Is there a class which is to APX what BPP is to P?
I'm looking for a definition that is like the following:
"For $r > 0$, an $r$-RPCA (randomized polynomial-time constant-factor approximation) algorithm for a function problem $T : \Sigma^* \to \mathbb{N}$ is a probabilistic Turing machine $A$ with the following property: $A$ runs in time $poly(|x|)$ and has $\mathbb{P}( r^{-1} T(x) \leq A(x) \leq r T(x)) \geq 2/3$."
I think that either a class like this exists and has a standard name, or there is something wrong with it. I'm looking for a similar definition with which to cleanly state a result.