1 /**
2 * Implementation of the gamma and beta functions, and their integrals.
3 *
4 * License: $(HTTP boost.org/LICENSE_1_0.txt, Boost License 1.0).
5 * Copyright: Based on the CEPHES math library, which is
6 * Copyright (C) 1994 Stephen L. Moshier (moshier@world.std.com).
7 * Authors: Stephen L. Moshier (original C code). Conversion to D by Don Clugston
8 *
9 *
10 Macros:
11 * TABLE_SV = <table border="1" cellpadding="4" cellspacing="0">
12 * <caption>Special Values</caption>
13 * $0</table>
14 * SVH = $(TR $(TH $1) $(TH $2))
15 * SV = $(TR $(TD $1) $(TD $2))
16 * GAMMA = Γ
17 * INTEGRATE = $(BIG ∫<sub>$(SMALL $1)</sub><sup>$2</sup>)
18 * POWER = $1<sup>$2</sup>
19 * NAN = $(RED NAN)
20 */
21 module std.internal.math.gammafunction;
22 import std.internal.math.errorfunction;
23 import std.math;
24
25 pure:
26 nothrow:
27 @safe:
28 @nogc:
29
30 private {
31
32 enum real SQRT2PI = 2.50662827463100050242E0L; // sqrt(2pi)
33 immutable real EULERGAMMA = 0.57721_56649_01532_86060_65120_90082_40243_10421_59335_93992L; /** Euler-Mascheroni constant 0.57721566.. */
34
35 // Polynomial approximations for gamma and loggamma.
36
37 immutable real[8] GammaNumeratorCoeffs = [ 1.0,
38 0x1.acf42d903366539ep-1, 0x1.73a991c8475f1aeap-2, 0x1.c7e918751d6b2a92p-4,
39 0x1.86d162cca32cfe86p-6, 0x1.0c378e2e6eaf7cd8p-8, 0x1.dc5c66b7d05feb54p-12,
40 0x1.616457b47e448694p-15
41 ];
42
43 immutable real[9] GammaDenominatorCoeffs = [ 1.0,
44 0x1.a8f9faae5d8fc8bp-2, -0x1.cb7895a6756eebdep-3, -0x1.7b9bab006d30652ap-5,
45 0x1.c671af78f312082ep-6, -0x1.a11ebbfaf96252dcp-11, -0x1.447b4d2230a77ddap-10,
46 0x1.ec1d45bb85e06696p-13,-0x1.d4ce24d05bd0a8e6p-17
47 ];
48
49 immutable real[9] GammaSmallCoeffs = [ 1.0,
50 0x1.2788cfc6fb618f52p-1, -0x1.4fcf4026afa2f7ecp-1, -0x1.5815e8fa24d7e306p-5,
51 0x1.5512320aea2ad71ap-3, -0x1.59af0fb9d82e216p-5, -0x1.3b4b61d3bfdf244ap-7,
52 0x1.d9358e9d9d69fd34p-8, -0x1.38fc4bcbada775d6p-10
53 ];
54
55 immutable real[9] GammaSmallNegCoeffs = [ -1.0,
56 0x1.2788cfc6fb618f54p-1, 0x1.4fcf4026afa2bc4cp-1, -0x1.5815e8fa2468fec8p-5,
57 -0x1.5512320baedaf4b6p-3, -0x1.59af0fa283baf07ep-5, 0x1.3b4a70de31e05942p-7,
58 0x1.d9398be3bad13136p-8, 0x1.291b73ee05bcbba2p-10
59 ];
60
61 immutable real[7] logGammaStirlingCoeffs = [
62 0x1.5555555555553f98p-4, -0x1.6c16c16c07509b1p-9, 0x1.a01a012461cbf1e4p-11,
63 -0x1.3813089d3f9d164p-11, 0x1.b911a92555a277b8p-11, -0x1.ed0a7b4206087b22p-10,
64 0x1.402523859811b308p-8
65 ];
66
67 immutable real[7] logGammaNumerator = [
68 -0x1.0edd25913aaa40a2p+23, -0x1.31c6ce2e58842d1ep+24, -0x1.f015814039477c3p+23,
69 -0x1.74ffe40c4b184b34p+22, -0x1.0d9c6d08f9eab55p+20, -0x1.54c6b71935f1fc88p+16,
70 -0x1.0e761b42932b2aaep+11
71 ];
72
73 immutable real[8] logGammaDenominator = [
74 -0x1.4055572d75d08c56p+24, -0x1.deeb6013998e4d76p+24, -0x1.106f7cded5dcc79ep+24,
75 -0x1.25e17184848c66d2p+22, -0x1.301303b99a614a0ap+19, -0x1.09e76ab41ae965p+15,
76 -0x1.00f95ced9e5f54eep+9, 1.0
77 ];
78
79 /*
80 * Helper function: Gamma function computed by Stirling's formula.
81 *
82 * Stirling's formula for the gamma function is:
83 *
84 * $(GAMMA)(x) = sqrt(2 π) x<sup>x-0.5</sup> exp(-x) (1 + 1/x P(1/x))
85 *
86 */
gammaStirling(real x)87 real gammaStirling(real x)
88 {
89 // CEPHES code Copyright 1994 by Stephen L. Moshier
90
91 static immutable real[9] SmallStirlingCoeffs = [
92 0x1.55555555555543aap-4, 0x1.c71c71c720dd8792p-9, -0x1.5f7268f0b5907438p-9,
93 -0x1.e13cd410e0477de6p-13, 0x1.9b0f31643442616ep-11, 0x1.2527623a3472ae08p-14,
94 -0x1.37f6bc8ef8b374dep-11,-0x1.8c968886052b872ap-16, 0x1.76baa9c6d3eeddbcp-11
95 ];
96
97 static immutable real[7] LargeStirlingCoeffs = [ 1.0L,
98 8.33333333333333333333E-2L, 3.47222222222222222222E-3L,
99 -2.68132716049382716049E-3L, -2.29472093621399176955E-4L,
100 7.84039221720066627474E-4L, 6.97281375836585777429E-5L
101 ];
102
103 real w = 1.0L/x;
104 real y = exp(x);
105 if ( x > 1024.0L )
106 {
107 // For large x, use rational coefficients from the analytical expansion.
108 w = poly(w, LargeStirlingCoeffs);
109 // Avoid overflow in pow()
110 real v = pow( x, 0.5L * x - 0.25L );
111 y = v * (v / y);
112 }
113 else
114 {
115 w = 1.0L + w * poly( w, SmallStirlingCoeffs);
116 static if (floatTraits!(real).realFormat == RealFormat.ieeeDouble)
117 {
118 // Avoid overflow in pow() for 64-bit reals
119 if (x > 143.0)
120 {
121 real v = pow( x, 0.5 * x - 0.25 );
122 y = v * (v / y);
123 }
124 else
125 {
126 y = pow( x, x - 0.5 ) / y;
127 }
128 }
129 else
130 {
131 y = pow( x, x - 0.5L ) / y;
132 }
133 }
134 y = SQRT2PI * y * w;
135 return y;
136 }
137
138 /*
139 * Helper function: Incomplete gamma function computed by Temme's expansion.
140 *
141 * This is a port of igamma_temme_large from Boost.
