Proof with auto with typeclass_instances.
intros x Hx.
destruct (ln_beta beta x) as (ex, He).
specialize (He Hx).
apply relative_error with (ex - 1)%Z...
intros k _.
apply relative_error_FLX_aux.
apply He.
Qed.

(* Specifications: *)
(** 1+#&epsilon;# property in any rounding in FLX *)
Theorem relative_error_FLX_ex :
forall x,
exists eps,
(Rabs eps < bpow (-prec + 1))%R /\ round beta (FLX_exp prec) rnd x = (x * (1 + eps))%R.
(** 1+#&epsilon;# property in any rounding in FLX *)
Theorem relative_error_FLX_ex :
forall x,
exists eps,
(Rabs eps < bpow (-prec + 1))%R /\ round beta (FLX_exp prec) rnd x = (x * (1 + eps))%R.