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gammafunction.d
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gammafunction.d
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/**
* Implementation of the gamma and beta functions, and their integrals.
*
* License: $(HTTP boost.org/LICENSE_1_0.txt, Boost License 1.0).
* Copyright: Based on the CEPHES math library, which is
* Copyright (C) 1994 Stephen L. Moshier (moshier@world.std.com).
* Authors: Stephen L. Moshier (original C code). Conversion to D by Don Clugston
*
*
Macros:
* TABLE_SV = <table border="1" cellpadding="4" cellspacing="0">
* <caption>Special Values</caption>
* $0</table>
* SVH = $(TR $(TH $1) $(TH $2))
* SV = $(TR $(TD $1) $(TD $2))
* GAMMA = Γ
* INTEGRATE = $(BIG ∫<sub>$(SMALL $1)</sub><sup>$2</sup>)
* POWER = $1<sup>$2</sup>
* NAN = $(RED NAN)
*/
module std.internal.math.gammafunction;
import std.internal.math.errorfunction;
import std.math;
import core.math : fabs, sin, sqrt;
pure:
nothrow:
@safe:
@nogc:
private {
enum real SQRT2PI = 2.50662827463100050242E0L; // sqrt(2pi)
immutable real EULERGAMMA = 0.57721_56649_01532_86060_65120_90082_40243_10421_59335_93992L; /** Euler-Mascheroni constant 0.57721566.. */
// Polynomial approximations for gamma and loggamma.
immutable real[8] GammaNumeratorCoeffs = [ 1.0L,
0x1.acf42d903366539ep-1L, 0x1.73a991c8475f1aeap-2L, 0x1.c7e918751d6b2a92p-4L,
0x1.86d162cca32cfe86p-6L, 0x1.0c378e2e6eaf7cd8p-8L, 0x1.dc5c66b7d05feb54p-12L,
0x1.616457b47e448694p-15L
];
immutable real[9] GammaDenominatorCoeffs = [ 1.0L,
0x1.a8f9faae5d8fc8bp-2L, -0x1.cb7895a6756eebdep-3L, -0x1.7b9bab006d30652ap-5L,
0x1.c671af78f312082ep-6L, -0x1.a11ebbfaf96252dcp-11L, -0x1.447b4d2230a77ddap-10L,
0x1.ec1d45bb85e06696p-13L,-0x1.d4ce24d05bd0a8e6p-17L
];
immutable real[9] GammaSmallCoeffs = [ 1.0L,
0x1.2788cfc6fb618f52p-1L, -0x1.4fcf4026afa2f7ecp-1L, -0x1.5815e8fa24d7e306p-5L,
0x1.5512320aea2ad71ap-3L, -0x1.59af0fb9d82e216p-5L, -0x1.3b4b61d3bfdf244ap-7L,
0x1.d9358e9d9d69fd34p-8L, -0x1.38fc4bcbada775d6p-10L
];
immutable real[9] GammaSmallNegCoeffs = [ -1.0L,
0x1.2788cfc6fb618f54p-1L, 0x1.4fcf4026afa2bc4cp-1L, -0x1.5815e8fa2468fec8p-5L,
-0x1.5512320baedaf4b6p-3L, -0x1.59af0fa283baf07ep-5L, 0x1.3b4a70de31e05942p-7L,
0x1.d9398be3bad13136p-8L, 0x1.291b73ee05bcbba2p-10L
];
immutable real[7] logGammaStirlingCoeffs = [
0x1.5555555555553f98p-4L, -0x1.6c16c16c07509b1p-9L, 0x1.a01a012461cbf1e4p-11L,
-0x1.3813089d3f9d164p-11L, 0x1.b911a92555a277b8p-11L, -0x1.ed0a7b4206087b22p-10L,
0x1.402523859811b308p-8L
];
immutable real[7] logGammaNumerator = [
-0x1.0edd25913aaa40a2p+23L, -0x1.31c6ce2e58842d1ep+24L, -0x1.f015814039477c3p+23L,
-0x1.74ffe40c4b184b34p+22L, -0x1.0d9c6d08f9eab55p+20L, -0x1.54c6b71935f1fc88p+16L,
-0x1.0e761b42932b2aaep+11L
];
immutable real[8] logGammaDenominator = [
-0x1.4055572d75d08c56p+24L, -0x1.deeb6013998e4d76p+24L, -0x1.106f7cded5dcc79ep+24L,
-0x1.25e17184848c66d2p+22L, -0x1.301303b99a614a0ap+19L, -0x1.09e76ab41ae965p+15L,
-0x1.00f95ced9e5f54eep+9L, 1.0L
];
/*
* Helper function: Gamma function computed by Stirling's formula.
