← All modules

Gamma, factorial, binomial coefficients, Bernoulli numbers and the Euler–Mascheroni constant — exact integers and rationals where the answer is exact, floating point where it cannot be.

Version

0.1.1 zef:antononcube

Depends

nothing outside the core

License

Artistic-2.0

Its own test suite

4 files, green

Checked

2026-09-15 against Raku++ 3.28.0 and Rakudo 2026.08

Where it lives

raku.land · source

Install it #

$ rakupp install Math::SpecialFunctions

zef install Math::SpecialFunctions writes the same store; either installer leaves the module usable by both engines.

What it is for #

The handful of functions that turn up the moment a program does analysis rather than arithmetic: the gamma function that extends the factorial to the whole plane, the exact factorial and binomial coefficient a combinatorial count needs, the Bernoulli numbers behind Euler–Maclaurin summation and the zeta function at even integers, and γ, the constant that relates the harmonic numbers to the logarithm.

Raku's own arithmetic makes an unusual split possible here, and this distribution takes it: the answers that are exact come back exact — factorial as an Int of any size, the Bernoulli numbers as FatRat fractions — and only gamma is floating point, because it must be.

Exact where it can be #

use Math::SpecialFunctions;

say factorial(20);
say factorial(30);
say binomial(52, 5);
say bernoulli-b(12), ' = ', bernoulli-b(12).nude.join('/');
say bernoulli-b(2).^name, ' ', bernoulli-b(0).^name;

say gamma(0.5);
say gamma(0.5) ** 2;
say gamma(5);
say factorial(4);

2432902008176640000
265252859812191058636308480000000
2598960
-0.253114 = -691/2730
FatRat Int
1.7724538509055163
3.1415926535897944
23.999999999999986
24

Γ(½) is the square root of π, and squaring it back gets π to fifteen digits. The last two lines are the same quantity computed two ways: gamma(5) is 4 factorial, and it is not 24 — it is a floating approximation of 24. Reach for factorial whenever the argument is a non-negative integer and you want the exact value; reach for gamma when it is not.

The one thing to know #

Every Bernoulli number with an odd index above 1 is exactly zero. This implementation returns the previous even one instead:

use Math::SpecialFunctions;

for 2..9 -> $n {
    say $n, ' -> ', bernoulli-b($n).nude.join('/'),
        ($n %% 2 ?? '' !! '   (should be 0)');
}
say bernoulli-b(3) == bernoulli-b(2);
say bernoulli-b(7) == bernoulli-b(6);

2 -> 1/6
3 -> 1/6   (should be 0)
4 -> -1/30
5 -> -1/30   (should be 0)
6 -> 1/42
7 -> 1/42   (should be 0)
8 -> -1/30
9 -> -1/30   (should be 0)
True
True

The even-index values are all correct, and every odd one from 3 up is the even one below it repeated. It is a quiet wrong answer — a plausible rational, not a NaN and not an exception — so a Euler–Maclaurin sum that runs over every index silently picks up terms that should have vanished. Sum over even indices only, or test $n %% 2 yourself and substitute zero.

Two smaller traps in the same family. factorial(-1) returns 1 rather than failing, and binomial returns 0 for every case it does not like, including binomial(-5, 2), whose generalised value is 15. And euler-gamma's :prec is really a two-state switch: up to 16 it answers a Num, and from 17 up it answers one fixed Rat whatever you ask for.