142 *
143 */
igammaTemmeLarge(real a,real x)144 real igammaTemmeLarge(real a, real x)
145 {
146 static immutable real[][13] coef = [
147 [ -0.333333333333333333333, 0.0833333333333333333333,
148 -0.0148148148148148148148, 0.00115740740740740740741,
149 0.000352733686067019400353, -0.0001787551440329218107,
150 0.39192631785224377817e-4, -0.218544851067999216147e-5,
151 -0.18540622107151599607e-5, 0.829671134095308600502e-6,
152 -0.176659527368260793044e-6, 0.670785354340149858037e-8,
153 0.102618097842403080426e-7, -0.438203601845335318655e-8,
154 0.914769958223679023418e-9, -0.255141939949462497669e-10,
155 -0.583077213255042506746e-10, 0.243619480206674162437e-10,
156 -0.502766928011417558909e-11 ],
157 [ -0.00185185185185185185185, -0.00347222222222222222222,
158 0.00264550264550264550265, -0.000990226337448559670782,
159 0.000205761316872427983539, -0.40187757201646090535e-6,
160 -0.18098550334489977837e-4, 0.764916091608111008464e-5,
161 -0.161209008945634460038e-5, 0.464712780280743434226e-8,
162 0.137863344691572095931e-6, -0.575254560351770496402e-7,
163 0.119516285997781473243e-7, -0.175432417197476476238e-10,
164 -0.100915437106004126275e-8, 0.416279299184258263623e-9,
165 -0.856390702649298063807e-10 ],
166 [ 0.00413359788359788359788, -0.00268132716049382716049,
167 0.000771604938271604938272, 0.200938786008230452675e-5,
168 -0.000107366532263651605215, 0.529234488291201254164e-4,
169 -0.127606351886187277134e-4, 0.342357873409613807419e-7,
170 0.137219573090629332056e-5, -0.629899213838005502291e-6,
171 0.142806142060642417916e-6, -0.204770984219908660149e-9,
172 -0.140925299108675210533e-7, 0.622897408492202203356e-8,
173 -0.136704883966171134993e-8 ],
174 [ 0.000649434156378600823045, 0.000229472093621399176955,
175 -0.000469189494395255712128, 0.000267720632062838852962,
176 -0.756180167188397641073e-4, -0.239650511386729665193e-6,
177 0.110826541153473023615e-4, -0.56749528269915965675e-5,
178 0.142309007324358839146e-5, -0.278610802915281422406e-10,
179 -0.169584040919302772899e-6, 0.809946490538808236335e-7,
180 -0.191111684859736540607e-7 ],
181 [ -0.000861888290916711698605, 0.000784039221720066627474,
182 -0.000299072480303190179733, -0.146384525788434181781e-5,
183 0.664149821546512218666e-4, -0.396836504717943466443e-4,
184 0.113757269706784190981e-4, 0.250749722623753280165e-9,
185 -0.169541495365583060147e-5, 0.890750753220530968883e-6,
186 -0.229293483400080487057e-6],
187 [ -0.000336798553366358150309, -0.697281375836585777429e-4,
188 0.000277275324495939207873, -0.000199325705161888477003,
189 0.679778047793720783882e-4, 0.141906292064396701483e-6,
190 -0.135940481897686932785e-4, 0.801847025633420153972e-5,
191 -0.229148117650809517038e-5 ],
192 [ 0.000531307936463992223166, -0.000592166437353693882865,
193 0.000270878209671804482771, 0.790235323266032787212e-6,
194 -0.815396936756196875093e-4, 0.561168275310624965004e-4,
195 -0.183291165828433755673e-4, -0.307961345060330478256e-8,
196 0.346515536880360908674e-5, -0.20291327396058603727e-5,
197 0.57887928631490037089e-6 ],
198 [ 0.000344367606892377671254, 0.517179090826059219337e-4,
199 -0.000334931610811422363117, 0.000281269515476323702274,
200 -0.000109765822446847310235, -0.127410090954844853795e-6,
201 0.277444515115636441571e-4, -0.182634888057113326614e-4,
202 0.578769494973505239894e-5 ],
203 [ -0.000652623918595309418922, 0.000839498720672087279993,
204 -0.000438297098541721005061, -0.696909145842055197137e-6,
205 0.000166448466420675478374, -0.000127835176797692185853,
206 0.462995326369130429061e-4 ],
207 [ -0.000596761290192746250124, -0.720489541602001055909e-4,
208 0.000678230883766732836162, -0.0006401475260262758451,
209 0.000277501076343287044992 ],
210 [ 0.00133244544948006563713, -0.0019144384985654775265,
211 0.00110893691345966373396 ],
212 [ 0.00157972766073083495909, 0.000162516262783915816899,
213 -0.00206334210355432762645, 0.00213896861856890981541,
214 -0.00101085593912630031708 ],
215 [ -0.00407251211951401664727, 0.00640336283380806979482,
216 -0.00404101610816766177474 ]
217 ];
218
219 // avoid nans when one of the arguments is inf:
220 if (x == real.infinity && a != real.infinity)
221 return 0;
222
223 if (x != real.infinity && a == real.infinity)
224 return 1;
225
226 real sigma = (x - a) / a;
227 real phi = sigma - log(sigma + 1);
228
229 real y = a * phi;
230 real z = sqrt(2 * phi);
231 if (x < a)
232 z = -z;
233
234 real[13] workspace;
235 foreach (i; 0 .. coef.length)
236 workspace[i] = poly(z, coef[i]);
237
238 real result = poly(1 / a, workspace);
239 result *= exp(-y) / sqrt(2 * PI * a);
240 if (x < a)
241 result = -result;
242
243 result += erfc(sqrt(y)) / 2;
244
245 return result;
246 }
247
248 } // private
249
250 public:
251 /// The maximum value of x for which gamma(x) < real.infinity.
252 static if (floatTraits!(real).realFormat == RealFormat.ieeeQuadruple)
253 enum real MAXGAMMA = 1755.5483429L;
254 else static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended)
255 enum real MAXGAMMA = 1755.5483429L;
256 else static if (floatTraits!(real).realFormat == RealFormat.ieeeDouble)
257 enum real MAXGAMMA = 171.6243769L;
258 else
259 static assert(0, "missing MAXGAMMA for other real types");
260
261
262 /*****************************************************
263 * The Gamma function, $(GAMMA)(x)
264 *
265 * $(GAMMA)(x) is a generalisation of the factorial function
266 * to real and complex numbers.
267 * Like x!, $(GAMMA)(x+1) = x*$(GAMMA)(x).
268 *
269 * Mathematically, if z.re > 0 then
270 * $(GAMMA)(z) = $(INTEGRATE 0, ∞) $(POWER t, z-1)$(POWER e, -t) dt
271 *
272 * $(TABLE_SV
273 * $(SVH x, $(GAMMA)(x) )
274 * $(SV $(NAN), $(NAN) )
275 * $(SV ±0.0, ±∞)
276 * $(SV integer > 0, (x-1)! )
277 * $(SV integer < 0, $(NAN) )
278 * $(SV +∞, +∞ )
279 * $(SV -∞, $(NAN) )
280 * )
281 */
gamma(real x)282 real gamma(real x)
283 {
284 /* Based on code from the CEPHES library.
285 * CEPHES code Copyright 1994 by Stephen L. Moshier
286 *
287 * Arguments |x| <= 13 are reduced by recurrence and the function
288 * approximated by a rational function of degree 7/8 in the
289 * interval (2,3). Large arguments are handled by Stirling's
290 * formula. Large negative arguments are made positive using
291 * a reflection formula.
292 */
293
294 real q, z;
295 if (isNaN(x)) return x;
296 if (x == -x.infinity) return real.nan;
297 if ( fabs(x) > MAXGAMMA ) return real.infinity;
298 if (x == 0) return 1.0 / x; // +- infinity depending on sign of x, create an exception.
299
300 q = fabs(x);
301
302 if ( q > 13.0L )
303 {
304 // Large arguments are handled by Stirling's
305 // formula. Large negative arguments are made positive using
306 // the reflection formula.
307
308 if ( x < 0.0L )
309 {
310 if (x < -1/real.epsilon)
311 {
312 // Large negatives lose all precision
313 return real.nan;
314 }
315 int sgngam = 1; // sign of gamma.
316 long intpart = cast(long)(q);
317 if (q == intpart)
318 return real.nan; // poles for all integers <0.
319 real p = intpart;
320 if ( (intpart & 1) == 0 )
321 sgngam = -1;
322 z = q - p;
323 if ( z > 0.5L )
324 {
325 p += 1.0L;
326 z = q - p;
327 }
328 z = q * sin( PI * z );
329 z = fabs(z) * gammaStirling(q);
330 if ( z <= PI/real.max ) return sgngam * real.infinity;
331 return sgngam * PI/z;
332 }
333 else
334 {
335 return gammaStirling(x);
336 }
337 }
338
339 // Arguments |x| <= 13 are reduced by recurrence and the function
340 // approximated by a rational function of degree 7/8 in the
341 // interval (2,3).
342
343 z = 1.0L;
344 while ( x >= 3.0L )
345 {
346 x -= 1.0L;
347 z *= x;
348 }
349
350 while ( x < -0.03125L )
351 {
352 z /= x;
353 x += 1.0L;
354 }
355
356 if ( x <= 0.03125L )
357 {
358 if ( x == 0.0L )
359 return real.nan;
360 else
361 {
362 if ( x < 0.0L )
363 {
364 x = -x;
365 return z / (x * poly( x, GammaSmallNegCoeffs ));
366 }
367 else
368 {
369 return z / (x * poly( x, GammaSmallCoeffs ));
370 }
371 }
372 }
373
374 while ( x < 2.0L )
375 {
376 z /= x;
377 x += 1.0L;
378 }
379 if ( x == 2.0L ) return z;
380
381 x -= 2.0L;
382 return z * poly( x, GammaNumeratorCoeffs ) / poly( x, GammaDenominatorCoeffs );
383 }
384
385 @safe unittest
386 {
387 // gamma(n) = factorial(n-1) if n is an integer.