*
* Stirling's formula for the gamma function is:
*
* $(GAMMA)(x) = sqrt(2 π) x<sup>x-0.5</sup> exp(-x) (1 + 1/x P(1/x))
*
*/
real gammaStirling(real x)
{
// CEPHES code Copyright 1994 by Stephen L. Moshier
static immutable real[9] SmallStirlingCoeffs = [
0x1.55555555555543aap-4L, 0x1.c71c71c720dd8792p-9L, -0x1.5f7268f0b5907438p-9L,
-0x1.e13cd410e0477de6p-13L, 0x1.9b0f31643442616ep-11L, 0x1.2527623a3472ae08p-14L,
-0x1.37f6bc8ef8b374dep-11L,-0x1.8c968886052b872ap-16L, 0x1.76baa9c6d3eeddbcp-11L
];
static immutable real[7] LargeStirlingCoeffs = [ 1.0L,
8.33333333333333333333E-2L, 3.47222222222222222222E-3L,
-2.68132716049382716049E-3L, -2.29472093621399176955E-4L,
7.84039221720066627474E-4L, 6.97281375836585777429E-5L
];
real w = 1.0L/x;
real y = exp(x);
if ( x > 1024.0L )
{
// For large x, use rational coefficients from the analytical expansion.
w = poly(w, LargeStirlingCoeffs);
// Avoid overflow in pow()
real v = pow( x, 0.5L * x - 0.25L );
y = v * (v / y);
}
else
{
w = 1.0L + w * poly( w, SmallStirlingCoeffs);
static if (floatTraits!(real).realFormat == RealFormat.ieeeDouble)
{
// Avoid overflow in pow() for 64-bit reals
if (x > 143.0)
{
real v = pow( x, 0.5 * x - 0.25 );
y = v * (v / y);
}
else
{
y = pow( x, x - 0.5 ) / y;
}
}
else
{
y = pow( x, x - 0.5L ) / y;
}
}
y = SQRT2PI * y * w;
return y;
}
/*
* Helper function: Incomplete gamma function computed by Temme's expansion.
*
* This is a port of igamma_temme_large from Boost.
*
*/
real igammaTemmeLarge(real a, real x)
{
static immutable real[][13] coef = [
[ -0.333333333333333333333L, 0.0833333333333333333333L,
-0.0148148148148148148148L, 0.00115740740740740740741L,
0.000352733686067019400353L, -0.0001787551440329218107L,
0.39192631785224377817e-4L, -0.218544851067999216147e-5L,
-0.18540622107151599607e-5L, 0.829671134095308600502e-6L,
-0.176659527368260793044e-6L, 0.670785354340149858037e-8L,
0.102618097842403080426e-7L, -0.438203601845335318655e-8L,
0.914769958223679023418e-9L, -0.255141939949462497669e-10L,
-0.583077213255042506746e-10L, 0.243619480206674162437e-10L,
-0.502766928011417558909e-11L ],
[ -0.00185185185185185185185L, -0.00347222222222222222222L,
0.00264550264550264550265L, -0.000990226337448559670782L,
0.000205761316872427983539L, -0.40187757201646090535e-6L,
-0.18098550334489977837e-4L, 0.764916091608111008464e-5L,
-0.161209008945634460038e-5L, 0.464712780280743434226e-8L,
0.137863344691572095931e-6L, -0.575254560351770496402e-7L,
0.119516285997781473243e-7L, -0.175432417197476476238e-10L,
-0.100915437106004126275e-8L, 0.416279299184258263623e-9L,
-0.856390702649298063807e-10L ],
[ 0.00413359788359788359788L, -0.00268132716049382716049L,
0.000771604938271604938272L, 0.200938786008230452675e-5L,
-0.000107366532263651605215L, 0.529234488291201254164e-4L,
-0.127606351886187277134e-4L, 0.342357873409613807419e-7L,
0.137219573090629332056e-5L, -0.629899213838005502291e-6L,
0.142806142060642417916e-6L, -0.204770984219908660149e-9L,
-0.140925299108675210533e-7L, 0.622897408492202203356e-8L,
-0.136704883966171134993e-8L ],
[ 0.000649434156378600823045L, 0.000229472093621399176955L,
-0.000469189494395255712128L, 0.000267720632062838852962L,
-0.756180167188397641073e-4L, -0.239650511386729665193e-6L,
0.110826541153473023615e-4L, -0.56749528269915965675e-5L,
0.142309007324358839146e-5L, -0.278610802915281422406e-10L,
-0.169584040919302772899e-6L, 0.809946490538808236335e-7L,
-0.191111684859736540607e-7L ],
[ -0.000861888290916711698605L, 0.000784039221720066627474L,
