From jhedden@motown.ge.com Thu Nov  2 16:37:34 1995

The spigot algorithm for pi is:

          1          2          3                    k
pi = 2 + --- * (2 + --- * (2 + --- * (2 + ...  (2 + ---- * (2 + ... ))...)))
          3          5          7                   2k+1

The last term can be approximated by:

              k
	(2 + ---- * (4))
             2k+1

where k = n * log (10)
                 2

for a desired precision of n decimal digits.


-------------------

a remark by Jacques Gelinas:

the following
article that presents a PROOF of correctness for the spigot
algorithms for e and \pi.

/* 
   A spigot algorithm for the Digits of \pi, Stanley Rabinowitz
   and Stan Wagon, Am.Math.Monthly, March 1995, 195-203
*/

Footnote, page 197:

"Any digit-producing algorithm fo a presumed-normal number x suffers
 from the drawback that, although unlikely, can impinge on the result.
 If x is between 1 and 10 and the algorithm says that the first 100
 digits of x are, say, 4,6,5,0,7,...,3,9,9,9,9,9 then one cannot be sure 
 that the last 6 digits are correct. They will be the digits of a
 certain approximation to x that is within 5x10^-100 of the true value.
 One cannot simply go farther until a non-9 is reached, because memory
 allocation must be made in advance..."