Lemniscate Constant

2 minute read

The Lemniscate Constant

Please cite:
Kim, S. Normality Analysis of Current World Record Computations for Catalan’s Constant and Arc Length of a Lemniscate with a=1. arXiv Preprint arXiv:1908.08925
if this article or the calculated digits were useful.

This world record computation of 600,000,000,000 digits by Seungmin Kim was done from Thu Mar 21 10:33:00 2019 to Tue May 7 21:51:17 2019.
Validation file generated by y-cruncher v0.7.7 Build 9499:
Verified by Dr. Ian Cutress (https://web.archive.org/web/20190724111018/http://www.numberworld.org/y-cruncher/records/2019_5_21_lemniscate.txt)

This is a lemniscate. (https://commons.wikimedia.org/wiki/File:Lemniscate_of_Booth.png)

The lemniscate function first comes out during precalculus. Look at Wikipedia to understand what is a lemniscate.

The lemniscate constant described here is actually the arc length of a lemniscate with a=1 (OEIS A064853) instead of the first lemniscate constant (OEIS A085565) or the second lemniscate constant (OEIS A076390). See this Wolfram Mathworld entry for an in-depth explanation.

I have used y-cruncher by Mr. Alexander J. Yee for this computation. This program is commonly used for stress testing and benchmarking overclocked PC builds (obviously this program performs a very rigorous computation), along with fellow mathematical computing program Prime95.
It was also very hard for me to maintain this server stable, as this takes all components of a computer to the extreme.

System information:
Operating System: Linux 3.10.0-693.21.1.el7.x86_64 x86_64 (CentOS 7)
Name: Intel(R) Xeon(R) Gold 6140 CPU @ 2.30GHz
Logical Cores: 72
Physical Cores: 36
Sockets: 2
NUMA Nodes: 2
Base Frequency: 2,294,553,184 Hz

I have used two CPUs that are compatible with AVX-512 operations that are used crucially in vector operations like y-cruncher. However, y-cruncher has severe I/O bottlenecks, so the number of cores did not assist my operation well. I have used RAID scratch storage for my operations, but the speed of the disk was very slow compared to the required R/W operations.

Start Date: Thu Mar 21 10:33:00 2019
End Date: Tue May 7 21:51:17 2019
Total Computation Time: 3518500.534 seconds
Start-to-End Wall Time: 4097896.648 seconds
CPU Utilization: 167.04 % + 173.19 % kernel overhead
Multi-core Efficiency: 2.32 % + 2.41 % kernel overhead

Usable Memory: 201,226,489,856 ( 187 GiB)
Logical Peak Disk Usage: 3,504,857,581,840 (3.19 TiB)
Logical Disk Bytes Read: 970,596,570,010,300 ( 883 TiB)
Logical Disk Bytes Written: 854,315,820,415,436 ( 777 TiB)

Selection of an inadequate algorithm (the AGM-Pi algorithm) contributed more to the bottleneck, as the disk operations were around two times of my world record computation of Catalan’s constant.

Dr. Ian Cutress finished the verification in only under 6 days. He had the better CPU, and most importantly, he used what was assumed to be an Intel Optane DIMM to decrease I/O bottleneck as much as possible.

If you want to take a look at the digits for the arc length of a lemniscate with a=1 (OEIS A064853), you can download it from This Link (Over 1 TB total but don’t worry, it will just redirect to a registry with a link to download).

Note that digits are released as an Attribution-NonCommercial-NoDerivatives 4.0 International License, meaning no commercial purposes and you cannot distribute a remixed, transformed, or built upon version without my consent. You must also give appropriate credit, provide a link to the license, and indicate if changes were made even if it is not a prohibited use case.

Archive for computation results in the y-cruncher website:
Special thanks to Mr. Alexander J. Yee for developing and releasing y-cruncher and providing advice, AnandTech.com Senior Editor Dr. Ian Cutress for verifying my computation, and the Internet Archive for hosting the computed digits.