HP Prime and Casio fx-5800p Approximating the Factorial Function

HP Prime and Casio fx-5800p Approximating the Factorial Function

A quick way to estimate the factorial function, which is good for all real numbers (and complex numbers with the HP Prime) is determined by Gergő Nemes Ph. D (Mathematics, University of Edinburgh):

N! ≈ N^N * √(2*π*N) * e^(1/(12*N+2/(5*N+53/(42*N)))-N)

The error is the order of 1 + O(N^-8).   Like the Sterling approximation formula, this formula is a better approximation as N increases. 

Casio fx-5800p Program:  GERGO

“GERGO RSKEY.ORG”

“N”? → N

N^(N)*√(2πN)*e^(

1÷(12N+2÷(5N+53÷

(42N)))-N)

HP Prime:  GERGO

EXPORT GERGO(N)

BEGIN

// rskey.org 2016-03-02

RETURN N^N*√(2*N*π)*

e^(1/(12*N+2/(5*N+53/(42*N)))

-N);

END;

How accurate is it?

Here a test of some random values to compare accuracy.

Values

N	N! (Determined by Wolfram Alpha)	N! approximation
1.25	1.13300309631…	1.133039736
3.08	6.64025496878…	6.640255733
5	120	120.0000005
6.64	2460.94013688180…	2460.940138
8.27	72172.53628421024…	72172.53629
11.5	1.368433654655… x 10^8	136843365.5

Source:

“Sterling’s Approximation”  Wikipedia – Page February 26, 2016 https://en.wikipedia.org/wiki/Stirling%27s_approximation#cite_note-Nemes2010-10Retrieved March 1, 2016

Toth, Viktor T.  “The Gamma Function”  R/S Programmable Calculators  http://www.rskey.org/CMS/the-library?id=11  Retrieved March 1, 2016