range error - or "How to take the factorial of a huge

[Jim Witte]

> 2.
> Work with logs of expressions.

   The MathWorld page on Stirling's approximation gives a formula for 
(log n!) as well. The first formula gives an *exect* (I think) 
derivation for ln n! as

ln n! = sum(k=1..n, ln k)

which is approximately (by changing that summation to an integral from 
1..n and simplifying it, to give

ln n! ~ n ln n - n

It also gives an (exact) integral definition of factorial:

n! = integral(0, infinity, ( exp(-x)*(x^n) ), dx)

So you could write an definite-integration function, taking a small 
slice parameter for dx, and sum the integrand term:  This would be 
something around the following

function integrand (x, n)
	reuturn (exp(-x) * x^n)
end integrand

function fact (n)
	put 0 into min
	put 100 into max
	put 0.001 into dx
	put min into x
	put 0 into sum
	add integrand(x, n) to sum
	add dx to x
	if (x>max) then
		return sum
	end if
end fact

since exp(-x) gets very small as x gets large, 100 should provide a 
reasonable maximum.  I don't know how small dx should really be.  I 
don't know how much better (or worse) this is compared to the foluma I 
quoted earlier.

Jim