Notation bpow e := (bpow beta e).

Lemma relative_error_lt_conversion : forall x b, (0 < b)%R -> (x <> 0 -> Rabs (round beta fexp rnd x - x) < b * Rabs x)%R -> exists eps, (Rabs eps < b)%R /\ round beta fexp rnd x = (x * (1 + eps))%R.