Definition bits_of_binary_float (x : binary_float) :=
match x with
| B754_zero sx => join_bits sx 0 0
| B754_infinity sx => join_bits sx 0 (Zpower 2 ew - 1)
| B754_nan sx (exist plx _) => join_bits sx (Zpos plx) (Zpower 2 ew - 1)
| B754_finite sx mx ex _ =>
let m := (Zpos mx - Zpower 2 mw)%Z in
if Zle_bool 0 m then
join_bits sx m (ex - emin + 1)
else
join_bits sx (Zpos mx) 0
end.
Definition split_bits_of_binary_float (x : binary_float) :=
match x with
| B754_zero sx => (sx, 0, 0)%Z
| B754_infinity sx => (sx, 0, Zpower 2 ew - 1)%Z
| B754_nan sx (exist plx _) => (sx, Zpos plx, Zpower 2 ew - 1)%Z
| B754_finite sx mx ex _ =>
let m := (Zpos mx - Zpower 2 mw)%Z in
if Zle_bool 0 m then
(sx, m, ex - emin + 1)%Z
else
(sx, Zpos mx, 0)%Z
end.

Theorem bits_of_binary_float_range:
forall x, (0 <= bits_of_binary_float x < 2^(mw+ew+1))%Z.