Lemniscate Constant

The Lemniscate Constant

Update: This computation has been superseded by my own newer computation of 1,200,000,000,100 digits, which is twice the number of digits below. It was computed using the Gauss Formula from Wed Apr 27 15:08:28 2022 to Sun May 29 19:15:15 2022 and verified using Sebah’s formula from Mon May 30 08:26:23 2022 to Fri Jul 15 20:28:27 2022, both with y-cruncher v0.7.9 Build 9509. The digits and both validation files are available at This Link. The rest of this post describes the earlier 600,000,000,000-digit computation.

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 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:
https://web.archive.org/web/20190724100731/http://www.numberworld.org/y-cruncher/records/2019_5_7_lemniscate.txt
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 is introduced in precalculus. Look at Wikipedia to understand what a lemniscate is.

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). For a more detailed explanation of the lemniscate constant, see this Wolfram Mathworld entry.

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 the fellow mathematical computing program Prime95.
Maintaining server stability was challenging, as the computation pushed all components of the computer to their limits.

System information:
Operating System: Linux 3.10.0-693.21.1.el7.x86_64 x86_64 (CentOS 7)
Processor(s):
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-heavy programs 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

Memory:
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)

The use of an inadequate algorithm (the AGM-Pi algorithm) contributed to the bottleneck, resulting in disk operations that were around twice as many as those in my world record computation of Catalan’s constant.

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

To view the digits for the arc length of a lemniscate with a=1 (OEIS A064853), you can download them from This Link (note that the total size is over 1 TB, but the link will simply redirect to a registry with a download link). The digits of the newer 1,200,000,000,100-digit computation are at This Link instead.

Note that the digits are released under an Attribution-NonCommercial-NoDerivatives 4.0 International License, which prohibits commercial use and distribution of remixed, transformed, or built-upon versions without consent. Proper attribution and indication of changes are required even if it is not a prohibited use case.

Archive for computation results in the y-cruncher website:
https://web.archive.org/web/20190722034426/http://www.numberworld.org/y-cruncher/
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.