-0.000299072480303190179733L, -0.146384525788434181781e-5L,
0.664149821546512218666e-4L, -0.396836504717943466443e-4L,
0.113757269706784190981e-4L, 0.250749722623753280165e-9L,
-0.169541495365583060147e-5L, 0.890750753220530968883e-6L,
-0.229293483400080487057e-6L ],
[ -0.000336798553366358150309L, -0.697281375836585777429e-4L,
0.000277275324495939207873L, -0.000199325705161888477003L,
0.679778047793720783882e-4L, 0.141906292064396701483e-6L,
-0.135940481897686932785e-4L, 0.801847025633420153972e-5L,
-0.229148117650809517038e-5L ],
[ 0.000531307936463992223166L, -0.000592166437353693882865L,
0.000270878209671804482771L, 0.790235323266032787212e-6L,
-0.815396936756196875093e-4L, 0.561168275310624965004e-4L,
-0.183291165828433755673e-4L, -0.307961345060330478256e-8L,
0.346515536880360908674e-5L, -0.20291327396058603727e-5L,
0.57887928631490037089e-6L ],
[ 0.000344367606892377671254L, 0.517179090826059219337e-4L,
-0.000334931610811422363117L, 0.000281269515476323702274L,
-0.000109765822446847310235L, -0.127410090954844853795e-6L,
0.277444515115636441571e-4L, -0.182634888057113326614e-4L,
0.578769494973505239894e-5L ],
[ -0.000652623918595309418922L, 0.000839498720672087279993L,
-0.000438297098541721005061L, -0.696909145842055197137e-6L,
0.000166448466420675478374L, -0.000127835176797692185853L,
0.462995326369130429061e-4L ],
[ -0.000596761290192746250124L, -0.720489541602001055909e-4L,
0.000678230883766732836162L, -0.0006401475260262758451L,
0.000277501076343287044992L ],
[ 0.00133244544948006563713L, -0.0019144384985654775265L,
0.00110893691345966373396L ],
[ 0.00157972766073083495909L, 0.000162516262783915816899L,
-0.00206334210355432762645L, 0.00213896861856890981541L,
-0.00101085593912630031708L ],
[ -0.00407251211951401664727L, 0.00640336283380806979482L,
-0.00404101610816766177474L ]
];
// avoid nans when one of the arguments is inf:
if (x == real.infinity && a != real.infinity)
return 0;
if (x != real.infinity && a == real.infinity)
return 1;
real sigma = (x - a) / a;
real phi = sigma - log(sigma + 1);
real y = a * phi;
real z = sqrt(2 * phi);
if (x < a)
z = -z;
real[13] workspace;
foreach (i; 0 .. coef.length)
workspace[i] = poly(z, coef[i]);
real result = poly(1 / a, workspace);
result *= exp(-y) / sqrt(2 * PI * a);
if (x < a)
result = -result;
result += erfc(sqrt(y)) / 2;
return result;
}
} // private
public:
/// The maximum value of x for which gamma(x) < real.infinity.
static if (floatTraits!(real).realFormat == RealFormat.ieeeQuadruple)
enum real MAXGAMMA = 1755.5483429L;
else static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended)
enum real MAXGAMMA = 1755.5483429L;
else static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended53)
enum real MAXGAMMA = 1755.5483429L;
else static if (floatTraits!(real).realFormat == RealFormat.ieeeDouble)
enum real MAXGAMMA = 171.6243769L;
else
static assert(0, "missing MAXGAMMA for other real types");
/*****************************************************
* The Gamma function, $(GAMMA)(x)
*
* $(GAMMA)(x) is a generalisation of the factorial function
* to real and complex numbers.
* Like x!, $(GAMMA)(x+1) = x*$(GAMMA)(x).
*
* Mathematically, if z.re > 0 then
* $(GAMMA)(z) = $(INTEGRATE 0, ∞) $(POWER t, z-1)$(POWER e, -t) dt
*
* $(TABLE_SV
* $(SVH x, $(GAMMA)(x) )
* $(SV $(NAN), $(NAN) )
* $(SV ±0.0, ±∞)
* $(SV integer > 0, (x-1)! )
* $(SV integer < 0, $(NAN) )
* $(SV +∞, +∞ )
* $(SV -∞, $(NAN) )
* )
*/
real gamma(real x)
{
/* Based on code from the CEPHES library.
