Proof with auto with typeclass_instances.
intros m x.
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_F2R_emin.
Qed.

(* Specifications: *)
Theorem relative_error_N_round :
(0 < p)%Z ->
forall x,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs (round beta fexp (Znearest choice) x))%R.
Theorem relative_error_N_round :
(0 < p)%Z ->
forall x,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs (round beta fexp (Znearest choice) x))%R.