What branch of mathematics formally describes operations like converting FP32 ↔ FP64?

I’m trying to understand which area of mathematics deals with operations such as converting between FP32 (single precision) and FP64 (double precision) numbers.

Conceptually, FP32→FP64 is an exact embedding (injective mapping) between two finite subsets of ℝ, while FP64→FP32 is a rounding or projection that loses information.

So from a mathematical standpoint, what field studies this kind of operation?
Is it part of numerical analysis, set theory, abstract algebra (homomorphisms between number systems), or maybe category theory (as morphisms between finite approximations of ℝ)?

I’m not asking about implementation details, but about the mathematical framework that formally describes these conversions.