* CEPHES code Copyright 1994 by Stephen L. Moshier
*
* Arguments |x| <= 13 are reduced by recurrence and the function
* approximated by a rational function of degree 7/8 in the
* interval (2,3). Large arguments are handled by Stirling's
* formula. Large negative arguments are made positive using
* a reflection formula.
*/
real q, z;
if (isNaN(x)) return x;
if (x == -x.infinity) return real.nan;
if ( fabs(x) > MAXGAMMA ) return real.infinity;
if (x == 0) return 1.0 / x; // +- infinity depending on sign of x, create an exception.
q = fabs(x);
if ( q > 13.0L )
{
// Large arguments are handled by Stirling's
// formula. Large negative arguments are made positive using
// the reflection formula.
if ( x < 0.0L )
{
if (x < -1/real.epsilon)
{
// Large negatives lose all precision
return real.nan;
}
int sgngam = 1; // sign of gamma.
long intpart = cast(long)(q);
if (q == intpart)
return real.nan; // poles for all integers <0.
real p = intpart;
if ( (intpart & 1) == 0 )
sgngam = -1;
z = q - p;
if ( z > 0.5L )
{
p += 1.0L;
z = q - p;
}
z = q * sin( PI * z );
z = fabs(z) * gammaStirling(q);
if ( z <= PI/real.max ) return sgngam * real.infinity;
return sgngam * PI/z;
}
else
{
return gammaStirling(x);
}
}
// Arguments |x| <= 13 are reduced by recurrence and the function
// approximated by a rational function of degree 7/8 in the
// interval (2,3).
z = 1.0L;
while ( x >= 3.0L )
{
x -= 1.0L;
z *= x;
}
while ( x < -0.03125L )
{
z /= x;
x += 1.0L;
}
if ( x <= 0.03125L )
{
if ( x == 0.0L )
return real.nan;
else
{
if ( x < 0.0L )
{
x = -x;
return z / (x * poly( x, GammaSmallNegCoeffs ));
}
else
{
return z / (x * poly( x, GammaSmallCoeffs ));
}
}
}
while ( x < 2.0L )
{
z /= x;
x += 1.0L;
}
if ( x == 2.0L ) return z;
x -= 2.0L;
return z * poly( x, GammaNumeratorCoeffs ) / poly( x, GammaDenominatorCoeffs );
}
@safe unittest
{
// gamma(n) = factorial(n-1) if n is an integer.
real fact = 1.0L;
for (int i=1; fact<real.max; ++i)
{
// Require exact equality for small factorials
if (i<14) assert(gamma(i*1.0L) == fact);
assert(feqrel(gamma(i*1.0L), fact) >= real.mant_dig-15);
fact *= (i*1.0L);
}
assert(gamma(0.0) == real.infinity);
assert(gamma(-0.0) == -real.infinity);
assert(isNaN(gamma(-1.0)));
assert(isNaN(gamma(-15.0)));
assert(isIdentical(gamma(NaN(0xABC)), NaN(0xABC)));
assert(gamma(real.infinity) == real.infinity);
assert(gamma(real.max) == real.infinity);
assert(isNaN(gamma(-real.infinity)));
assert(gamma(real.min_normal*real.epsilon) == real.infinity);
assert(gamma(MAXGAMMA)< real.infinity);
assert(gamma(MAXGAMMA*2) == real.infinity);
// Test some high-precision values (50 decimal digits)
real SQRT_PI = 1.77245385090551602729816748334114518279754945612238L;
assert(feqrel(gamma(0.5L), SQRT_PI) >= real.mant_dig-1);
assert(feqrel(gamma(17.25L), 4.224986665692703551570937158682064589938e13L) >= real.mant_dig-4);
assert(feqrel(gamma(1.0 / 3.0L), 2.67893853470774763365569294097467764412868937795730L) >= real.mant_dig-2);
assert(feqrel(gamma(0.25L),
3.62560990822190831193068515586767200299516768288006L) >= real.mant_dig-1);
assert(feqrel(gamma(1.0 / 5.0L),
4.59084371199880305320475827592915200343410999829340L) >= real.mant_dig-1);
}
/*****************************************************
* Natural logarithm of gamma function.
