Wednesday, March 2, 2016

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-Nemes2010-10 Retrieved 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

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