New record on the computation of digits of Gamma :
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Number of digits computed (and verified) : 7286255
When : 25/12/97 -> 26/12/97
Verification : 03/01/98 -> 04/01/98
Machine : sgi r10000, 256 Mo of memory.
Total timing : 47 hours and 36 minutes
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Formula used:
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3 steps :
1 - Computation of S0 = sum_{k>0} (-N)^k/k!
2 - Computation of R0 = e^(-N)/N (sum_{k=0}^N k!/N^k)
3 - Computation of log(N)
We used the formula : S0+R0-log(N) = gamma + O(e^(-2N)/N), with N = 2^23.
Details on the computations of step 1 and 2 :
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A variant of the Brent binary-splitting algorithm was used to
compute the series. Thanks to this approach, the complexity of
the computation is reduced to 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.
Details on the computations of step 3 :
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step 3 was done using the Brent-Salamin sequence to compute log(N), with
the same program as the one used for the record of 58 millions digits of log(2).
Details of the timings:
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Timing of step 1 and 2 : 43h41'
Timing of step 3 : 3h55'
Verification:
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The verification was done using the same formula with N=2^23+1.
The timings were approximately the same.