Variable choice : Z -> bool.

(* Specifications: *)
Theorem relative_error_N :
forall x,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs x)%R.
(** 1+#&epsilon;# property in rounding to nearest *)
Theorem relative_error_N_ex :
forall x,
(bpow emin <= Rabs x)%R ->
exists eps,
(Rabs eps <= /2 * bpow (-p + 1))%R /\ round beta fexp (Znearest choice) x = (x * (1 + eps))%R.
Theorem relative_error_N :
forall x,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs x)%R.
(** 1+#&epsilon;# property in rounding to nearest *)
Theorem relative_error_N_ex :
forall x,
(bpow emin <= Rabs x)%R ->
exists eps,
(Rabs eps <= /2 * bpow (-p + 1))%R /\ round beta fexp (Znearest choice) x = (x * (1 + eps))%R.