*
* Returns the base e (2.718...) logarithm of the absolute
* value of the gamma function of the argument.
*
* For reals, logGamma is equivalent to log(fabs(gamma(x))).
*
* $(TABLE_SV
* $(SVH x, logGamma(x) )
* $(SV $(NAN), $(NAN) )
* $(SV integer <= 0, +∞ )
* $(SV ±∞, +∞ )
* )
*/
real logGamma(real x)
{
/* Based on code from the CEPHES library.
* CEPHES code Copyright 1994 by Stephen L. Moshier
*
* For arguments greater than 33, the logarithm of the gamma
* function is approximated by the logarithmic version of
* Stirling's formula using a polynomial approximation of
* degree 4. Arguments between -33 and +33 are reduced by
* recurrence to the interval [2,3] of a rational approximation.
* The cosecant reflection formula is employed for arguments
* less than -33.
*/
real q, w, z, f, nx;
if (isNaN(x)) return x;
if (fabs(x) == x.infinity) return x.infinity;
if ( x < -34.0L )
{
q = -x;
w = logGamma(q);
real p = floor(q);
if ( p == q )
return real.infinity;
int intpart = cast(int)(p);
real sgngam = 1;
if ( (intpart & 1) == 0 )
sgngam = -1;
z = q - p;
if ( z > 0.5L )
{
p += 1.0L;
z = p - q;
}
z = q * sin( PI * z );
if ( z == 0.0L )
return sgngam * real.infinity;
/* z = LOGPI - logl( z ) - w; */
z = log( PI/z ) - w;
return z;
}
if ( x < 13.0L )
{
z = 1.0L;
nx = floor( x + 0.5L );
f = x - nx;
while ( x >= 3.0L )
{
nx -= 1.0L;
x = nx + f;
z *= x;
}
while ( x < 2.0L )
{
if ( fabs(x) <= 0.03125L )
{
if ( x == 0.0L )
return real.infinity;
if ( x < 0.0L )
{
x = -x;
q = z / (x * poly( x, GammaSmallNegCoeffs));
} else
q = z / (x * poly( x, GammaSmallCoeffs));
return log( fabs(q) );
}
z /= nx + f;
nx += 1.0L;
x = nx + f;
}
z = fabs(z);
if ( x == 2.0L )
return log(z);
x = (nx - 2.0L) + f;
real p = x * rationalPoly( x, logGammaNumerator, logGammaDenominator);
return log(z) + p;
}
// const real MAXLGM = 1.04848146839019521116e+4928L;
// if ( x > MAXLGM ) return sgngaml * real.infinity;
const real LOGSQRT2PI = 0.91893853320467274178L; // log( sqrt( 2*pi ) )
q = ( x - 0.5L ) * log(x) - x + LOGSQRT2PI;
if (x > 1.0e10L) return q;
real p = 1.0L / (x*x);
q += poly( p, logGammaStirlingCoeffs ) / x;
return q ;
}
@safe unittest
{
assert(isIdentical(logGamma(NaN(0xDEF)), NaN(0xDEF)));
assert(logGamma(real.infinity) == real.infinity);
assert(logGamma(-1.0) == real.infinity);
assert(logGamma(0.0) == real.infinity);
assert(logGamma(-50.0) == real.infinity);
assert(isIdentical(0.0L, logGamma(1.0L)));
assert(isIdentical(0.0L, logGamma(2.0L)));
assert(logGamma(real.min_normal*real.epsilon) == real.infinity);
assert(logGamma(-real.min_normal*real.epsilon) == real.infinity);
// x, correct loggamma(x), correct d/dx loggamma(x).
immutable static real[] testpoints = [
8.0L, 8.525146484375L + 1.48766904143001655310E-5, 2.01564147795560999654E0L,
8.99993896484375e-1L, 6.6375732421875e-2L + 5.11505711292524166220E-6L, -7.54938684259372234258E-1,
7.31597900390625e-1L, 2.2369384765625e-1 + 5.21506341809849792422E-6L, -1.13355566660398608343E0L,
2.31639862060546875e-1L, 1.3686676025390625L + 1.12609441752996145670E-5L, -4.56670961813812679012E0,
1.73162841796875L, -8.88214111328125e-2L + 3.36207740803753034508E-6L, 2.33339034686200586920E-1L,
1.23162841796875L, -9.3902587890625e-2L + 1.28765089229009648104E-5L, -2.49677345775751390414E-1L,
7.3786976294838206464e19L, 3.301798506038663053312e21L - 1.656137564136932662487046269677E5L,
4.57477139169563904215E1L,
1.08420217248550443401E-19L, 4.36682586669921875e1L + 1.37082843669932230418E-5L,
-9.22337203685477580858E18L,
1.0L, 0.0L, -5.77215664901532860607E-1L,
2.0L, 0.0L, 4.22784335098467139393E-1L,
-0.5L, 1.2655029296875L + 9.19379714539648894580E-6L, 3.64899739785765205590E-2L,
-1.5L, 8.6004638671875e-1L + 6.28657731014510932682E-7L, 7.03156640645243187226E-1L,
-2.5L, -5.6243896484375E-2L + 1.79986700949327405470E-7, 1.10315664064524318723E0L,
-3.5L, -1.30902099609375L + 1.43111007079536392848E-5L, 1.38887092635952890151E0L
];
// TODO: test derivatives as well.
