After your editing of the original question:
You will not find an easy formula for the exact evaluation of factorials, except using $n! = \Gamma(n+1)$. If you want to compute faster than just multiply the numbers, use e.g. the prime power table approch.
A basic fast algorithm is described in P.B. Borwein, On the complexity of calculating factorials, Journal of Algorithms, Vol.6, pp. 376-380, 1985. Available as http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P29.pdf
Here the four steps of Borwein's algorithm for $n!$:
Construct the prime table $2 \le p_k \le n\;$ (e.g. using a sieve method)
Compute the exponents of the $p_k$ in $n!\;$ (e.g. Legendre's formula)
Compute $O(\log n)$ numbers $\alpha_i$ (for details see the paper)
Compute $n! = \prod \alpha_i$
For actual implementations see e.g. the mentioned FastFactorial Functions page.