Given a
multivariate (real or complex) polynomial $p$ and a (real or complex)
domain $\mathcal{D}$, we would like to decide if there is an algorithm
that can evaluate $p$ accurately (i.e., with relative error less than
$1$), using rounded (real or complex) "traditional model" arithmetic. We
consider this problem both in the classical setting (where the
operations are $+$, $-$, $\times$) and in a black-box setting (where
other polynomial operations are allowed). We obtain necessary and
sufficient conditions for $p$ to be accurately evaluable on $\mathcal{D}$
when $\mathcal{D} = \mathbb{R}^n$ or $\mathbb{C}^n $, and also for some
smaller domains. I will also indicate progress toward constructing a
complete decision procedure.