388 real fact = 1.0L;
389 for (int i=1; fact<real.max; ++i)
390 {
391 // Require exact equality for small factorials
392 if (i<14) assert(gamma(i*1.0L) == fact);
393 assert(feqrel(gamma(i*1.0L), fact) >= real.mant_dig-15);
394 fact *= (i*1.0L);
395 }
396 assert(gamma(0.0) == real.infinity);
397 assert(gamma(-0.0) == -real.infinity);
398 assert(isNaN(gamma(-1.0)));
399 assert(isNaN(gamma(-15.0)));
400 assert(isIdentical(gamma(NaN(0xABC)), NaN(0xABC)));
401 assert(gamma(real.infinity) == real.infinity);
402 assert(gamma(real.max) == real.infinity);
403 assert(isNaN(gamma(-real.infinity)));
404 assert(gamma(real.min_normal*real.epsilon) == real.infinity);
405 assert(gamma(MAXGAMMA)< real.infinity);
406 assert(gamma(MAXGAMMA*2) == real.infinity);
407
408 // Test some high-precision values (50 decimal digits)
409 real SQRT_PI = 1.77245385090551602729816748334114518279754945612238L;
410
411
412 assert(feqrel(gamma(0.5L), SQRT_PI) >= real.mant_dig-1);
413 assert(feqrel(gamma(17.25L), 4.224986665692703551570937158682064589938e13L) >= real.mant_dig-4);
414
415 assert(feqrel(gamma(1.0 / 3.0L), 2.67893853470774763365569294097467764412868937795730L) >= real.mant_dig-2);
416 assert(feqrel(gamma(0.25L),
417 3.62560990822190831193068515586767200299516768288006L) >= real.mant_dig-1);
418 assert(feqrel(gamma(1.0 / 5.0L),
419 4.59084371199880305320475827592915200343410999829340L) >= real.mant_dig-1);
420 }
421
422 /*****************************************************
423 * Natural logarithm of gamma function.
424 *
425 * Returns the base e (2.718...) logarithm of the absolute
426 * value of the gamma function of the argument.
427 *
428 * For reals, logGamma is equivalent to log(fabs(gamma(x))).
429 *
430 * $(TABLE_SV
431 * $(SVH x, logGamma(x) )
432 * $(SV $(NAN), $(NAN) )
433 * $(SV integer <= 0, +∞ )
434 * $(SV ±∞, +∞ )
435 * )
436 */
logGamma(real x)437 real logGamma(real x)
438 {
439 /* Based on code from the CEPHES library.
440 * CEPHES code Copyright 1994 by Stephen L. Moshier
441 *
442 * For arguments greater than 33, the logarithm of the gamma
443 * function is approximated by the logarithmic version of
444 * Stirling's formula using a polynomial approximation of
445 * degree 4. Arguments between -33 and +33 are reduced by
446 * recurrence to the interval [2,3] of a rational approximation.
447 * The cosecant reflection formula is employed for arguments
448 * less than -33.
449 */
450 real q, w, z, f, nx;
451
452 if (isNaN(x)) return x;
453 if (fabs(x) == x.infinity) return x.infinity;
454
455 if ( x < -34.0L )
456 {
457 q = -x;
458 w = logGamma(q);
459 real p = floor(q);
460 if ( p == q )
461 return real.infinity;
462 int intpart = cast(int)(p);
463 real sgngam = 1;
464 if ( (intpart & 1) == 0 )
465 sgngam = -1;
466 z = q - p;
467 if ( z > 0.5L )
468 {
469 p += 1.0L;
470 z = p - q;
471 }
472 z = q * sin( PI * z );
473 if ( z == 0.0L )
474 return sgngam * real.infinity;
475 /* z = LOGPI - logl( z ) - w; */
476 z = log( PI/z ) - w;
477 return z;
478 }
479
480 if ( x < 13.0L )
481 {
482 z = 1.0L;
483 nx = floor( x + 0.5L );
484 f = x - nx;
485 while ( x >= 3.0L )
486 {
487 nx -= 1.0L;
488 x = nx + f;
489 z *= x;
490 }
491 while ( x < 2.0L )
492 {
493 if ( fabs(x) <= 0.03125 )
494 {
495 if ( x == 0.0L )
496 return real.infinity;
497 if ( x < 0.0L )
498 {
499 x = -x;
500 q = z / (x * poly( x, GammaSmallNegCoeffs));
501 } else
502 q = z / (x * poly( x, GammaSmallCoeffs));
503 return log( fabs(q) );
504 }
505 z /= nx + f;
506 nx += 1.0L;
507 x = nx + f;
508 }
509 z = fabs(z);
510 if ( x == 2.0L )
511 return log(z);
512 x = (nx - 2.0L) + f;
513 real p = x * rationalPoly( x, logGammaNumerator, logGammaDenominator);
514 return log(z) + p;
515 }
516
517 // const real MAXLGM = 1.04848146839019521116e+4928L;
518 // if ( x > MAXLGM ) return sgngaml * real.infinity;
519
520 const real LOGSQRT2PI = 0.91893853320467274178L; // log( sqrt( 2*pi ) )
521
522 q = ( x - 0.5L ) * log(x) - x + LOGSQRT2PI;
523 if (x > 1.0e10L) return q;
524 real p = 1.0L / (x*x);
525 q += poly( p, logGammaStirlingCoeffs ) / x;
526 return q ;
527 }
528
529 @safe unittest
530 {
531 assert(isIdentical(logGamma(NaN(0xDEF)), NaN(0xDEF)));
532 assert(logGamma(real.infinity) == real.infinity);
533 assert(logGamma(-1.0) == real.infinity);
534 assert(logGamma(0.0) == real.infinity);
535 assert(logGamma(-50.0) == real.infinity);
536 assert(isIdentical(0.0L, logGamma(1.0L)));
537 assert(isIdentical(0.0L, logGamma(2.0L)));
538 assert(logGamma(real.min_normal*real.epsilon) == real.infinity);
539 assert(logGamma(-real.min_normal*real.epsilon) == real.infinity);
540
541 // x, correct loggamma(x), correct d/dx loggamma(x).
542 immutable static real[] testpoints = [
543 8.0L, 8.525146484375L + 1.48766904143001655310E-5, 2.01564147795560999654E0L,
544 8.99993896484375e-1L, 6.6375732421875e-2L + 5.11505711292524166220E-6L, -7.54938684259372234258E-1,
545 7.31597900390625e-1L, 2.2369384765625e-1 + 5.21506341809849792422E-6L, -1.13355566660398608343E0L,
546 2.31639862060546875e-1L, 1.3686676025390625L + 1.12609441752996145670E-5L, -4.56670961813812679012E0,
547 1.73162841796875L, -8.88214111328125e-2L + 3.36207740803753034508E-6L, 2.33339034686200586920E-1L,
548 1.23162841796875L, -9.3902587890625e-2L + 1.28765089229009648104E-5L, -2.49677345775751390414E-1L,
549 7.3786976294838206464e19L, 3.301798506038663053312e21L - 1.656137564136932662487046269677E5L,
550 4.57477139169563904215E1L,
551 1.08420217248550443401E-19L, 4.36682586669921875e1L + 1.37082843669932230418E-5L,
552 -9.22337203685477580858E18L,
553 1.0L, 0.0L, -5.77215664901532860607E-1L,
554 2.0L, 0.0L, 4.22784335098467139393E-1L,
555 -0.5L, 1.2655029296875L + 9.19379714539648894580E-6L, 3.64899739785765205590E-2L,
556 -1.5L, 8.6004638671875e-1L + 6.28657731014510932682E-7L, 7.03156640645243187226E-1L,
557 -2.5L, -5.6243896484375E-2L + 1.79986700949327405470E-7, 1.10315664064524318723E0L,
558 -3.5L, -1.30902099609375L + 1.43111007079536392848E-5L, 1.38887092635952890151E0L
559 ];
560 // TODO: test derivatives as well.
561 for (int i=0; i<testpoints.length; i+=3)
562 {
563 assert( feqrel(logGamma(testpoints[i]), testpoints[i+1]) > real.mant_dig-5);
564 if (testpoints[i]<MAXGAMMA)
565 {
566 assert( feqrel(log(fabs(gamma(testpoints[i]))), testpoints[i+1]) > real.mant_dig-5);
567 }
568 }
569 assert(logGamma(-50.2) == log(fabs(gamma(-50.2))));
570 assert(logGamma(-0.008) == log(fabs(gamma(-0.008))));
571 assert(feqrel(logGamma(-38.8),log(fabs(gamma(-38.8)))) > real.mant_dig-4);
572 static if (real.mant_dig >= 64) // incl. 80-bit reals
573 assert(feqrel(logGamma(1500.0L),log(gamma(1500.0L))) > real.mant_dig-2);
574 else static if (real.mant_dig >= 53) // incl. 64-bit reals
575 assert(feqrel(logGamma(150.0L),log(gamma(150.0L))) > real.mant_dig-2);
576 }
577
578
579 private {
580 /*
581 * These value can be calculated like this:
582 * 1) Get exact real.max/min_normal/epsilon from compiler:
583 * writefln!"%a"(real.max/min_normal_epsilon)
584 * 2) Convert for Wolfram Alpha
585 * 0xf.fffffffffffffffp+16380 ==> (f.fffffffffffffff base 16) * 2^16380
586 * 3) Calculate result on wofram alpha:
587 * http://www.wolframalpha.com/input/?i=ln((1.ffffffffffffffffffffffffffff+base+16)+*+2%5E16383)+in+base+2
588 * 4) Convert to proper format:
589 * string mantissa = "1.011...";
590 * write(mantissa[0 .. 2]); mantissa = mantissa[2 .. $];
591 * for (size_t i = 0; i < mantissa.length/4; i++)
592 * {
593 * writef!"%x"(to!ubyte(mantissa[0 .. 4], 2)); mantissa = mantissa[4 .. $];
594 * }
595 */
596 static if (floatTraits!(real).realFormat == RealFormat.ieeeQuadruple)
597 {
598 enum real MAXLOG = 0x1.62e42fefa39ef35793c7673007e6p+13; // log(real.max)
599 enum real MINLOG = -0x1.6546282207802c89d24d65e96274p+13; // log(real.min_normal*real.epsilon) = log(smallest denormal)
600 }
601 else static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended)
602 {
603 enum real MAXLOG = 0x1.62e42fefa39ef358p+13L; // log(real.max)
604 enum real MINLOG = -0x1.6436716d5406e6d8p+13L; // log(real.min_normal*real.epsilon) = log(smallest denormal)
605 }
606 else static if (floatTraits!(real).realFormat == RealFormat.ieeeDouble)
607 {
608 enum real MAXLOG = 0x1.62e42fefa39efp+9L; // log(real.max)
609 enum real MINLOG = -0x1.74385446d71c3p+9L; // log(real.min_normal*real.epsilon) = log(smallest denormal)
610 }
611 else
612 static assert(0, "missing MAXLOG and MINLOG for other real types");
613
614 enum real BETA_BIG = 9.223372036854775808e18L;
615 enum real BETA_BIGINV = 1.084202172485504434007e-19L;
616 }
617
618 /** Incomplete beta integral
619 *
620 * Returns incomplete beta integral of the arguments, evaluated
621 * from zero to x. The regularized incomplete beta function is defined as
622 *
623 * betaIncomplete(a, b, x) = Γ(a+b)/(Γ(a) Γ(b)) *
624 * $(INTEGRATE 0, x) $(POWER t, a-1)$(POWER (1-t),b-1) dt
625 *
626 * and is the same as the the cumulative distribution function.
