Any evidence for the claims?

This sqrtApprox() function is not universal. It is only used for hill shading, which is very approximate and rough in itself. So the accuracy argument is irrelevant. The function is accurate enough for the purpose. It won't be used to send rockets to the moon.

The speed argument is also irrelevant. Do your testing on Xperia devices, for example. The reasons can be this or that, it doesn't matter. Bit-manipulating sqrtApprox() is about 10x faster on some $$$ Xperias. On some other devices the difference in speed will be much, much smaller, but Math.sqrt() will still not be faster. In other words, sqrtApprox() should be fast on almost any device (prove otherwise?), while Math.sqrt() will be extremely slow on some (proven).

Want to use Math.sqrt()? Who's stopping you from overriding the StandardClasyHillShading.sqrt() function?