for (int i=0; i<testpoints.length; i+=3)
{
assert( feqrel(logGamma(testpoints[i]), testpoints[i+1]) > real.mant_dig-5);
if (testpoints[i]<MAXGAMMA)
{
assert( feqrel(log(fabs(gamma(testpoints[i]))), testpoints[i+1]) > real.mant_dig-5);
}
}
assert(feqrel(logGamma(-50.2L),log(fabs(gamma(-50.2L)))) > real.mant_dig-2);
assert(feqrel(logGamma(-0.008L),log(fabs(gamma(-0.008L)))) > real.mant_dig-2);
assert(feqrel(logGamma(-38.8L),log(fabs(gamma(-38.8L)))) > real.mant_dig-4);
static if (real.mant_dig >= 64) // incl. 80-bit reals
assert(feqrel(logGamma(1500.0L),log(gamma(1500.0L))) > real.mant_dig-2);
else static if (real.mant_dig >= 53) // incl. 64-bit reals
assert(feqrel(logGamma(150.0L),log(gamma(150.0L))) > real.mant_dig-2);
}
private {
/*
* These value can be calculated like this:
* 1) Get exact real.max/min_normal/epsilon from compiler:
* writefln!"%a"(real.max/min_normal_epsilon)
* 2) Convert for Wolfram Alpha
* 0xf.fffffffffffffffp+16380 ==> (f.fffffffffffffff base 16) * 2^16380
* 3) Calculate result on wofram alpha:
* http://www.wolframalpha.com/input/?i=ln((1.ffffffffffffffffffffffffffff+base+16)+*+2%5E16383)+in+base+2
* 4) Convert to proper format:
* string mantissa = "1.011...";
* write(mantissa[0 .. 2]); mantissa = mantissa[2 .. $];
* for (size_t i = 0; i < mantissa.length/4; i++)
* {
* writef!"%x"(to!ubyte(mantissa[0 .. 4], 2)); mantissa = mantissa[4 .. $];
* }
*/
static if (floatTraits!(real).realFormat == RealFormat.ieeeQuadruple)
{
enum real MAXLOG = 0x1.62e42fefa39ef35793c7673007e6p+13L; // log(real.max)
enum real MINLOG = -0x1.6546282207802c89d24d65e96274p+13L; // log(real.min_normal*real.epsilon) = log(smallest denormal)
}
else static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended)
{
enum real MAXLOG = 0x1.62e42fefa39ef358p+13L; // log(real.max)
enum real MINLOG = -0x1.6436716d5406e6d8p+13L; // log(real.min_normal*real.epsilon) = log(smallest denormal)
}
else static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended53)
{
enum real MAXLOG = 0x1.62e42fefa39ef358p+13L; // log(real.max)
enum real MINLOG = -0x1.6436716d5406e6d8p+13L; // log(real.min_normal*real.epsilon) = log(smallest denormal)
}
else static if (floatTraits!(real).realFormat == RealFormat.ieeeDouble)
{
enum real MAXLOG = 0x1.62e42fefa39efp+9L; // log(real.max)
enum real MINLOG = -0x1.74385446d71c3p+9L; // log(real.min_normal*real.epsilon) = log(smallest denormal)
}
else
static assert(0, "missing MAXLOG and MINLOG for other real types");
enum real BETA_BIG = 9.223372036854775808e18L;
enum real BETA_BIGINV = 1.084202172485504434007e-19L;
}
/** Incomplete beta integral
*
* Returns incomplete beta integral of the arguments, evaluated
* from zero to x. The regularized incomplete beta function is defined as
*
* betaIncomplete(a, b, x) = Γ(a+b)/(Γ(a) Γ(b)) *
* $(INTEGRATE 0, x) $(POWER t, a-1)$(POWER (1-t),b-1) dt
*
* and is the same as the cumulative distribution function.
