Definition: Probability-one output

Let $U$ be a circuit on an input register $I$, an output register $Y$, and work registers $W$. All registers except $I$ start in zero. For a basis input $N$, write

Let ${\Pi }_{\mathrm{bad},N}$ project onto basis states whose output register does not encode the complete prime factorization of $N$. The circuit is exact on $N$ if