Definition Bdiv div_nan m x y :=
let f pl := B754_nan (fst pl) (snd pl) in
match x, y with
| B754_nan _ _, _ | _, B754_nan _ _ => f (div_nan x y)
| B754_infinity sx, B754_infinity sy => f (div_nan x y)
| B754_infinity sx, B754_finite sy _ _ _ => B754_infinity (xorb sx sy)
| B754_finite sx _ _ _, B754_infinity sy => B754_zero (xorb sx sy)
| B754_infinity sx, B754_zero sy => B754_infinity (xorb sx sy)
| B754_zero sx, B754_infinity sy => B754_zero (xorb sx sy)
| B754_finite sx _ _ _, B754_zero sy => B754_infinity (xorb sx sy)
| B754_zero sx, B754_finite sy _ _ _ => B754_zero (xorb sx sy)
| B754_zero sx, B754_zero sy => f (div_nan x y)
| B754_finite sx mx ex _, B754_finite sy my ey _ =>
FF2B _ (proj1 (Bdiv_correct_aux m sx mx ex sy my ey))
end.

Theorem Bdiv_correct :
forall div_nan m x y,
B2R y <> 0%R ->
if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x / B2R y))) (bpow radix2 emax) then
B2R (Bdiv div_nan m x y) = round radix2 fexp (round_mode m) (B2R x / B2R y) /\
is_finite (Bdiv div_nan m x y) = is_finite x /\
(is_nan (Bdiv div_nan m x y) = false ->
Bsign (Bdiv div_nan m x y) = xorb (Bsign x) (Bsign y))
else
B2FF (Bdiv div_nan m x y) = binary_overflow m (xorb (Bsign x) (Bsign y)).