This place is web 1.0 levels of under construction. If you feel this page missing something, please let me know. It is likely in my google drive awaiting migration here—if it is something **not** in my google drive, I would very much like to know about it.

In 2001 an approximation of a reciprocal square root (y = 1/sqrt(x)) leaked to the internet, was attributed to a charismatic hacker, and attained the kind of fame normally reserved for soap opera stars or particularly photogenic zoo animals. This fame allows us to, quite incidentally, follow the normally invisible history of a mathematical approximation over time, across many organizations, and through co-creation of formal knowledge.

Working with oral history interviews, archival material, and source code, this project is an attempt to follow this trace which, far from being born from the brow of a 90s programmer, spans most of the history of the digital computer.

Hang on, isn’t the magic constant “0x5F3759DF”?

That’s right! This page is named 0x5f37642f, Lomont’s optimal constant for the approximation derived without the next step of Newton-Raphson (NR) in mind. Considering the next step of NR, Lomont’s constant is outperformed by 0x5F3759DF, which is the one that leaked to the internet.

Resources on Quake III’s Fast Inverse Square Root

Resources are linked below where links are available and known live. Otherwise copies are hosted on this site as fair use.

Data and code on the FISR

This repository contains code to reproduce versions of the FISR. Also contains data and R code to plot data.

Whitepapers about the Quake III FISR

• First explanation in English by Chris Eberly, January 2002.
• Explanation by Chris Lomont with a newly computed “magic” constant, February 2003
• Charles McEniry on the mathematics of the constant, August 2007
• Matthew Robertson’s BS thesis, April 2012
• Ben Self’s BS thesis, May 2012
• Thomas Nelson’s survey of the field, July 2017

More recent works which situate the FISR

• Jerome Coonen’s explainer with history of the FISR, April 2022
• Noah Hellman’s June 2020 masters thesis demonstrating modern approximate computing on a logarithmic number system names the method used in the FISR as Mitchell’s
• My slide deck on the FISR, given at UW HCDE’s autumn seminar series 2022: What’s in a Number?

The Wikipedia article

• Fast inverse square root. It is not up to date completely with all post-2010 changes in understanding. A source and place to find sources. (I wrote the first version but no longer maintain it.)

Jean-Michel Muller’s excellent survey

• Jean-Michel Muller, ”Elementary Functions and Approximate Computing,” in Proceedings of the IEEE, vol. 108, no. 12, pp. 2136-2149, Dec. 2020

A “Reciproot” on the Manchester Mark I

• One of the few remaining subroutines from the Manchester Mark I is a reciprocal square root. Martin Campbell-Kelly notes that in a manual written for the Mark I, “A total of ten sub-routines are named here; half were for input/output and half were mathematical functions.” (Programming the Mark I: Early programming activity at the University of Manchester [Archive Link], p. 149) Kelly notes that the routine was first written in 1949. The extant copy archived here is dated September 7, 1951.

On Kahan and K-C Ng’s 1986 version

• An implementation with explanatory comments in C
• The unpublished paper from 1986
• An error analysis of Kahan-Ng by Ren-Cang Li, one of Kahan’s PhD students. This is the only paper I have found which cites the unpublished Kahan-Ng work.