Derivatives III: I Ran Out of Rez

One more note on derivatives in GLSL shaders: derivatives can run into precision problems that the underlying expressions don't have.

Recall that derivatives are typically calculated by taking the actual difference of two values in two nearby pixels. This means that the derivative is subject to the precision limits between two pixels.

Consider a UV map spread over a really huge distance, say, 5 km in-game. The texture is 1024 x 1024 and therefore each texel is about 10m in size.

What happens if we zoom way the heck in so that one game pixel (5m) is covering nearly all of the screen?

We need about 10 bits of precision to select our texel; any remaining precision can be used to take an interpolated position between texels for filtering. If our monitor res is about 1024x768, we're using 20 bits of precision in our UV map, and we have only three bits left. If we zoom in any more, we may reach a point where our interpolated UV map doesn't have enough precision to provide a UV position for each pixel.

(In other words, if we have less than 10 bits of precision left between the left and right side of the screen, then some adjacent pixels will have the same UV coordinates!)

Now this generally doesn't matter for texture sampling. We're sampling 1024 unique mixes between two pixels, and we can only show 256 shades on the screen - in practice if two pixels have the same UV coordinates, it doesn't matter, because the amount of RGB change per pixel is not perceivable anyway.

But for our derivatives, it's a different story: some pairs of pixels will have a zero derivative, and some will have a non-zero derivative! Even if we don't run out res, we're very low on res and our derivatives may be 'chunky' or in other ways screwed up. If we need to reconstruct basis vectors from our derivatives, those basis vectors are going to be a train wreck.

The only solution I have found to this problem has been to replace GLSL built-in derivatives with an algorithmic derivative function. Fortunately the only cases where we have ridiculous UV mapping is when the texture coordinates are generated by formula, and thus a similar formula can be used to create the derivatives.