Fastest way to compute a factorial in a numba nopython function

Question

I have a function that I want to compile with numba, however I need to calculate a factorial inside that function. Unfortunatly numba doesn't support math.factorial:

import math
import numba as nb

@nb.njit
def factorial1(x):
    return math.factorial(x)

factorial1(10)
# UntypedAttributeError: Failed at nopython (nopython frontend)

I saw that it supported math.gamma (which could be used to calculate the factorial), however contrary to the real math.gamma function it doesn't return floats that represent "integral values":

@nb.njit
def factorial2(x):
    return math.gamma(x+1)

factorial2(10)
# 3628799.9999999995  <-- not exact

math.gamma(11)
# 3628800.0  <-- exact

and it's slow compared to math.factorial:

%timeit factorial2(10)
# 1.12 µs ± 11.3 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)
%timeit math.factorial(10)
# 321 ns ± 6.12 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)

So I decided to define my own function:

@nb.njit
def factorial3(x):
    n = 1
    for i in range(2, x+1):
        n *= i
    return n

factorial3(10)
# 3628800

%timeit factorial3(10)
# 821 ns ± 12.2 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)

It's still slower than math.factorial but it's faster than a math.gamma based numba function and the value is "exact".

So I'm looking for the fastest way to compute the factorial of a positive integer number (<= 20; to avoid overflow) inside a nopython numba function.

Answer 1

For values <= 20, python is using a lookup table, as was suggested in the comments. https://github.com/python/cpython/blob/3.6/Modules/mathmodule.c#L1452

LOOKUP_TABLE = np.array([
    1, 1, 2, 6, 24, 120, 720, 5040, 40320,
    362880, 3628800, 39916800, 479001600,
    6227020800, 87178291200, 1307674368000,
    20922789888000, 355687428096000, 6402373705728000,
    121645100408832000, 2432902008176640000], dtype='int64')

@nb.jit
def fast_factorial(n):
    if n > 20:
        raise ValueError
    return LOOKUP_TABLE[n]

Called from python it's slightly slower than the python version due to the numba dispatch overhead.

In [58]: %timeit math.factorial(10)
10000000 loops, best of 3: 79.4 ns per loop

In [59]: %timeit fast_factorial(10)
10000000 loops, best of 3: 173 ns per loop

But called inside another numba function can be much faster.

def loop_python():
    for i in range(10000):
        for n in range(21):
            math.factorial(n)

@nb.njit
def loop_numba():
    for i in range(10000):
        for n in range(21):
            fast_factorial(n)

In [65]: %timeit loop_python()
10 loops, best of 3: 36.7 ms per loop

In [66]: %timeit loop_numba()
10000000 loops, best of 3: 73.6 ns per loop