Proof with auto with typeclass_instances.
intros x.
apply relative_error_lt_conversion...
apply bpow_gt_0.
now apply relative_error_FLX.
Qed.

Theorem relative_error_FLX_round :
forall x,
(x <> 0)%R ->
(Rabs (round beta (FLX_exp prec) rnd x - x) < bpow (-prec + 1) * Rabs (round beta (FLX_exp prec) rnd x))%R.