Here’s something new that isn’t mentioned and might be worth adding to the repo. (Edit I’m wrong, it is there, I just didn’t recognize it.) The magic number in the famous code snippet is not the optimal constant. You can do maybe 0.5% better relative error using a different constant. Maybe at the time it was infeasible to search for the absolutely optimal number, but now it’s relatively easy. I also went down this rabbit hole at some point, so I have a Jupyter notebook for finding the optimal magic number for this (1/x^2) and also for (1/x). Anyone wanna know what the optimal magic number is?
Good point, I hadn’t read Lomont’s paper and should have. I read the section in the Wikipedia article talking about it, and did try the constant that it suggests, however it depends on doing extra Newton iterations and I looked at the relative error of the initial guess without Newton. I can see in the paper he found something within 1 bit of what I found. I’m not certain mine’s better, but my python script claims it is.
For no Newton iterations, I thought I found 0x5f37641f had the lowest relative error, and I measure it at 3.4211. Of course I’m not super certain, Lomont’s effort is way more complete than mine. Lomont’s paper mentions 0x5f37642f with a relative error of 3.42128 and Wikipedia and the paper both talk about 0x5f375a86 being best when using Newton iterations.