627 *
628 * The domain of definition is 0 <= x <= 1. In this
629 * implementation a and b are restricted to positive values.
630 * The integral from x to 1 may be obtained by the symmetry
631 * relation
632 *
633 * betaIncompleteCompl(a, b, x ) = betaIncomplete( b, a, 1-x )
634 *
635 * The integral is evaluated by a continued fraction expansion
636 * or, when b*x is small, by a power series.
637 */
betaIncomplete(real aa,real bb,real xx)638 real betaIncomplete(real aa, real bb, real xx )
639 {
640 if ( !(aa>0 && bb>0) )
641 {
642 if ( isNaN(aa) ) return aa;
643 if ( isNaN(bb) ) return bb;
644 return real.nan; // domain error
645 }
646 if (!(xx>0 && xx<1.0))
647 {
648 if (isNaN(xx)) return xx;
649 if ( xx == 0.0L ) return 0.0;
650 if ( xx == 1.0L ) return 1.0;
651 return real.nan; // domain error
652 }
653 if ( (bb * xx) <= 1.0L && xx <= 0.95L)
654 {
655 return betaDistPowerSeries(aa, bb, xx);
656 }
657 real x;
658 real xc; // = 1 - x
659
660 real a, b;
661 int flag = 0;
662
663 /* Reverse a and b if x is greater than the mean. */
664 if ( xx > (aa/(aa+bb)) )
665 {
666 // here x > aa/(aa+bb) and (bb*x>1 or x>0.95)
667 flag = 1;
668 a = bb;
669 b = aa;
670 xc = xx;
671 x = 1.0L - xx;
672 }
673 else
674 {
675 a = aa;
676 b = bb;
677 xc = 1.0L - xx;
678 x = xx;
679 }
680
681 if ( flag == 1 && (b * x) <= 1.0L && x <= 0.95L)
682 {
683 // here xx > aa/(aa+bb) and ((bb*xx>1) or xx>0.95) and (aa*(1-xx)<=1) and xx > 0.05
684 return 1.0 - betaDistPowerSeries(a, b, x); // note loss of precision
685 }
686
687 real w;
688 // Choose expansion for optimal convergence
689 // One is for x * (a+b+2) < (a+1),
690 // the other is for x * (a+b+2) > (a+1).
691 real y = x * (a+b-2.0L) - (a-1.0L);
692 if ( y < 0.0L )
693 {
694 w = betaDistExpansion1( a, b, x );
695 }
696 else
697 {
698 w = betaDistExpansion2( a, b, x ) / xc;
699 }
700
701 /* Multiply w by the factor
702 a b
703 x (1-x) Gamma(a+b) / ( a Gamma(a) Gamma(b) ) . */
704
705 y = a * log(x);
706 real t = b * log(xc);
707 if ( (a+b) < MAXGAMMA && fabs(y) < MAXLOG && fabs(t) < MAXLOG )
708 {
709 t = pow(xc,b);
710 t *= pow(x,a);
711 t /= a;
712 t *= w;
713 t *= gamma(a+b) / (gamma(a) * gamma(b));
714 }
715 else
716 {
717 /* Resort to logarithms. */
718 y += t + logGamma(a+b) - logGamma(a) - logGamma(b);
719 y += log(w/a);
720
721 t = exp(y);
722 /+
723 // There seems to be a bug in Cephes at this point.
724 // Problems occur for y > MAXLOG, not y < MINLOG.
725 if ( y < MINLOG )
726 {
727 t = 0.0L;
728 }
729 else
730 {
731 t = exp(y);
732 }
733 +/
734 }
735 if ( flag == 1 )
736 {
737 /+ // CEPHES includes this code, but I think it is erroneous.
738 if ( t <= real.epsilon )
739 {
740 t = 1.0L - real.epsilon;
741 } else
742 +/
743 t = 1.0L - t;
744 }
745 return t;
746 }
747
748 /** Inverse of incomplete beta integral
749 *
750 * Given y, the function finds x such that
751 *
752 * betaIncomplete(a, b, x) == y
753 *
754 * Newton iterations or interval halving is used.
755 */
betaIncompleteInv(real aa,real bb,real yy0)756 real betaIncompleteInv(real aa, real bb, real yy0 )
757 {
758 real a, b, y0, d, y, x, x0, x1, lgm, yp, di, dithresh, yl, yh, xt;
759 int i, rflg, dir, nflg;
760
761 if (isNaN(yy0)) return yy0;
762 if (isNaN(aa)) return aa;
763 if (isNaN(bb)) return bb;
764 if ( yy0 <= 0.0L )
765 return 0.0L;
766 if ( yy0 >= 1.0L )
767 return 1.0L;
768 x0 = 0.0L;
769 yl = 0.0L;
770 x1 = 1.0L;
771 yh = 1.0L;
772 if ( aa <= 1.0L || bb <= 1.0L )
773 {
774 dithresh = 1.0e-7L;
775 rflg = 0;
776 a = aa;
777 b = bb;
778 y0 = yy0;
779 x = a/(a+b);
780 y = betaIncomplete( a, b, x );
781 nflg = 0;
782 goto ihalve;
783 }
784 else
785 {
786 nflg = 0;
787 dithresh = 1.0e-4L;
788 }
789
790 // approximation to inverse function
791
792 yp = -normalDistributionInvImpl( yy0 );
793
794 if ( yy0 > 0.5L )
795 {
796 rflg = 1;
797 a = bb;
798 b = aa;
799 y0 = 1.0L - yy0;
800 yp = -yp;
801 }
802 else
803 {
804 rflg = 0;
805 a = aa;
806 b = bb;
807 y0 = yy0;
808 }
809
810 lgm = (yp * yp - 3.0L)/6.0L;
811 x = 2.0L/( 1.0L/(2.0L * a-1.0L) + 1.0L/(2.0L * b - 1.0L) );
812 d = yp * sqrt( x + lgm ) / x
813 - ( 1.0L/(2.0L * b - 1.0L) - 1.0L/(2.0L * a - 1.0L) )
814 * (lgm + (5.0L/6.0L) - 2.0L/(3.0L * x));
815 d = 2.0L * d;
816 if ( d < MINLOG )
817 {
818 x = 1.0L;
819 goto under;
820 }
821 x = a/( a + b * exp(d) );
822 y = betaIncomplete( a, b, x );
823 yp = (y - y0)/y0;
824 if ( fabs(yp) < 0.2 )
825 goto newt;
826
827 /* Resort to interval halving if not close enough. */
828 ihalve:
829
830 dir = 0;
831 di = 0.5L;
832 for ( i=0; i<400; i++ )
833 {
834 if ( i != 0 )
835 {
836 x = x0 + di * (x1 - x0);
837 if ( x == 1.0L )
838 {
839 x = 1.0L - real.epsilon;
840 }
841 if ( x == 0.0L )
842 {
843 di = 0.5;
844 x = x0 + di * (x1 - x0);
845 if ( x == 0.0 )
846 goto under;
847 }
848 y = betaIncomplete( a, b, x );
849 yp = (x1 - x0)/(x1 + x0);
850 if ( fabs(yp) < dithresh )
851 goto newt;
852 yp = (y-y0)/y0;
853 if ( fabs(yp) < dithresh )
854 goto newt;
855 }
856 if ( y < y0 )
857 {
858 x0 = x;
859 yl = y;
860 if ( dir < 0 )
861 {
862 dir = 0;
863 di = 0.5L;
864 } else if ( dir > 3 )
865 di = 1.0L - (1.0L - di) * (1.0L - di);
866 else if ( dir > 1 )
867 di = 0.5L * di + 0.5L;
868 else
869 di = (y0 - y)/(yh - yl);
870 dir += 1;
871 if ( x0 > 0.95L )
872 {
873 if ( rflg == 1 )
874 {
875 rflg = 0;
876 a = aa;
877 b = bb;
878 y0 = yy0;
879 }
880 else
881 {
882 rflg = 1;
883 a = bb;
884 b = aa;
885 y0 = 1.0 - yy0;
886 }
887 x = 1.0L - x;
888 y = betaIncomplete( a, b, x );
889 x0 = 0.0;
890 yl = 0.0;
891 x1 = 1.0;
892 yh = 1.0;
893 goto ihalve;
894 }
895 }
896 else
897 {
898 x1 = x;
899 if ( rflg == 1 && x1 < real.epsilon )
900 {
901 x = 0.0L;
902 goto done;
903 }
