Notation bpow e := (bpow beta e).

Lemma relative_error_le_conversion : forall x b, (0 <= b)%R -> (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.