Proof with auto with typeclass_instances.
intros rnd Zrnd x y Hx Hy.
destruct (Req_dec y 0) as [Zy|Zy].
now rewrite Zy, Rmult_0_r, Rminus_0_r.
destruct (Req_dec (round beta (FLX_exp prec) rnd (x/y)) 0) as [Hr|Hr].
rewrite Hr; ring_simplify (x-0*y)%R; assumption.
assert (Zx: x <> R0).
contradict Hr.
rewrite Hr.
unfold Rdiv.
now rewrite Rmult_0_l, round_0.
destruct (ex_Fexp_canonic _ x Hx) as (fx,(Hx1,Hx2)).
destruct (ex_Fexp_canonic _ y Hy) as (fy,(Hy1,Hy2)).
destruct (ex_Fexp_canonic (FLX_exp prec) (round beta (FLX_exp prec) rnd (x / y))) as (fr,(Hr1,Hr2)).
apply generic_format_round...
unfold Rminus; apply generic_format_plus_prec with fx (Fopp beta (Fmult beta fr fy)); trivial.
intros e; apply Zle_refl.
now rewrite F2R_opp, F2R_mult, <- Hr1, <- Hy1.
(* *)
destruct (relative_error_FLX_ex beta prec (prec_gt_0 prec) rnd (x / y)%R) as (eps,(Heps1,Heps2)).
rewrite Heps2.
rewrite <- Rabs_Ropp.
replace (-(x + - (x / y * (1 + eps) * y)))%R with (x * eps)%R by now field.
rewrite Rabs_mult.
apply Rlt_le_trans with (Rabs x * 1)%R.
apply Rmult_lt_compat_l.
now apply Rabs_pos_lt.
apply Rlt_le_trans with (1 := Heps1).
change 1%R with (bpow 0).
apply bpow_le.
generalize (prec_gt_0 prec).
clear ; omega.
rewrite Rmult_1_r.
rewrite Hx2.
unfold canonic_exp.
destruct (ln_beta beta x) as (ex, Hex).
simpl.
specialize (Hex Zx).
apply Rlt_le.
apply Rlt_le_trans with (1 := proj2 Hex).
apply bpow_le.
unfold FLX_exp.
ring_simplify.
apply Zle_refl.
(* *)
replace (Fexp (Fopp beta (Fmult beta fr fy))) with (Fexp fr + Fexp fy)%Z.
2: unfold Fopp, Fmult; destruct fr; destruct fy; now simpl.
replace (x + - (round beta (FLX_exp prec) rnd (x / y) * y))%R with
(y * (-(round beta (FLX_exp prec) rnd (x / y) - x/y)))%R.
2: field; assumption.
rewrite Rabs_mult.
apply Rlt_le_trans with (Rabs y * bpow (Fexp fr))%R.
apply Rmult_lt_compat_l.
now apply Rabs_pos_lt.
rewrite Rabs_Ropp.
replace (bpow (Fexp fr)) with (ulp beta (FLX_exp prec) (F2R fr)).
rewrite <- Hr1.
apply error_lt_ulp_round...
apply Rmult_integral_contrapositive_currified; try apply Rinv_neq_0_compat; assumption.
rewrite ulp_neq_0.
2: now rewrite <- Hr1.
apply f_equal.
now rewrite Hr2, <- Hr1.
replace (prec+(Fexp fr+Fexp fy))%Z with ((prec+Fexp fy)+Fexp fr)%Z by ring.
rewrite bpow_plus.
apply Rmult_le_compat_r.
apply bpow_ge_0.
rewrite Hy2; unfold canonic_exp, FLX_exp.
ring_simplify (prec + (ln_beta beta y - prec))%Z.
destruct (ln_beta beta y); simpl.
left; now apply a.
Qed.
(** Remainder of the square in FLX (with p>1) and rounding to nearest *)
Variable Hp1 : Zlt 1 prec.

Theorem sqrt_error_FLX_N :
forall x, format x ->
format (x - Rsqr (round beta (FLX_exp prec) (Znearest choice) (sqrt x)))%R.