Can the Banach-Tarski paradox be "realized" by floating-point round-off?
This seems to have little to do with Banach-Tarski.
In your setting, f is simply not an isometry due to floating-point errors, and in particular there must be a single piece $i$ such that $\mathrm{Vol}(f(P_i))&gt;\mathrm{Vol}(P_i)$, so no need to cut into several pieces.

Banach-Tarski works because the notion of volume is not well-defined on the pieces. This clashes with your requirement that the pieces are defined by "a finite amount of geometric data", and with your writing of $\mathrm{Vol}(P_1)+\dots+\mathrm{Vol}(P_n)$.