904 yh = y;
905 if ( dir > 0 )
906 {
907 dir = 0;
908 di = 0.5L;
909 }
910 else if ( dir < -3 )
911 di = di * di;
912 else if ( dir < -1 )
913 di = 0.5L * di;
914 else
915 di = (y - y0)/(yh - yl);
916 dir -= 1;
917 }
918 }
919 if ( x0 >= 1.0L )
920 {
921 // partial loss of precision
922 x = 1.0L - real.epsilon;
923 goto done;
924 }
925 if ( x <= 0.0L )
926 {
927 under:
928 // underflow has occurred
929 x = real.min_normal * real.min_normal;
930 goto done;
931 }
932
933 newt:
934
935 if ( nflg )
936 {
937 goto done;
938 }
939 nflg = 1;
940 lgm = logGamma(a+b) - logGamma(a) - logGamma(b);
941
942 for ( i=0; i<15; i++ )
943 {
944 /* Compute the function at this point. */
945 if ( i != 0 )
946 y = betaIncomplete(a,b,x);
947 if ( y < yl )
948 {
949 x = x0;
950 y = yl;
951 }
952 else if ( y > yh )
953 {
954 x = x1;
955 y = yh;
956 }
957 else if ( y < y0 )
958 {
959 x0 = x;
960 yl = y;
961 }
962 else
963 {
964 x1 = x;
965 yh = y;
966 }
967 if ( x == 1.0L || x == 0.0L )
968 break;
969 /* Compute the derivative of the function at this point. */
970 d = (a - 1.0L) * log(x) + (b - 1.0L) * log(1.0L - x) + lgm;
971 if ( d < MINLOG )
972 {
973 goto done;
974 }
975 if ( d > MAXLOG )
976 {
977 break;
978 }
979 d = exp(d);
980 /* Compute the step to the next approximation of x. */
981 d = (y - y0)/d;
982 xt = x - d;
983 if ( xt <= x0 )
984 {
985 y = (x - x0) / (x1 - x0);
986 xt = x0 + 0.5L * y * (x - x0);
987 if ( xt <= 0.0L )
988 break;
989 }
990 if ( xt >= x1 )
991 {
992 y = (x1 - x) / (x1 - x0);
993 xt = x1 - 0.5L * y * (x1 - x);
994 if ( xt >= 1.0L )
995 break;
996 }
997 x = xt;
998 if ( fabs(d/x) < (128.0L * real.epsilon) )
999 goto done;
1000 }
1001 /* Did not converge. */
1002 dithresh = 256.0L * real.epsilon;
1003 goto ihalve;
1004
1005 done:
1006 if ( rflg )
1007 {
1008 if ( x <= real.epsilon )
1009 x = 1.0L - real.epsilon;
1010 else
1011 x = 1.0L - x;
1012 }
1013 return x;
1014 }
1015
1016 @safe unittest { // also tested by the normal distribution
1017 // check NaN propagation
1018 assert(isIdentical(betaIncomplete(NaN(0xABC),2,3), NaN(0xABC)));
1019 assert(isIdentical(betaIncomplete(7,NaN(0xABC),3), NaN(0xABC)));
1020 assert(isIdentical(betaIncomplete(7,15,NaN(0xABC)), NaN(0xABC)));
1021 assert(isIdentical(betaIncompleteInv(NaN(0xABC),1,17), NaN(0xABC)));
1022 assert(isIdentical(betaIncompleteInv(2,NaN(0xABC),8), NaN(0xABC)));
1023 assert(isIdentical(betaIncompleteInv(2,3, NaN(0xABC)), NaN(0xABC)));
1024
1025 assert(isNaN(betaIncomplete(-1, 2, 3)));
1026
1027 assert(betaIncomplete(1, 2, 0)==0);
1028 assert(betaIncomplete(1, 2, 1)==1);
1029 assert(isNaN(betaIncomplete(1, 2, 3)));
1030 assert(betaIncompleteInv(1, 1, 0)==0);
1031 assert(betaIncompleteInv(1, 1, 1)==1);
1032
1033 // Test against Mathematica betaRegularized[z,a,b]
1034 // These arbitrary points are chosen to give good code coverage.
1035 assert(feqrel(betaIncomplete(8, 10, 0.2), 0.010_934_315_234_099_2L) >= real.mant_dig - 5);
1036 assert(feqrel(betaIncomplete(2, 2.5, 0.9), 0.989_722_597_604_452_767_171_003_59L) >= real.mant_dig - 1);
1037 static if (real.mant_dig >= 64) // incl. 80-bit reals
1038 assert(feqrel(betaIncomplete(1000, 800, 0.5), 1.179140859734704555102808541457164E-06L) >= real.mant_dig - 13);
1039 else
1040 assert(feqrel(betaIncomplete(1000, 800, 0.5), 1.179140859734704555102808541457164E-06L) >= real.mant_dig - 14);
1041 assert(feqrel(betaIncomplete(0.0001, 10000, 0.0001), 0.999978059362107134278786L) >= real.mant_dig - 18);
1042 assert(betaIncomplete(0.01, 327726.7, 0.545113) == 1.0);
1043 assert(feqrel(betaIncompleteInv(8, 10, 0.010_934_315_234_099_2L), 0.2L) >= real.mant_dig - 2);
1044 assert(feqrel(betaIncomplete(0.01, 498.437, 0.0121433), 0.99999664562033077636065L) >= real.mant_dig - 1);
1045 assert(feqrel(betaIncompleteInv(5, 10, 0.2000002972865658842), 0.229121208190918L) >= real.mant_dig - 3);
1046 assert(feqrel(betaIncompleteInv(4, 7, 0.8000002209179505L), 0.483657360076904L) >= real.mant_dig - 3);
1047
1048 // Coverage tests. I don't have correct values for these tests, but
1049 // these values cover most of the code, so they are useful for
1050 // regression testing.
1051 // Extensive testing failed to increase the coverage. It seems likely that about
1052 // half the code in this function is unnecessary; there is potential for
1053 // significant improvement over the original CEPHES code.
1054 static if (real.mant_dig == 64) // 80-bit reals
1055 {
1056 assert(betaIncompleteInv(0.01, 8e-48, 5.45464e-20) == 1-real.epsilon);
1057 assert(betaIncompleteInv(0.01, 8e-48, 9e-26) == 1-real.epsilon);
1058
1059 // Beware: a one-bit change in pow() changes almost all digits in the result!
1060 assert(feqrel(
1061 betaIncompleteInv(0x1.b3d151fbba0eb18p+1, 1.2265e-19, 2.44859e-18),
1062 0x1.c0110c8531d0952cp-1L
1063 ) > 10);
1064 // This next case uncovered a one-bit difference in the FYL2X instruction
1065 // between Intel and AMD processors. This difference gets magnified by 2^^38.
1066 // WolframAlpha crashes attempting to calculate this.
1067 assert(feqrel(betaIncompleteInv(0x1.ff1275ae5b939bcap-41, 4.6713e18, 0.0813601),
1068 0x1.f97749d90c7adba8p-63L) >= real.mant_dig - 39);
1069 real a1 = 3.40483;
1070 assert(betaIncompleteInv(a1, 4.0640301659679627772e19L, 0.545113) == 0x1.ba8c08108aaf5d14p-109);
1071 real b1 = 2.82847e-25;
1072 assert(feqrel(betaIncompleteInv(0.01, b1, 9e-26), 0x1.549696104490aa9p-830L) >= real.mant_dig-10);
1073
1074 // --- Problematic cases ---
1075 // This is a situation where the series expansion fails to converge
1076 assert( isNaN(betaIncompleteInv(0.12167, 4.0640301659679627772e19L, 0.0813601)));
1077 // This next result is almost certainly erroneous.
