Monte-Carlo Pi

[C.Ret]

(01-13-2022 01:04 AM)ttw Wrote:  The use of multiplies 113 (and some other numbers) comes from the theory of continued fractions. Any number X may be approximated by a fraction P/Q (possibly improper). Trivially the error e=|X-P/Q| is less than 1/2Q. For each X, there exists a sequence of Qs with an error of less than 1/2Q^2. The sequence for Pi is 7, 106, 113, 33102, 33215,...

If one takes samples of size 33000, one gets at best errors of size 1/33000; with size 33215, there may be some errors of size 1/1103236225. Some samples are extraordinarily good.

Of course, in "real -ife" computation, one wouldn't know what Q to use.

Thank you for this explanation, I will now have to deepen this somewhat and perfect my modest knowledge on this subject ...

In the meantime, I have found a way to shorten VERSION#2 while making it even faster.
But I won't say anything about it here, I hope that other HP-15C enthusiasts have already found how to do it.