*
* The domain of definition is 0 <= x <= 1. In this
* implementation a and b are restricted to positive values.
* The integral from x to 1 may be obtained by the symmetry
* relation
*
* betaIncompleteCompl(a, b, x ) = betaIncomplete( b, a, 1-x )
*
* The integral is evaluated by a continued fraction expansion
* or, when b*x is small, by a power series.
*/
real betaIncomplete(real aa, real bb, real xx )
{
if ( !(aa>0 && bb>0) )
{
if ( isNaN(aa) ) return aa;
if ( isNaN(bb) ) return bb;
return real.nan; // domain error
}
if (!(xx>0 && xx<1.0))
{
if (isNaN(xx)) return xx;
if ( xx == 0.0L ) return 0.0;
if ( xx == 1.0L ) return 1.0;
return real.nan; // domain error
}
if ( (bb * xx) <= 1.0L && xx <= 0.95L)
{
return betaDistPowerSeries(aa, bb, xx);
}
real x;
real xc; // = 1 - x
real a, b;
int flag = 0;
/* Reverse a and b if x is greater than the mean. */
if ( xx > (aa/(aa+bb)) )
{
// here x > aa/(aa+bb) and (bb*x>1 or x>0.95)
flag = 1;
a = bb;
b = aa;
xc = xx;
x = 1.0L - xx;
}
else
{
a = aa;
b = bb;
xc = 1.0L - xx;
x = xx;
}
if ( flag == 1 && (b * x) <= 1.0L && x <= 0.95L)
{
// here xx > aa/(aa+bb) and ((bb*xx>1) or xx>0.95) and (aa*(1-xx)<=1) and xx > 0.05
return 1.0 - betaDistPowerSeries(a, b, x); // note loss of precision
}
real w;
// Choose expansion for optimal convergence
// One is for x * (a+b+2) < (a+1),
// the other is for x * (a+b+2) > (a+1).
real y = x * (a+b-2.0L) - (a-1.0L);
if ( y < 0.0L )
{
w = betaDistExpansion1( a, b, x );
}
else
{
w = betaDistExpansion2( a, b, x ) / xc;
}
/* Multiply w by the factor
a b
x (1-x) Gamma(a+b) / ( a Gamma(a) Gamma(b) ) . */
y = a * log(x);
real t = b * log(xc);
if ( (a+b) < MAXGAMMA && fabs(y) < MAXLOG && fabs(t) < MAXLOG )
{
t = pow(xc,b);
t *= pow(x,a);
t /= a;
t *= w;
t *= gamma(a+b) / (gamma(a) * gamma(b));
}
else
{
/* Resort to logarithms. */
y += t + logGamma(a+b) - logGamma(a) - logGamma(b);
y += log(w/a);
t = exp(y);
/+
// There seems to be a bug in Cephes at this point.
// Problems occur for y > MAXLOG, not y < MINLOG.
if ( y < MINLOG )
{
t = 0.0L;
}
else
{
t = exp(y);
}
+/
}
if ( flag == 1 )
{
/+ // CEPHES includes this code, but I think it is erroneous.
if ( t <= real.epsilon )
{
t = 1.0L - real.epsilon;
} else
+/
t = 1.0L - t;
}
return t;
}
/** Inverse of incomplete beta integral
*
* Given y, the function finds x such that
*
* betaIncomplete(a, b, x) == y
*
* Newton iterations or interval halving is used.
