Proof with auto with typeclass_instances.
intros x Hx.
apply relative_error_le_conversion...
apply Rlt_le.
apply Rmult_lt_0_compat.
apply Rinv_0_lt_compat.
now apply (Z2R_lt 0 2).
apply bpow_gt_0.
now apply relative_error_N_FLT.
Qed.

(* Specifications: *)
Theorem relative_error_N_FLT_round :
forall x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
(Rabs (round beta (FLT_exp emin prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs (round beta (FLT_exp emin prec) (Znearest choice) x))%R.
Theorem relative_error_N_FLT_round :
forall x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
(Rabs (round beta (FLT_exp emin prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs (round beta (FLT_exp emin prec) (Znearest choice) x))%R.