Statistical approximations, calculator style (3)

Guest

A simple approximation to x!, valid for x>=0
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While there is no lack of simple approximations to the factorial, they
are generally not performing well for small positive or 0 arguments.
The following approximation to x! is valid for x>=0 (x not necessarily
an integer):

f(x) = t^(x+1/2) e^(-t) sqrt(2 pi)

where:

t = x + c + d/(x+1)

and the max. |rel. error| is for conveniently rounded versions of
(c,d):

<1.045E-5 for (c,d) = (0.788208,0.031083)
<1.060E-5 for (c,d) = (0.78821,0.03108)
<1.395E-5 for (c,d) = (0.7882,0.0311)

A little extra accuracy can be gained by adjusting the value 1 in the
denominator, but hardly worth the effort if we wish to keep the formula
simple.

Derivation:
The Stirling and Burnside approximations (regarding the latter see
contributions to this forum by David W. Cantrell) can be written on the
common form:

f(x) = (x+a)^(x+1/2) e^(-x-a) sqrt(2 pi)

where a=0 for Stirling's and a=1/2 for Burnside's formula.
The optimal value of a depends on x approximately as follows:

a = c + d/x

where (c,d) are constants.

Regards,

Knud Thomsen
(Aarhus, Denmark)

Guest

Obviously, in the 'Derivation' section, the formula for a should read:
'a = c + d/(x+1)' .
Knud Thomsen

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