Computation of Catalan constant :
Number of decimal digits : 12 500 000
When : 30/12/97 -> 31/12/97
Verification : 03/01/98 -> 04/01/98
Machine : sgi r10000, 256 Mo of memory
by Xavier Gourdon.
3 steps :
1 - Computation of S0 = sum_{n>=0} n!/[(n+1)..(2n+1)]/(2n+1)
Timing : 18h1m37.78s
The technique used was a variant of Brent algorithm to compute
such a series in O(n log(n)^3).
The key of the speed was to use a fast multiplication of big numbers,
with an efficient FFT modulo two prime numbers of 59 bits.
A verification has been done using a different splitting in
the Brent technique. Timing of verif: 23h58'36"
2 - Computation of Pi
Timings : 2h42'10" (using FFT and a quartiquely convergent sequence)
A verification has been done using the Gauss-Salamin sequence.
3 - Computation of log(2+sqrt(3))
Timing : 9h09'19" (using the Brent-Salamin AGM formulae)
A verification has been done by computing log(7+4sqrt(3)) = 2 log(2+sqrt(3)).
The final formula is :
Catalan = 3/8 S0 + Pi/8 log(2+sqrt(3))
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Total timing (without verification) : 29h52'