*/
real betaIncompleteInv(real aa, real bb, real yy0 )
{
real a, b, y0, d, y, x, x0, x1, lgm, yp, di, dithresh, yl, yh, xt;
int i, rflg, dir, nflg;
if (isNaN(yy0)) return yy0;
if (isNaN(aa)) return aa;
if (isNaN(bb)) return bb;
if ( yy0 <= 0.0L )
return 0.0L;
if ( yy0 >= 1.0L )
return 1.0L;
x0 = 0.0L;
yl = 0.0L;
x1 = 1.0L;
yh = 1.0L;
if ( aa <= 1.0L || bb <= 1.0L )
{
dithresh = 1.0e-7L;
rflg = 0;
a = aa;
b = bb;
y0 = yy0;
x = a/(a+b);
y = betaIncomplete( a, b, x );
nflg = 0;
goto ihalve;
}
else
{
nflg = 0;
dithresh = 1.0e-4L;
}
// approximation to inverse function
yp = -normalDistributionInvImpl( yy0 );
if ( yy0 > 0.5L )
{
rflg = 1;
a = bb;
b = aa;
y0 = 1.0L - yy0;
yp = -yp;
}
else
{
rflg = 0;
a = aa;
b = bb;
y0 = yy0;
}
lgm = (yp * yp - 3.0L)/6.0L;
x = 2.0L/( 1.0L/(2.0L * a-1.0L) + 1.0L/(2.0L * b - 1.0L) );
d = yp * sqrt( x + lgm ) / x
- ( 1.0L/(2.0L * b - 1.0L) - 1.0L/(2.0L * a - 1.0L) )
* (lgm + (5.0L/6.0L) - 2.0L/(3.0L * x));
d = 2.0L * d;
if ( d < MINLOG )
{
x = 1.0L;
goto under;
}
x = a/( a + b * exp(d) );
y = betaIncomplete( a, b, x );
yp = (y - y0)/y0;
if ( fabs(yp) < 0.2 )
goto newt;
/* Resort to interval halving if not close enough. */
ihalve:
dir = 0;
di = 0.5L;
for ( i=0; i<400; i++ )
{
if ( i != 0 )
{
x = x0 + di * (x1 - x0);
if ( x == 1.0L )
{
x = 1.0L - real.epsilon;
}
if ( x == 0.0L )
{
di = 0.5;
x = x0 + di * (x1 - x0);
if ( x == 0.0 )
goto under;
}
y = betaIncomplete( a, b, x );
yp = (x1 - x0)/(x1 + x0);
if ( fabs(yp) < dithresh )
goto newt;
yp = (y-y0)/y0;
if ( fabs(yp) < dithresh )
goto newt;
}
if ( y < y0 )
{
x0 = x;
yl = y;
if ( dir < 0 )
{
dir = 0;
di = 0.5L;
} else if ( dir > 3 )
di = 1.0L - (1.0L - di) * (1.0L - di);
else if ( dir > 1 )
di = 0.5L * di + 0.5L;
else
di = (y0 - y)/(yh - yl);
dir += 1;
if ( x0 > 0.95L )
{
if ( rflg == 1 )
{
rflg = 0;
a = aa;
b = bb;
y0 = yy0;
}
else
{
rflg = 1;
a = bb;
b = aa;
y0 = 1.0 - yy0;
}
x = 1.0L - x;
y = betaIncomplete( a, b, x );
x0 = 0.0;
yl = 0.0;
x1 = 1.0;
yh = 1.0;
goto ihalve;
}
}
else
{
x1 = x;
if ( rflg == 1 && x1 < real.epsilon )
{
x = 0.0L;
goto done;
}
yh = y;
if ( dir > 0 )
{
dir = 0;
di = 0.5L;
}
else if ( dir < -3 )
di = di * di;
else if ( dir < -1 )
di = 0.5L * di;
else
di = (y - y0)/(yh - yl);
dir -= 1;
}
}
if ( x0 >= 1.0L )
{
// partial loss of precision
x = 1.0L - real.epsilon;
goto done;
}
if ( x <= 0.0L )
{
under:
// underflow has occurred
x = real.min_normal * real.min_normal;
goto done;
}
newt:
if ( nflg )
{
goto done;
}
nflg = 1;
lgm = logGamma(a+b) - logGamma(a) - logGamma(b);
for ( i=0; i<15; i++ )
{
/* Compute the function at this point. */
if ( i != 0 )
y = betaIncomplete(a,b,x);
if ( y < yl )
{
x = x0;
y = yl;
}
else if ( y > yh )
{
x = x1;
y = yh;
}
else if ( y < y0 )
{
x0 = x;
yl = y;
}
else
{
x1 = x;
yh = y;
}
if ( x == 1.0L || x == 0.0L )
break;
/* Compute the derivative of the function at this point. */
d = (a - 1.0L) * log(x) + (b - 1.0L) * log(1.0L - x) + lgm;
if ( d < MINLOG )
{
goto done;
}
if ( d > MAXLOG )
{
break;
}
d = exp(d);
/* Compute the step to the next approximation of x. */
d = (y - y0)/d;
xt = x - d;
if ( xt <= x0 )
{
y = (x - x0) / (x1 - x0);
xt = x0 + 0.5L * y * (x - x0);
if ( xt <= 0.0L )
break;
}
if ( xt >= x1 )
{
y = (x1 - x) / (x1 - x0);