1078 // Mathematica states: "(cannot be determined by current methods)"
1079 assert(betaIncomplete(1.16251e20, 2.18e39, 5.45e-20) == -real.infinity);
1080 // WolframAlpha gives no result for this, though indicates that it approximately 1.0 - 1.3e-9
1081 assert(1 - betaIncomplete(0.01, 328222, 4.0375e-5) == 0x1.5f62926b4p-30);
1082 }
1083 }
1084
1085
1086 private {
1087 // Implementation functions
1088
1089 // Continued fraction expansion #1 for incomplete beta integral
1090 // Use when x < (a+1)/(a+b+2)
betaDistExpansion1(real a,real b,real x)1091 real betaDistExpansion1(real a, real b, real x )
1092 {
1093 real xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
1094 real k1, k2, k3, k4, k5, k6, k7, k8;
1095 real r, t, ans;
1096 int n;
1097
1098 k1 = a;
1099 k2 = a + b;
1100 k3 = a;
1101 k4 = a + 1.0L;
1102 k5 = 1.0L;
1103 k6 = b - 1.0L;
1104 k7 = k4;
1105 k8 = a + 2.0L;
1106
1107 pkm2 = 0.0L;
1108 qkm2 = 1.0L;
1109 pkm1 = 1.0L;
1110 qkm1 = 1.0L;
1111 ans = 1.0L;
1112 r = 1.0L;
1113 n = 0;
1114 const real thresh = 3.0L * real.epsilon;
1115 do
1116 {
1117 xk = -( x * k1 * k2 )/( k3 * k4 );
1118 pk = pkm1 + pkm2 * xk;
1119 qk = qkm1 + qkm2 * xk;
1120 pkm2 = pkm1;
1121 pkm1 = pk;
1122 qkm2 = qkm1;
1123 qkm1 = qk;
1124
1125 xk = ( x * k5 * k6 )/( k7 * k8 );
1126 pk = pkm1 + pkm2 * xk;
1127 qk = qkm1 + qkm2 * xk;
1128 pkm2 = pkm1;
1129 pkm1 = pk;
1130 qkm2 = qkm1;
1131 qkm1 = qk;
1132
1133 if ( qk != 0.0L )
1134 r = pk/qk;
1135 if ( r != 0.0L )
1136 {
1137 t = fabs( (ans - r)/r );
1138 ans = r;
1139 }
1140 else
1141 {
1142 t = 1.0L;
1143 }
1144
1145 if ( t < thresh )
1146 return ans;
1147
1148 k1 += 1.0L;
1149 k2 += 1.0L;
1150 k3 += 2.0L;
1151 k4 += 2.0L;
1152 k5 += 1.0L;
1153 k6 -= 1.0L;
1154 k7 += 2.0L;
1155 k8 += 2.0L;
1156
1157 if ( (fabs(qk) + fabs(pk)) > BETA_BIG )
1158 {
1159 pkm2 *= BETA_BIGINV;
1160 pkm1 *= BETA_BIGINV;
1161 qkm2 *= BETA_BIGINV;
1162 qkm1 *= BETA_BIGINV;
1163 }
1164 if ( (fabs(qk) < BETA_BIGINV) || (fabs(pk) < BETA_BIGINV) )
1165 {
1166 pkm2 *= BETA_BIG;
1167 pkm1 *= BETA_BIG;
1168 qkm2 *= BETA_BIG;
1169 qkm1 *= BETA_BIG;
1170 }
1171 }
1172 while ( ++n < 400 );
1173 // loss of precision has occurred
1174 // mtherr( "incbetl", PLOSS );
1175 return ans;
1176 }
1177
1178 // Continued fraction expansion #2 for incomplete beta integral
1179 // Use when x > (a+1)/(a+b+2)
betaDistExpansion2(real a,real b,real x)1180 real betaDistExpansion2(real a, real b, real x )
1181 {
1182 real xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
1183 real k1, k2, k3, k4, k5, k6, k7, k8;
1184 real r, t, ans, z;
1185
1186 k1 = a;
1187 k2 = b - 1.0L;
1188 k3 = a;
1189 k4 = a + 1.0L;
1190 k5 = 1.0L;
1191 k6 = a + b;
1192 k7 = a + 1.0L;
1193 k8 = a + 2.0L;
1194
1195 pkm2 = 0.0L;
1196 qkm2 = 1.0L;
1197 pkm1 = 1.0L;
1198 qkm1 = 1.0L;
1199 z = x / (1.0L-x);
1200 ans = 1.0L;
1201 r = 1.0L;
1202 int n = 0;
1203 const real thresh = 3.0L * real.epsilon;
1204 do
1205 {
1206 xk = -( z * k1 * k2 )/( k3 * k4 );
1207 pk = pkm1 + pkm2 * xk;
1208 qk = qkm1 + qkm2 * xk;
1209 pkm2 = pkm1;
1210 pkm1 = pk;
1211 qkm2 = qkm1;
1212 qkm1 = qk;
1213
1214 xk = ( z * k5 * k6 )/( k7 * k8 );
1215 pk = pkm1 + pkm2 * xk;
1216 qk = qkm1 + qkm2 * xk;
1217 pkm2 = pkm1;
1218 pkm1 = pk;
1219 qkm2 = qkm1;
1220 qkm1 = qk;
1221
1222 if ( qk != 0.0L )
1223 r = pk/qk;
1224 if ( r != 0.0L )
1225 {
1226 t = fabs( (ans - r)/r );
1227 ans = r;
1228 } else
1229 t = 1.0L;
1230
1231 if ( t < thresh )
1232 return ans;
1233 k1 += 1.0L;
1234 k2 -= 1.0L;
1235 k3 += 2.0L;
1236 k4 += 2.0L;
1237 k5 += 1.0L;
1238 k6 += 1.0L;
1239 k7 += 2.0L;
1240 k8 += 2.0L;
1241
1242 if ( (fabs(qk) + fabs(pk)) > BETA_BIG )
1243 {
1244 pkm2 *= BETA_BIGINV;
1245 pkm1 *= BETA_BIGINV;
1246 qkm2 *= BETA_BIGINV;
1247 qkm1 *= BETA_BIGINV;
1248 }
1249 if ( (fabs(qk) < BETA_BIGINV) || (fabs(pk) < BETA_BIGINV) )
1250 {
1251 pkm2 *= BETA_BIG;
1252 pkm1 *= BETA_BIG;
1253 qkm2 *= BETA_BIG;
1254 qkm1 *= BETA_BIG;
1255 }
1256 } while ( ++n < 400 );
1257 // loss of precision has occurred
1258 //mtherr( "incbetl", PLOSS );
1259 return ans;
1260 }
1261
1262 /* Power series for incomplete gamma integral.
1263 Use when b*x is small. */
betaDistPowerSeries(real a,real b,real x)1264 real betaDistPowerSeries(real a, real b, real x )
1265 {
1266 real ai = 1.0L / a;
1267 real u = (1.0L - b) * x;
1268 real v = u / (a + 1.0L);
1269 real t1 = v;
1270 real t = u;
1271 real n = 2.0L;
1272 real s = 0.0L;
1273 real z = real.epsilon * ai;
1274 while ( fabs(v) > z )
1275 {
1276 u = (n - b) * x / n;
1277 t *= u;
1278 v = t / (a + n);
1279 s += v;
1280 n += 1.0L;
1281 }
1282 s += t1;
1283 s += ai;
1284
1285 u = a * log(x);
1286 if ( (a+b) < MAXGAMMA && fabs(u) < MAXLOG )
1287 {
1288 t = gamma(a+b)/(gamma(a)*gamma(b));
1289 s = s * t * pow(x,a);
1290 }
1291 else
1292 {
1293 t = logGamma(a+b) - logGamma(a) - logGamma(b) + u + log(s);
1294
1295 if ( t < MINLOG )
1296 {
1297 s = 0.0L;
1298 } else
1299 s = exp(t);
1300 }
1301 return s;
1302 }
1303
1304 }
1305
1306 /***************************************
1307 * Incomplete gamma integral and its complement
1308 *
1309 * These functions are defined by
1310 *
1311 * gammaIncomplete = ( $(INTEGRATE 0, x) $(POWER e, -t) $(POWER t, a-1) dt )/ $(GAMMA)(a)
1312 *
1313 * gammaIncompleteCompl(a,x) = 1 - gammaIncomplete(a,x)
1314 * = ($(INTEGRATE x, ∞) $(POWER e, -t) $(POWER t, a-1) dt )/ $(GAMMA)(a)
1315 *
1316 * In this implementation both arguments must be positive.
1317 * The integral is evaluated by either a power series or
1318 * continued fraction expansion, depending on the relative
1319 * values of a and x.
