Notation bpow e := (bpow beta e).(* floating-point format with abrupt underflow *)
Definition FTZ_format (x : R) :=
  exists f : float beta,
  x = F2R f /\ (x <> R0 -> Zpower beta (prec - 1) <= Zabs (Fnum f) < Zpower beta prec)%Z /\
  (emin <= Fexp f)%Z.Definition FTZ_exp e := if Zlt_bool (e - prec) emin then (emin + prec - 1)%Z else (e - prec)%Z.(** Properties of the FTZ format *)
Global Instance FTZ_exp_valid : Valid_exp FTZ_exp.

(* Specifications: *)
Theorem FLXN_format_FTZ : forall x, FTZ_format x -> FLXN_format beta prec x.
Theorem generic_format_FTZ : forall x, FTZ_format x -> generic_format beta FTZ_exp x.
Theorem FTZ_format_generic : forall x, generic_format beta FTZ_exp x -> FTZ_format x.
Theorem FTZ_format_satisfies_any : satisfies_any FTZ_format.
Theorem FTZ_format_FLXN : forall x : R, (bpow (emin + prec - 1) <= Rabs x)%R -> FLXN_format beta prec x -> FTZ_format x.
Theorem ulp_FTZ_0: ulp beta FTZ_exp 0 = bpow (emin+prec-1).
Theorem FLXN_format_FTZ : forall x, FTZ_format x -> FLXN_format beta prec x.
Theorem generic_format_FTZ : forall x, FTZ_format x -> generic_format beta FTZ_exp x.
Theorem FTZ_format_generic : forall x, generic_format beta FTZ_exp x -> FTZ_format x.
Theorem FTZ_format_satisfies_any : satisfies_any FTZ_format.
Theorem FTZ_format_FLXN : forall x : R, (bpow (emin + prec - 1) <= Rabs x)%R -> FLXN_format beta prec x -> FTZ_format x.
Theorem ulp_FTZ_0: ulp beta FTZ_exp 0 = bpow (emin+prec-1).