#!/usr/bin/ruby
# Daniel "Trizen" Șuteu
# License: GPLv3
# Date: 27 September 2016
# Website: https://github.com/trizen
# The inverse of n factorial, based on the inverse of Stirling's approximation.
define τ = Num.tau
define e = Num.e
define S = τ.sqrt.ln
define T = τ.root(-2*e)
func inv_fac_W(n) {
    var l = (n.ln - S)
    l / lambert_w(l / e) - 1/2
}
func inv_fac_lgrt(n) {
    lgrt(T * n.root(e)) * e - 1/2
}
var tests = [
    [3, 6],
    [4, 24],
    [5, 120],
    [10, 3628800],
    [15, 1307674368000],
]
for n,f in tests {
    var a = inv_fac_W(f)
    var b = inv_fac_lgrt(f)
    printf("F(%2s!) =~ (%.10g, %.10g)\n", n, a, b)
    if (a.round(-20) != b.round(-20)) {
        die "#{a} != #{b}"
    }
    if (a.round(0) != n) {
        die "a=#{a} is incorrect!"
    }
}
var x = inv_fac_W(-123456789)
var y = inv_fac_lgrt(-123456789)
assert_eq(x.round(-20), y.round(-20))
assert((sqrt(2*Num.pi*x) * (x/e)**x) / -123456789 -> is_between(0.9, 0.99))