stirling_factorial_correction_terms.sf

#!/usr/bin/ruby

# Correction terms for Stirling's asymptotic formula for n!

# See also:
#   https://arxiv.org/pdf/1002.3894.pdf
#   https://en.wikipedia.org/wiki/Stirling%27s_approximation

func factorial_correction_terms(n, r = n) {
    1..r -> sum {|k|
        bernoulli(2*k) / (2*k * (2*k - 1) * n**(2*k - 1))
    }
}

func log_factorial_approx(n){
   n*(log(n) - 1) + log(Num.tau * n)/2 + factorial_correction_terms(n)
}

say "=> Approximation of log(100!):"
say lngamma(100 + 1)
say log_factorial_approx(100)

__END__
=> Approximation of log(100!):
363.739375555563490144079993369655638027823921063
363.739375555563490144079993369655638027823921063