Proof with auto with typeclass_instances.
intros x Hx.
apply relative_error_lt_conversion...
apply bpow_gt_0.
intros _.
now apply relative_error.
Qed.

(* Specifications: *)
Theorem relative_error_F2R_emin :
forall m, let x := F2R (Float beta m emin) in
(x <> 0)%R ->
(Rabs (round beta fexp rnd x - x) < bpow (-p + 1) * Rabs x)%R.
Theorem relative_error_F2R_emin_ex :
forall m, let x := F2R (Float beta m emin) in
exists eps,
(Rabs eps < bpow (-p + 1))%R /\ round beta fexp rnd x = (x * (1 + eps))%R.
Theorem relative_error_F2R_emin :
forall m, let x := F2R (Float beta m emin) in
(x <> 0)%R ->
(Rabs (round beta fexp rnd x - x) < bpow (-p + 1) * Rabs x)%R.
Theorem relative_error_F2R_emin_ex :
forall m, let x := F2R (Float beta m emin) in
exists eps,
(Rabs eps < bpow (-p + 1))%R /\ round beta fexp rnd x = (x * (1 + eps))%R.