Definition split_bits x :=
  let mm := Zpower 2 mw in
  let em := Zpower 2 ew in
  (Zle_bool (mm * em) x, Zmod x mm, Zmod (Zdiv x mm) em)%Z.

Theorem join_split_bits : forall x, (0 <= x < Zpower 2 (mw + ew + 1))%Z -> let '(s, m, e) := split_bits x in join_bits s m e = x.