Section relative_error_conversion.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

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.