Dear Simon Plouffe,
I send you Euler's number e = Exp(1) to 200,000,579 decimal digits.
I have done this record because of the following three reasons:
1. I want to start some experiences with the article [1]
of H. J. Brothers and J. A. Knox.
2. I have done many improvements in my arithmetic Piologie [3]
and need an exercise to test it.
3. It is a confirmation that the calculation of Apery's constant to
256 millions of decimals can be done in reasonable time.
Best regards,
Sebastian Wedeniwski (wedeniws@de.ibm.com)
Timings:
========
The computation (with no line of assembler) was done on February 26, 1999 and
it took 76 hours and 9 minutes on the following 64-bit emulating workstations:
1. IBM Power2 SC 135 MHz, 2 GB RAM, GNU C++ 2.8.0, AIX 4.1.5.
2. IBM PowerPC 604e 233 MHz, 1 GB RAM, GNU C++ 2.8.0, AIX 4.1.5.
Thanks:
=======
This computational record was possible with the machine support of
IBM Deutschland Entwicklung GmbH, Boeblingen.
Therefore, many thanks are going to my managers
Dr. Michael von Carlowitz, Reinhold Krause and Joerg Thielges
who have made a serie of computations possible and have been helpful
coordinators.
Thanks also to Henry Koplien for his very cooperative and administrative aid.
He has put the last step of the calculation of zeta(3) to 128 millions of
decimals on 16 CD's, so that it will be possible to continue the calculation
without any recomputation.
And last but not least, many thanks are going to my ex-manager
Dr. Oskar von Dungern
who set these computations on foot.
Verification:
=============
I used two different binary splitting and FFT algorithm.
The verification was done on March 8, 1999 and
it took 81 hours and 9 minutes on the following 32-bit workstations:
1. IBM Power2 SC 135 MHz, 2 GB RAM, GNU C++ 2.8.0, AIX 4.1.5.
2. IBM PowerPC 604e 233 MHz, 1 GB RAM, GNU C++ 2.8.0, AIX 4.1.5.
Unfortunately, there are only 25,277,126 digits correct of the 100 millions
of decimals of the previous record by Bruno Haible.
Technical Details:
==================
The computation of Exp(1) was splitted up in 26 processes on 2 processors.
During the computation it takes 2 GB hard disc splitted up on 12 folders.
- partial computations:
(0) in 6524 sec
(1) in 6822 sec
(2) in 6939 sec
(3) in 11032 sec
(4) in 11050 sec
(5) in 11114 sec
(6) in 11206 sec
(7) in 11240 sec
- combinations (each of them parallelized in two processes):
(0)-(1) in 1476 sec / 1470 sec
(2)-(3) in 1473 sec / 1468 sec
(4)-(5) in 1479 sec / 1472 sec
(6)-(7) in 1478 sec / 1476 sec
- divisions:
(0)-(1) in 54413 sec
(2)-(3) in 54582 sec
(4)-(5) in 54569 sec
(6)-(7) in 54681 sec
- last combinations with divisions:
(4)-(7) in 42836 sec + 1 sec
(2)-(7) in 42833 sec + 1 sec
(0)-(7) in 51098 sec + 1 sec
The verification of Exp(1) was splitted up in 98 processes on 2 processors.
During the computation it takes 5 GB hard disc splitted up on 60 folders.
- partial computations:
(0) in 971 sec
(1) in 1158 sec
(2) in 1173 sec
(3) in 1196 sec
(4) in 1525 sec
(5) in 1619 sec
(6) in 1458 sec
(7) in 1210 sec
(8) in 1232 sec
(9) in 1412 sec
(10) in 1671 sec
(11) in 1263 sec
(12) in 1281 sec
(13) in 1230 sec
(14) in 1190 sec
(15) in 1187 sec
(16) in 1213 sec
(17) in 1199 sec
(18) in 1207 sec
(19) in 1223 sec
(20) in 1217 sec
(21) in 1214 sec
(22) in 1223 sec
(23) in 1231 sec
(24) in 1240 sec
(25) in 1236 sec
(26) in 1243 sec
(27) in 1250 sec
(28) in 1253 sec
(29) in 1260 sec
(30) in 1257 sec
(31) in 1245 sec
- combinations (each of them parallelized in two processes):
(0)- (1) in 201 sec / 198 sec
(2)- (3) in 200 sec / 198 sec
(4)- (5) in 197 sec / 198 sec
(6)- (7) in 198 sec / 198 sec
(8)- (9) in 209 sec / 210 sec
(10)-(11) in 198 sec / 199 sec
(12)-(13) in 199 sec / 196 sec
(14)-(15) in 198 sec / 197 sec
(16)-(17) in 196 sec / 195 sec
(18)-(19) in 197 sec / 197 sec
(20)-(21) in 228 sec / 226 sec
(22)-(23) in 229 sec / 227 sec
(24)-(25) in 228 sec / 229 sec
(26)-(27) in 227 sec / 227 sec
(28)-(29) in 228 sec / 228 sec
(30)-(31) in 228 sec / 226 sec
(0)- (3) in 414 sec / 412 sec
(4)- (7) in 408 sec / 408 sec
(8)-(11) in 423 sec / 412 sec
(12)-(15) in 431 sec / 408 sec
(16)-(19) in 464 sec / 463 sec
(20)-(23) in 466 sec / 602 sec
(24)-(27) in 464 sec / 461 sec
(28)-(31) in 464 sec / 461 sec
(0)- (7) in 990 sec / 986 sec
(8)-(15) in 1002 sec / 973 sec
(16)-(23) in 1532 sec / 1517 sec
(24)-(31) in 980 sec / 974 sec
- divisions:
(0)- (7) in 54523 sec
(8)-(15) in 54763 sec
(16)-(23) in 54499 sec
(24)-(31) in 54544 sec
- last combinations with divisions:
(16)-(31) in 42973 sec + 2 sec
(8)-(31) in 42962 sec + 2 sec
(0)-(31) in 51364 sec + 2 sec
I have taken advantage of the binary splitting method [2] with some
extentions and of an improved version of my arithmetic Piologie 1.2.1 [3].
References:
===========
[1] H. J. Brothers, J. A. Knox: New Closed-Form Approximations to the
Logarithmic Constant e, The Mathematical Intelligencer 20 (1998), 25-29.
[2] B. Haible, T. Papanikolaou: Fast multiprecision evaluation of series of
rational numbers, Technical Report TI-97-7, Darmstadt University of
Technology, April 1997.
[3] S. Wedeniwski: Piologie - Eine exakte arithmetische Bibliothek in C++,
Technical Report WSI 96-35, Tuebingen University, available by anonymous
ftp from "ftp://ftp.informatik.uni-tuebingen.de/pub/CA/software/Piologie/"
or "ftp://ruediger.informatik.uni-tuebingen.de/Piologie/".