Proof with auto with typeclass_instances.
intros m x.
apply relative_error_le_conversion...
apply Rmult_le_pos.
apply Rlt_le.
apply (RinvN_pos 1).
apply bpow_ge_0.
now apply relative_error_N_FLT_F2R_emin.
Qed.

(* Specifications: *)
Theorem relative_error_N_FLT_round_F2R_emin :
forall m, let x := F2R (Float beta m emin) in
(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_F2R_emin :
forall m, let x := F2R (Float beta m emin) in
(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.