1320 */
gammaIncomplete(real a,real x)1321 real gammaIncomplete(real a, real x )
1322 in {
1323 assert(x >= 0);
1324 assert(a > 0);
1325 }
1326 body {
1327 /* left tail of incomplete gamma function:
1328 *
1329 * inf. k
1330 * a -x - x
1331 * x e > ----------
1332 * - -
1333 * k=0 | (a+k+1)
1334 *
1335 */
1336 if (x == 0)
1337 return 0.0L;
1338
1339 if ( (x > 1.0L) && (x > a ) )
1340 return 1.0L - gammaIncompleteCompl(a,x);
1341
1342 real ax = a * log(x) - x - logGamma(a);
1343 /+
1344 if ( ax < MINLOGL ) return 0; // underflow
1345 // { mtherr( "igaml", UNDERFLOW ); return( 0.0L ); }
1346 +/
1347 ax = exp(ax);
1348
1349 /* power series */
1350 real r = a;
1351 real c = 1.0L;
1352 real ans = 1.0L;
1353
1354 do
1355 {
1356 r += 1.0L;
1357 c *= x/r;
1358 ans += c;
1359 } while ( c/ans > real.epsilon );
1360
1361 return ans * ax/a;
1362 }
1363
1364 /** ditto */
gammaIncompleteCompl(real a,real x)1365 real gammaIncompleteCompl(real a, real x )
1366 in {
1367 assert(x >= 0);
1368 assert(a > 0);
1369 }
1370 body {
1371 if (x == 0)
1372 return 1.0L;
1373 if ( (x < 1.0L) || (x < a) )
1374 return 1.0L - gammaIncomplete(a,x);
1375
1376 // DAC (Cephes bug fix): This is necessary to avoid
1377 // spurious nans, eg
1378 // log(x)-x = NaN when x = real.infinity
1379 const real MAXLOGL = 1.1356523406294143949492E4L;
1380 if (x > MAXLOGL)
1381 return igammaTemmeLarge(a, x);
1382
1383 real ax = a * log(x) - x - logGamma(a);
1384 //const real MINLOGL = -1.1355137111933024058873E4L;
1385 // if ( ax < MINLOGL ) return 0; // underflow;
1386 ax = exp(ax);
1387
1388
1389 /* continued fraction */
1390 real y = 1.0L - a;
1391 real z = x + y + 1.0L;
1392 real c = 0.0L;
1393
1394 real pk, qk, t;
1395
1396 real pkm2 = 1.0L;
1397 real qkm2 = x;
1398 real pkm1 = x + 1.0L;
1399 real qkm1 = z * x;
1400 real ans = pkm1/qkm1;
1401
1402 do
1403 {
1404 c += 1.0L;
1405 y += 1.0L;
1406 z += 2.0L;
1407 real yc = y * c;
1408 pk = pkm1 * z - pkm2 * yc;
1409 qk = qkm1 * z - qkm2 * yc;
1410 if ( qk != 0.0L )
1411 {
1412 real r = pk/qk;
1413 t = fabs( (ans - r)/r );
1414 ans = r;
1415 }
1416 else
1417 {
1418 t = 1.0L;
1419 }
1420 pkm2 = pkm1;
1421 pkm1 = pk;
1422 qkm2 = qkm1;
1423 qkm1 = qk;
1424
1425 const real BIG = 9.223372036854775808e18L;
1426
1427 if ( fabs(pk) > BIG )
1428 {
1429 pkm2 /= BIG;
1430 pkm1 /= BIG;
1431 qkm2 /= BIG;
1432 qkm1 /= BIG;
1433 }
1434 } while ( t > real.epsilon );
1435
1436 return ans * ax;
1437 }
1438
1439 /** Inverse of complemented incomplete gamma integral
1440 *
1441 * Given a and p, the function finds x such that
1442 *
1443 * gammaIncompleteCompl( a, x ) = p.
1444 *
1445 * Starting with the approximate value x = a $(POWER t, 3), where
1446 * t = 1 - d - normalDistributionInv(p) sqrt(d),
1447 * and d = 1/9a,
1448 * the routine performs up to 10 Newton iterations to find the
1449 * root of incompleteGammaCompl(a,x) - p = 0.
1450 */
gammaIncompleteComplInv(real a,real p)1451 real gammaIncompleteComplInv(real a, real p)
1452 in {
1453 assert(p >= 0 && p <= 1);
1454 assert(a>0);
1455 }
1456 body {
1457 if (p == 0) return real.infinity;
1458
1459 real y0 = p;
1460 const real MAXLOGL = 1.1356523406294143949492E4L;
1461 real x0, x1, x, yl, yh, y, d, lgm, dithresh;
1462 int i, dir;
1463
1464 /* bound the solution */
1465 x0 = real.max;
1466 yl = 0.0L;
1467 x1 = 0.0L;
1468 yh = 1.0L;
1469 dithresh = 4.0 * real.epsilon;
1470
1471 /* approximation to inverse function */
1472 d = 1.0L/(9.0L*a);
1473 y = 1.0L - d - normalDistributionInvImpl(y0) * sqrt(d);
1474 x = a * y * y * y;
1475
1476 lgm = logGamma(a);
1477
1478 for ( i=0; i<10; i++ )
1479 {
1480 if ( x > x0 || x < x1 )
1481 goto ihalve;
1482 y = gammaIncompleteCompl(a,x);
1483 if ( y < yl || y > yh )
1484 goto ihalve;
1485 if ( y < y0 )
1486 {
1487 x0 = x;
1488 yl = y;
1489 }
1490 else
1491 {
1492 x1 = x;
1493 yh = y;
1494 }
1495 /* compute the derivative of the function at this point */
1496 d = (a - 1.0L) * log(x0) - x0 - lgm;
1497 if ( d < -MAXLOGL )
1498 goto ihalve;
1499 d = -exp(d);
1500 /* compute the step to the next approximation of x */
1501 d = (y - y0)/d;
1502 x = x - d;
1503 if ( i < 3 ) continue;
1504 if ( fabs(d/x) < dithresh ) return x;
1505 }
1506
1507 /* Resort to interval halving if Newton iteration did not converge. */
1508 ihalve:
1509 d = 0.0625L;
1510 if ( x0 == real.max )
1511 {
1512 if ( x <= 0.0L )
1513 x = 1.0L;
1514 while ( x0 == real.max )
1515 {
1516 x = (1.0L + d) * x;
1517 y = gammaIncompleteCompl( a, x );
1518 if ( y < y0 )
1519 {
1520 x0 = x;
1521 yl = y;
1522 break;
1523 }
1524 d = d + d;
1525 }
1526 }
1527 d = 0.5L;
1528 dir = 0;
1529
1530 for ( i=0; i<400; i++ )
1531 {
1532 x = x1 + d * (x0 - x1);
1533 y = gammaIncompleteCompl( a, x );
1534 lgm = (x0 - x1)/(x1 + x0);
1535 if ( fabs(lgm) < dithresh )
1536 break;
1537 lgm = (y - y0)/y0;
1538 if ( fabs(lgm) < dithresh )
1539 break;
1540 if ( x <= 0.0L )
1541 break;
1542 if ( y > y0 )
1543 {
1544 x1 = x;
1545 yh = y;
1546 if ( dir < 0 )
1547 {
1548 dir = 0;
1549 d = 0.5L;
1550 } else if ( dir > 1 )
1551 d = 0.5L * d + 0.5L;
1552 else
1553 d = (y0 - yl)/(yh - yl);
1554 dir += 1;
1555 }
1556 else
1557 {
1558 x0 = x;
1559 yl = y;
1560 if ( dir > 0 )
1561 {
1562 dir = 0;
1563 d = 0.5L;
1564 } else if ( dir < -1 )
1565 d = 0.5L * d;
1566 else
1567 d = (y0 - yl)/(yh - yl);
1568 dir -= 1;
1569 }
1570 }
1571 /+
1572 if ( x == 0.0L )
1573 mtherr( "igamil", UNDERFLOW );
1574 +/
1575 return x;
1576 }
1577
1578 @safe unittest
1579 {
1580 //Values from Excel's GammaInv(1-p, x, 1)
1581 assert(fabs(gammaIncompleteComplInv(1, 0.5) - 0.693147188044814) < 0.00000005);
1582 assert(fabs(gammaIncompleteComplInv(12, 0.99) - 5.42818075054289) < 0.00000005);
1583 assert(fabs(gammaIncompleteComplInv(100, 0.8) - 91.5013985848288L) < 0.000005);
1584 assert(gammaIncomplete(1, 0)==0);
1585 assert(gammaIncompleteCompl(1, 0)==1);
1586 assert(gammaIncomplete(4545, real.infinity)==1);
1587
1588 // Values from Excel's (1-GammaDist(x, alpha, 1, TRUE))
1589
1590 assert(fabs(1.0L-gammaIncompleteCompl(0.5, 2) - 0.954499729507309L) < 0.00000005);
1591 assert(fabs(gammaIncomplete(0.5, 2) - 0.954499729507309L) < 0.00000005);
1592 // Fixed Cephes bug:
1593 assert(gammaIncompleteCompl(384, real.infinity)==0);
1594 assert(gammaIncompleteComplInv(3, 0)==real.infinity);
1595 // Fixed a bug that caused gammaIncompleteCompl to return a wrong value when
1596 // x was larger than a, but not by much, and both were large:
1597 // The value is from WolframAlpha (Gamma[100000, 100001, inf] / Gamma[100000])
1598 static if (real.mant_dig >= 64) // incl. 80-bit reals
1599 assert(fabs(gammaIncompleteCompl(100000, 100001) - 0.49831792109) < 0.000000000005);
1600 else
1601 assert(fabs(gammaIncompleteCompl(100000, 100001) - 0.49831792109) < 0.00000005);
1602 }
1603
1604
1605 // DAC: These values are Bn / n for n=2,4,6,8,10,12,14.
