# A new formula to compute the n'th binary digit of pi

Fabrice Bellard

January 20, 1997

We describe here a mean to find formulas similar to those in [1]. We show in particular that

 (1)

This formula is very interesting because, with the algorithm described in [1], it enables us to compute the th binary digit of 43% faster than the previous known formula [1]:

 (2)

The method to get formulas such as (1) is in fact very simple. We use that

and

for real.

In particular

 (3)

and
 (4)

With in (3) we get

 (5)

which is mentionned in [1].

Some classical arctangent relations give interesting results:

 (6) (7) (8) (9)

In particular, we obtain from (6) and (3)

which gives (1) by reordering the terms.

The existence of a formula faster than (1) to calculate the th binary digit of remains an open question.

## Bibliography

1
David H. Bailey, Peter B. Borwein and Simon Plouffe, On the Rapid Computation of Various Polylogarithmic Constants, to appear in April 1997 in Mathematics of Computation.

Thu Feb 15 23:31:20 CET 2007
Fabrice Bellard (http://bellard.org)