Also of interest: Herbie analyzes your math expressions and helps you figure out where FP error accumulates. https://herbie.uwplse.org/demo/

We use floats as a trade-off between speed and accuracy. IEEE 754 is a very reasonable trade-off for a wide range of applications, but if you can figure out where you need to trade speed to get more accuracy with e.g. one of the methods mentioned here, Herbie's gotcha covered.

I remember seeing some research about switching between formats, but I don't have anything to cite right now.