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.
Qed.

Theorem relative_error_N_F2R_emin :
forall m, let x := F2R (Float beta m emin) in
(Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs x)%R.