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.

Theorem FTZ_format_FLXN : forall x : R, (bpow (emin + prec - 1) <= Rabs x)%R -> FLXN_format beta prec x -> FTZ_format x.