# [Kattis] Stirling’s Approximation

Stirling’s approximation is a powerful mathematical formula which can be used to estimate the value of N!, and it is especially handy when it comes to large values of N. It was derived by the the French mathematician Abraham de Moivre (1667-1754) and the Scottish mathematician James Stirling (1692-1770). The approximation is given by this formula:

where **π** is roughly equal to **3.14**, and **e** (the Euler’s number) is roughly equal to** 2.718**.

Stirling’s approximation is an invaluable tool in various fields, including statistics, physics, and engineering, where large factorials frequently arise. It becomes even more accurate as N increases.

This post is associated with a programming puzzle from Kattis. Here you can find more about the puzzle and its solution:

And here you will find the code for the solution:

# Libraries from math import log, pi, e # function to calculate the logarithm of S(n) def log_sn(n): return 0.5 * (log(2) + log(pi) + log(n)) + n * (log(n) - log(e)) # function to calculate the logarithm of the factorial def log_fact(n): return sum(log(i) for i in range(2, n+1)) # The puzzle while True: # receive n n = int(input()) # the exit condition if n == 0: break else: print(e ** (log_fact(n) - log_sn(n))