digits in a factorial number result

digits_factorial.py

# See: https://mathoverflow.net/questions/19170/how-good-is-kamenetskys-formula-for-the-number-of-digits-in-n-factorial/44927#44927
# See: https://math.stackexchange.com/questions/661227/factorial-length

import math
import functools

_LIMIT = 5*pow(10, 7)

def digitsInFactorial(N: int) -> int:
    cache_ln = functools.lru_cache(maxsize=None)(math.log)
    cache_log = functools.lru_cache(maxsize=None)(math.log10)
    sum_log = functools.lru_cache(maxsize=None)(lambda n: sum(cache_log(i) for i in range(2, n+1)))
    fact_exact = lambda n: math.floor(sum_log(n)) + 1
    fact_approx = lambda n: math.floor(((cache_ln(2*math.pi*n)/2)+n*(cache_ln(n)-1))/cache_ln(10))+1
    return fact_approx(N) if N < _LIMIT else fact_exact(N)