HP
Prime and Casio fx5800p 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
fx5800p 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 20160302
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_noteNemes201010
Retrieved March 1, 2016
Toth,
Viktor T. “The Gamma Function” R/S Programmable Calculators http://www.rskey.org/CMS/thelibrary?id=11 Retrieved March 1, 2016
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