factorial_approximation_from_incomplete_gamma.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 02 February 2019
# https://github.com/trizen

# A new asymptotic formula for approximating n-factorial.

# See also:
#   https://en.wikipedia.org/wiki/Incomplete_gamma_function

const π = Num.pi
const e = Num.e

func incomplete_gamma(s, x) {

    if (s == 1) {
        return exp(-x)
    }

    ((s-1) * __FUNC__(s-1, x)) + (x**(s-1) * exp(-x))
}

func incomplete_e (n) {
    e * incomplete_gamma(n+1, 1) / gamma(n+1)
}

func incomplete_e_asymptotic (n) {
    (-sqrt(1/n)/sqrt(2*π) - (11 * (1/n)**(3/2))/(12*sqrt(2*π))) * exp((1 - log(n))*n)
}

# Identity for gamma(n+1) = n!
func approximate_factorial(n) {

    var a = (e - incomplete_e(n))
    var b = incomplete_e(n-1)

    1 / (e - (a+b))
}

# Concept only.
func approximate_factorial_asymptotic_1(n) {

    var a = (e - incomplete_e_asymptotic(n+1))
    var b = incomplete_e_asymptotic(n)

    1 / (e - (a + b))
}

# Useful asymptotic formula for n! (derived from #1).
func approximate_factorial_asymptotic_2(n) {
    (12*sqrt(2*π)) / (exp(n) * (n**(-n - 3/2) * (12*n + 11) - (e * (n + 1)**(-n - 5/2) * (12*n + 23))))
}

func stirling_approximation(n) {
    sqrt(2*π*n) * (n/e)**n
}

var n = 20

say n!
say approximate_factorial(n)
say approximate_factorial_asymptotic_1(n)
say approximate_factorial_asymptotic_2(n)
say stirling_approximation(n)

__END__
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