I love these "what if you only had X" puzzles. The constraint here (no bit access, only FP multiply and add) sounds impossible until you realize rounding behavior carries information.

The edge cases around negative zero and infinities make sense. Those values break the mathematical properties you'd need to distinguish them.

Recommended readings:

Jim McCann, Tom Murphy VII, The fluint8 Software Integer Library. https://tom7.org/papers/fluint.pdf

Tom Murphy VII, GradIEEEnt half decent. https://tom7.org/grad/murphy2023grad.pdf