Re: Early calculators that could do x! for x > 69 ? Message #13 Posted by Bart (UK) on 26 May 2010, 5:34 p.m., in response to message #12 by Bart (UK)
OK, so this got me thinking, how to do a factorial using this "feature" on the SR7919. Just doing simple 1*2*3... doesn't help much of course and only gets to 70 and is errorprone (I think human error increases as the tediousness of repetition increases).
So, in my search, Wikipedea was my friend and on their factorial page I found the following approximation by Srinivasa Ramanujan:
LN(n!) = n*LN(n)  n + (LN(n(1+4n(1+2n))))/6 + (LN(pi))/2
So all I have to do is calculate the right hand side (let's call it RHS) , and do e^x, except that the SR7919 doesn't do exponent overflow for the e^x function. The most complex function it does so for is x^2. Now I can see those cogs in your minds already at work and before you have finished reading this sentence, worked out that SQR(n!)=e^(RHS/2) then do x^2 and voila! I have n! up to n=120. Some results:
n SR7919 HP20S
70 1.1978 E00 1.1978572 E100
99 9.3327 E55 9.3326215 E155
120 6.6894 E98 6.6895029 E198 These are just a few sample points and accuracy may differ more at other points.
My next "challenge"  to find the least number of keystrokes to do this :).
(And why a 20S for comparing? It happened to be laying on the table and is capable of E499)
Edit: I used the 20S builtin n! function, whose accuracy I don't know either. It only accepts integers, so is not a Gamma aprroximation though.
Edited: 26 May 2010, 5:57 p.m.
