Definition binary_float_of_bits_aux x :=
  let '(sx, mx, ex) := split_bits x in
  if Zeq_bool ex 0 then
    match mx with
    | Z0 => F754_zero sx
    | Zpos px => F754_finite sx px emin
    | Zneg _ => F754_nan false xH (* dummy *)
    end
  else if Zeq_bool ex (Zpower 2 ew - 1) then
    match mx with
      | Z0 => F754_infinity sx
      | Zpos plx => F754_nan sx plx
      | Zneg _ => F754_nan false xH (* dummy *)
    end
  else
    match (mx + Zpower 2 mw)%Z with
    | Zpos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false xH (* dummy *)
    end.

Lemma binary_float_of_bits_aux_correct : forall x, valid_binary prec emax (binary_float_of_bits_aux x) = true.