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

Theorem Bmult_correct : forall mult_nan m x y, if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x * B2R y))) (bpow radix2 emax) then B2R (Bmult mult_nan m x y) = round radix2 fexp (round_mode m) (B2R x * B2R y) /\ is_finite (Bmult mult_nan m x y) = andb (is_finite x) (is_finite y) /\ (is_nan (Bmult mult_nan m x y) = false -> Bsign (Bmult mult_nan m x y) = xorb (Bsign x) (Bsign y)) else B2FF (Bmult mult_nan m x y) = binary_overflow m (xorb (Bsign x) (Bsign y)).