Corollary round_odd_fix:
  forall x p choice,
  prec > 1 ->
  0 <= p ->
  (bpow radix2 (prec + p + 1) <= Rabs x)%R ->
  round radix2 fexp (Znearest choice) (round radix2 (FIX_exp p) Zrnd_odd x) =
  round radix2 fexp (Znearest choice) x.Definition int_round_odd (x: Z) (p: Z) :=
  (if Z.eqb (x mod 2^p) 0 || Z.odd (x / 2^p) then x / 2^p else x / 2^p + 1) * 2^p.

Theorem BofZ_round_odd: forall x p, prec > 1 -> Z.abs x <= 2^emax - 2^(emax-prec) -> 0 <= p <= emax - prec -> 2^(prec + p + 1) <= Z.abs x -> BofZ x = BofZ (int_round_odd x p).