Article: Understanding Quake's Fast Inverse Square Root

This article pretty much confirms what I suspected, but the author explains it pretty well.
The only thing really missing is how the "magic number" was derived to minimize the error.
This is just speculation on my part, but I see two potential approaches.
The first would be to pick some mid point value that reduces the maximum error for any legal number (within the range of floating point numbers).
The second would be to pick a value that is the mid point for the most common values a program uses, thus minimizing the error for that program more than for any value.
The first would be easy to calculate, the 2nd would require a lot of number logging & crunching.
https://betterexplained.com/articles/understanding-quakes-fast-inverse-square-root/