Section Fprop_relative_FLT.
Variable emin prec : Z.
Variable Hp : Z.lt 0 prec.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Proof with auto with typeclass_instances.
intros x Hx.
apply relative_error with (emin + prec - 1)%Z...
apply relative_error_FLT_aux.
Qed.
Proof with auto with typeclass_instances.
intros m x Zx.
destruct (Rlt_or_le (Rabs x) (bpow (emin + prec - 1))) as [Hx|Hx].
rewrite round_generic...
unfold Rminus.
rewrite Rplus_opp_r, Rabs_R0.
apply Rmult_lt_0_compat.
apply bpow_gt_0.
now apply Rabs_pos_lt.
apply generic_format_FLT_FIX...
apply Rlt_le.
apply Rlt_le_trans with (1 := Hx).
apply bpow_le.
apply Zle_pred.
apply generic_format_FIX.
now exists (Float beta m emin).
now apply relative_error_FLT.
Qed.
Proof with auto with typeclass_instances.
intros m x.
apply relative_error_lt_conversion...
apply bpow_gt_0.
now apply relative_error_FLT_F2R_emin.
Qed.
Proof with auto with typeclass_instances.
intros x Hx.
apply relative_error_lt_conversion...
apply bpow_gt_0.
intros _; now apply relative_error_FLT.
Qed.
Variable choice : Z -> bool.
Proof with auto with typeclass_instances.
intros x Hx.
apply relative_error_N with (emin + prec - 1)%Z...
apply relative_error_FLT_aux.
Qed.
Proof with auto with typeclass_instances.
intros x Hx.
apply relative_error_le_conversion...
apply Rlt_le.
apply Rmult_lt_0_compat.
apply Rinv_0_lt_compat.
now apply IZR_lt.
apply bpow_gt_0.
now apply relative_error_N_FLT.
Qed.
Proof with auto with typeclass_instances.
intros x Hx.
apply relative_error_N_round with (emin + prec - 1)%Z...
apply relative_error_FLT_aux.
Qed.
Proof with auto with typeclass_instances.
intros m x.
destruct (Rlt_or_le (Rabs x) (bpow (emin + prec - 1))) as [Hx|Hx].
rewrite round_generic...
unfold Rminus.
rewrite Rplus_opp_r, Rabs_R0.
apply Rmult_le_pos.
apply Rmult_le_pos.
apply Rlt_le.
apply (RinvN_pos 1).
apply bpow_ge_0.
apply Rabs_pos.
apply generic_format_FLT_FIX...
apply Rlt_le.
apply Rlt_le_trans with (1 := Hx).
apply bpow_le.
apply Zle_pred.
apply generic_format_FIX.
now exists (Float beta m emin).
now apply relative_error_N_FLT.
Qed.
Proof with auto with typeclass_instances.
intros m x.
apply relative_error_le_conversion...
apply Rmult_le_pos.
apply Rlt_le.
apply (RinvN_pos 1).
apply bpow_ge_0.
now apply relative_error_N_FLT_F2R_emin.
Qed.
Proof with auto with typeclass_instances.
intros m x.
destruct (Rlt_or_le (Rabs x) (bpow (emin + prec - 1))) as [Hx|Hx].
rewrite round_generic...
unfold Rminus.
rewrite Rplus_opp_r, Rabs_R0.
apply Rmult_le_pos.
apply Rmult_le_pos.
apply Rlt_le.
apply (RinvN_pos 1).
apply bpow_ge_0.
apply Rabs_pos.
apply generic_format_FLT_FIX...
apply Rlt_le.
apply Rlt_le_trans with (1 := Hx).
apply bpow_le.
apply Zle_pred.
apply generic_format_FIX.
now exists (Float beta m emin).
apply relative_error_N_round with (emin := (emin + prec - 1)%Z)...
apply relative_error_FLT_aux.
Qed.

Lemma error_N_FLT_aux :
forall x,
(0 < x)%R ->
exists eps, exists  eta,
(Rabs eps <= /2 * bpow (-prec + 1))%R /\
(Rabs eta <= /2 * bpow (emin))%R      /\
(eps*eta=0)%R /\
round beta (FLT_exp emin prec) (Znearest choice) x = (x * (1 + eps) + eta)%R.