Is there any way to simplify $\lfloor x\rfloor-\lfloor y\rfloor$, preferably such that the result is in the form of $\lfloor f(x, y)\rfloor$ for some function $f$?
I attempted to replace $\lfloor x\rfloor$ with $x - (x\;\text{mod}\;1)$ and the same with $y$ to get $(x - y) - ((x\;\text{mod}\;1)-(y\;\text{mod}\;1))$, but I couldn't get past this.
For most reasonable conceptions of "nice," there isn't a nice function $f$ that satisfies your criterion. Generally, the form you have is considered the simplest.