Visualizing (inverse) square root optimizations

A few years ago I experimented with a few of these approximation functions, and then I decided not to use them for some reason. (I forgot--, maybe they weren't a performance gain at the time.) I remember vaguely getting weird (but cool) results with some approximation formula.

Then I encountered David Hulsman's RSqrtBenchmark by accident: a collection of sqrt approximations he benchmarked for performance and accuracy. I found this a very cool idea, and wondered how these different sqrt approximations would visualize.. (having this faint memory of cool visual artifacts).

But apparently all these approximations are extremely good! :P Visually indistinguishable even! In this image each row represents a different sqrt implementation, and a white to black gradient is drawn with it:

I tried different visual representations of course, where for some functions you can see very small differences along the edges (if you swap them real fast). If you view the images side-by-side you won't see anything.

Of course there are more differences, you see them when comparing, like for example.. (left: compare sqrt <> lomont's InvSqrt, right: sqrt <> carmack D3RSqrt)

But I wanted my anomolies! :P so I tried tracking down the function I might have used back then, but I couldn't find it anywhere in my code repositories but after searching the internet for a while as well I now think it probably was something based on this very crude approximation:

(And yes it really is a very crude approximation :P)

So I kind of wasted way too much of my time on this already, but while searching the internet I encountered a few functions and got curious how they would visualize. I added a few, and maybe i'll add them to RSqrtBenchmark later:

Some of these do actually have visual artifacts. You can see those more clearly in the zipped uncompressed images. << I included radial blur images (1024x1024 px) for all functions.

By the way--I thought this one I generated while with a programming mistake was also pretty cool, almost fractal-like

(Edit 29-12-2012: uploaded source to bitbucket repository)