Proof with auto with typeclass_instances.
intros x Hx.
destruct (ln_beta beta x) as (ex, He).
specialize (He Hx).
apply relative_error_round with (ex - 1)%Z...
intros k _.
apply relative_error_FLX_aux.
apply He.
Qed.
Variable choice : Z -> bool.

Theorem relative_error_N_FLX :
forall x,
(Rabs (round beta (FLX_exp prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs x)%R.