1606 immutable real [7] Bn_n = [
1607 1.0L/(6*2), -1.0L/(30*4), 1.0L/(42*6), -1.0L/(30*8),
1608 5.0L/(66*10), -691.0L/(2730*12), 7.0L/(6*14) ];
1609
1610 /** Digamma function
1611 *
1612 * The digamma function is the logarithmic derivative of the gamma function.
1613 *
1614 * digamma(x) = d/dx logGamma(x)
1615 *
1616 * References:
1617 * 1. Abramowitz, M., and Stegun, I. A. (1970).
1618 * Handbook of mathematical functions. Dover, New York,
1619 * pages 258-259, equations 6.3.6 and 6.3.18.
1620 */
digamma(real x)1621 real digamma(real x)
1622 {
1623 // Based on CEPHES, Stephen L. Moshier.
1624
1625 real p, q, nz, s, w, y, z;
1626 long i, n;
1627 int negative;
1628
1629 negative = 0;
1630 nz = 0.0;
1631
1632 if ( x <= 0.0 )
1633 {
1634 negative = 1;
1635 q = x;
1636 p = floor(q);
1637 if ( p == q )
1638 {
1639 return real.nan; // singularity.
1640 }
1641 /* Remove the zeros of tan(PI x)
1642 * by subtracting the nearest integer from x
1643 */
1644 nz = q - p;
1645 if ( nz != 0.5 )
1646 {
1647 if ( nz > 0.5 )
1648 {
1649 p += 1.0;
1650 nz = q - p;
1651 }
1652 nz = PI/tan(PI*nz);
1653 }
1654 else
1655 {
1656 nz = 0.0;
1657 }
1658 x = 1.0 - x;
1659 }
1660
1661 // check for small positive integer
1662 if ((x <= 13.0) && (x == floor(x)) )
1663 {
1664 y = 0.0;
1665 n = lrint(x);
1666 // DAC: CEPHES bugfix. Cephes did this in reverse order, which
1667 // created a larger roundoff error.
1668 for (i=n-1; i>0; --i)
1669 {
1670 y+=1.0L/i;
1671 }
1672 y -= EULERGAMMA;
1673 goto done;
1674 }
1675
1676 s = x;
1677 w = 0.0;
1678 while ( s < 10.0 )
1679 {
1680 w += 1.0/s;
1681 s += 1.0;
1682 }
1683
1684 if ( s < 1.0e17 )
1685 {
1686 z = 1.0/(s * s);
1687 y = z * poly(z, Bn_n);
1688 } else
1689 y = 0.0;
1690
1691 y = log(s) - 0.5L/s - y - w;
1692
1693 done:
1694 if ( negative )
1695 {
1696 y -= nz;
1697 }
1698 return y;
1699 }
1700
1701 @safe unittest
1702 {
1703 // Exact values
1704 assert(digamma(1.0)== -EULERGAMMA);
1705 assert(feqrel(digamma(0.25), -PI/2 - 3* LN2 - EULERGAMMA) >= real.mant_dig-7);
1706 assert(feqrel(digamma(1.0L/6), -PI/2 *sqrt(3.0L) - 2* LN2 -1.5*log(3.0L) - EULERGAMMA) >= real.mant_dig-7);
1707 assert(digamma(-5.0).isNaN());
1708 assert(feqrel(digamma(2.5), -EULERGAMMA - 2*LN2 + 2.0 + 2.0L/3) >= real.mant_dig-9);
1709 assert(isIdentical(digamma(NaN(0xABC)), NaN(0xABC)));
1710
1711 for (int k=1; k<40; ++k)
1712 {
1713 real y=0;
1714 for (int u=k; u >= 1; --u)
1715 {
1716 y += 1.0L/u;
1717 }
1718 assert(feqrel(digamma(k+1.0), -EULERGAMMA + y) >= real.mant_dig-2);
1719 }
1720 }
1721
1722 /** Log Minus Digamma function
1723 *
1724 * logmdigamma(x) = log(x) - digamma(x)
1725 *
1726 * References:
1727 * 1. Abramowitz, M., and Stegun, I. A. (1970).
1728 * Handbook of mathematical functions. Dover, New York,
1729 * pages 258-259, equations 6.3.6 and 6.3.18.
1730 */
logmdigamma(real x)1731 real logmdigamma(real x)
1732 {
1733 if (x <= 0.0)
1734 {
1735 if (x == 0.0)
1736 {
1737 return real.infinity;
1738 }
1739 return real.nan;
1740 }
1741
1742 real s = x;
1743 real w = 0.0;
1744 while ( s < 10.0 )
1745 {
1746 w += 1.0/s;
1747 s += 1.0;
1748 }
1749
1750 real y;
1751 if ( s < 1.0e17 )
1752 {
1753 immutable real z = 1.0/(s * s);
1754 y = z * poly(z, Bn_n);
1755 } else
1756 y = 0.0;
1757
1758 return x == s ? y + 0.5L/s : (log(x/s) + 0.5L/s + y + w);
1759 }
1760
1761 @safe unittest
1762 {
1763 assert(logmdigamma(-5.0).isNaN());
1764 assert(isIdentical(logmdigamma(NaN(0xABC)), NaN(0xABC)));
1765 assert(logmdigamma(0.0) == real.infinity);
1766 for (auto x = 0.01; x < 1.0; x += 0.1)
1767 assert(approxEqual(digamma(x), log(x) - logmdigamma(x)));
1768 for (auto x = 1.0; x < 15.0; x += 1.0)
1769 assert(approxEqual(digamma(x), log(x) - logmdigamma(x)));
1770 }
1771
1772 /** Inverse of the Log Minus Digamma function
1773 *
1774 * Returns x such $(D log(x) - digamma(x) == y).
1775 *
1776 * References:
1777 * 1. Abramowitz, M., and Stegun, I. A. (1970).
1778 * Handbook of mathematical functions. Dover, New York,
1779 * pages 258-259, equation 6.3.18.
1780 *
1781 * Authors: Ilya Yaroshenko
1782 */
logmdigammaInverse(real y)1783 real logmdigammaInverse(real y)
1784 {
1785 import std.numeric : findRoot;
1786 // FIXME: should be returned back to enum.
1787 // Fix requires CTFEable `log` on non-x86 targets (check both LDC and GDC).
1788 immutable maxY = logmdigamma(real.min_normal);
1789 assert(maxY > 0 && maxY <= real.max);
1790
1791 if (y >= maxY)
1792 {
1793 //lim x->0 (log(x)-digamma(x))*x == 1
1794 return 1 / y;
1795 }
1796 if (y < 0)
1797 {
1798 return real.nan;
1799 }
1800 if (y < real.min_normal)
1801 {
1802 //6.3.18
1803 return 0.5 / y;
1804 }
1805 if (y > 0)
1806 {
1807 // x/2 <= logmdigamma(1 / x) <= x, x > 0
1808 // calls logmdigamma ~6 times
1809 return 1 / findRoot((real x) => logmdigamma(1 / x) - y, y, 2*y);
1810 }
1811 return y; //NaN
1812 }
1813
1814 @safe unittest
1815 {
1816 import std.typecons;
1817 //WolframAlpha, 22.02.2015
1818 immutable Tuple!(real, real)[5] testData = [
1819 tuple(1.0L, 0.615556766479594378978099158335549201923L),
1820 tuple(1.0L/8, 4.15937801516894947161054974029150730555L),
1821 tuple(1.0L/1024, 512.166612384991507850643277924243523243L),
1822 tuple(0.000500083333325000003968249801594877323784632117L, 1000.0L),
1823 tuple(1017.644138623741168814449776695062817947092468536L, 1.0L/1024),
1824 ];
1825 foreach (test; testData)
1826 assert(approxEqual(logmdigammaInverse(test[0]), test[1], 2e-15, 0));
1827
1828 assert(approxEqual(logmdigamma(logmdigammaInverse(1)), 1, 1e-15, 0));
1829 assert(approxEqual(logmdigamma(logmdigammaInverse(real.min_normal)), real.min_normal, 1e-15, 0));
1830 assert(approxEqual(logmdigamma(logmdigammaInverse(real.max/2)), real.max/2, 1e-15, 0));
1831 assert(approxEqual(logmdigammaInverse(logmdigamma(1)), 1, 1e-15, 0));
1832 assert(approxEqual(logmdigammaInverse(logmdigamma(real.min_normal)), real.min_normal, 1e-15, 0));
1833 assert(approxEqual(logmdigammaInverse(logmdigamma(real.max/2)), real.max/2, 1e-15, 0));
1834 }
1835