Bailey, Borwein, and Plouffe in the article
``On the Rapid Computation of Various Polylogarithmic Constants''
give the following formula for 
,
which allows the computation of an individual binary digit in the
binary expansion of with small storage
Erich Kaltofen and C. Ryan Vinroot, following their integer relation approach (re)-discovered the following alternate formula:
A Maple V.4 session showing our derivation is here.
Adamchik and Wagon [Am. Math. Monthly, Nov. 1997; url] give the following pretty variant:
Their solution is dependent on the two given above as follows:
which is 2 times BBP plus 1 times our variant.
Fabrice Bellard has given a
formula
for base 
, which allows
a faster algorithm for computing the hexadecimal digits of .