/msun/bsdsrc/b_exp.c (revision 356105) +++ head/lib/msun/bsdsrc/b_exp.c (revision 356106) @@ -1,177 +1,173 @@ /*- - * SPDX-License-Identifier: BSD-4-Clause + * SPDX-License-Identifier: BSD-3-Clause * * 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 + * 3. 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. */ /* @(#)exp.c 8.1 (Berkeley) 6/4/93 */ #include __FBSDID("$FreeBSD$"); /* 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" static const double p1 = 0x1.555555555553ep-3; static const double p2 = -0x1.6c16c16bebd93p-9; static const double p3 = 0x1.1566aaf25de2cp-14; static const double p4 = -0x1.bbd41c5d26bf1p-20; static const double p5 = 0x1.6376972bea4d0p-25; static const double ln2hi = 0x1.62e42fee00000p-1; static const double ln2lo = 0x1.a39ef35793c76p-33; static const double lnhuge = 0x1.6602b15b7ecf2p9; static const double lntiny = -0x1.77af8ebeae354p9; static const 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); } Index: head/lib/msun/bsdsrc/b_log.c =================================================================== --- head/lib/msun/bsdsrc/b_log.c (revision 356105) +++ head/lib/msun/bsdsrc/b_log.c (revision 356106) @@ -1,472 +1,468 @@ /*- - * SPDX-License-Identifier: BSD-4-Clause + * SPDX-License-Identifier: BSD-3-Clause * * 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 + * 3. 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. */ /* @(#)log.c 8.2 (Berkeley) 11/30/93 */ #include __FBSDID("$FreeBSD$"); #include #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); } Index: head/lib/msun/bsdsrc/b_tgamma.c =================================================================== --- head/lib/msun/bsdsrc/b_tgamma.c (revision 356105) +++ head/lib/msun/bsdsrc/b_tgamma.c (revision 356106) @@ -1,319 +1,315 @@ /*- - * SPDX-License-Identifier: BSD-4-Clause + * SPDX-License-Identifier: BSD-3-Clause * * 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 + * 3. 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. */ /* @(#)gamma.c 8.1 (Berkeley) 6/4/93 */ #include __FBSDID("$FreeBSD$"); /* * This code by P. McIlroy, Oct 1992; * * The financial support of UUNET Communications Services is greatfully * acknowledged. */ #include #include "mathimpl.h" /* METHOD: * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)) * At negative integers, return NaN and raise invalid. * * 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: * -Inf: return NaN and raise invalid; * negative integer: return NaN and raise invalid; * other x ~< 177.79: return +-0 and raise underflow; * +-0: return +-Inf and raise divide-by-zero; * finite x ~> 171.63: return +Inf and raise overflow; * +Inf: return +Inf; * NaN: return NaN. * * Accuracy: tgamma(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 (x / 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) u.a = one - tiny; /* raise inexact */ return (one/x); } else if (!finite(x)) return (x - x); /* x is NaN or -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 = ceil(x); if (y == x) /* Negative integer. */ return ((x - x) / zero); z = y - x; if (z > 0.5) z = one - z; y = 0.5 * y; 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)); }