Section Fclosestp2.
Variable b : Fbound.
Variable radix : Z.
Variable precision : nat.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ radixMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.
intros p r H'; split.
apply oppBounded; auto.
case H'; auto.
intros f H'0.
rewrite Fopp_correct.
replace (- FtoR radix p - - r)%R with (- (FtoR radix p - r))%R;
[ idtac | ring ].
replace (FtoR radix f - - r)%R with (- (- FtoR radix f - r))%R;
[ idtac | ring ].
rewrite <- Fopp_correct.
repeat rewrite Rabs_Ropp.
case H'; auto with float.
Qed.
intros p r H'; case (Rle_or_lt 0 r); intros Rl0.
rewrite Rabs_right; auto with real.
replace (Fabs p) with p; auto.
unfold Fabs in |- *; apply floatEq; simpl in |- *; auto.
cut (0 <= Fnum p)%Z.
case (Fnum p); simpl in |- *; auto; intros p' H0; Contradict H0;
apply Zlt_not_le; red in |- *; simpl in |- *; auto with zarith.
apply LeR0Fnum with (radix := radix); auto.
apply
RleRoundedR0
with (b := b) (precision := precision) (P := Closest b radix) (r := r);
auto.
apply ClosestRoundedModeP with (precision := precision); auto with real.
rewrite Faux.Rabsolu_left1; auto.
replace (Fabs p) with (Fopp p).
apply ClosestOpp; auto.
unfold Fabs in |- *; apply floatEq; simpl in |- *; auto.
cut (Fnum p <= 0)%Z.
case (Fnum p); simpl in |- *; auto; intros p' H0; Contradict H0;
apply Zlt_not_le; red in |- *; simpl in |- *; auto with zarith.
apply R0LeFnum with (radix := radix); auto.
apply
RleRoundedLessR0
with (b := b) (precision := precision) (P := Closest b radix) (r := r);
auto.
apply ClosestRoundedModeP with (precision := precision); auto.
apply Rlt_le; auto.
apply Rlt_le; auto.
Qed.
intros p q H'.
case (Req_dec p q); intros Eqpq.
rewrite Eqpq.
replace (Rabs (q - q)) with 0%R;
[ rewrite Rmult_0_r
| replace (q - q)%R with 0%R; try ring; rewrite Rabs_right; auto with real ].
unfold Fulp in |- *; apply Rlt_le; auto with real arith.
replace (2%nat * Rabs (p - q))%R with (Rabs (p - q) + Rabs (p - q))%R;
[ idtac | simpl in |- *; ring ].
case ClosestMinOrMax with (1 := H'); intros H'1.
apply Rle_trans with (Rabs (p - q) + Rabs (FNSucc b radix precision q - p))%R.
apply Rplus_le_compat_l.
rewrite <- (Rabs_Ropp (p - q)).
rewrite Ropp_minus_distr.
elim H'; auto.
intros H'0 H'2; apply H'2; auto.
apply FcanonicBound with (radix := radix); auto with float arith.
rewrite Rabs_right.
rewrite Rabs_right.
replace (p - q + (FNSucc b radix precision q - p))%R with
(FNSucc b radix precision q - q)%R; [ idtac | ring ].
unfold FtoRradix in |- *; apply FulpSuc; auto.
case H'1; auto.
apply Rge_minus; apply Rle_ge; auto with real float.
case MinMax with (3 := pGivesBound) (r := p) (p := q); auto with arith.
intros H'0 H'2; elim H'2; intros H'3 H'4; apply H'3; clear H'2; auto.
apply Rge_minus; apply Rle_ge; auto with real float.
apply isMin_inv1 with (1 := H'1).
apply Rle_trans with (Rabs (p - q) + Rabs (p - FNPred b radix precision q))%R.
apply Rplus_le_compat_l.
rewrite <- (Rabs_Ropp (p - q));
rewrite <- (Rabs_Ropp (p - FNPred b radix precision q)).
repeat rewrite Ropp_minus_distr.
elim H'; auto.
intros H'0 H'2; apply H'2; auto.
apply FcanonicBound with (radix := radix); auto with float arith.
rewrite <- (Rabs_Ropp (p - q)); rewrite Ropp_minus_distr.
rewrite Rabs_right.
rewrite Rabs_right.
replace (q - p + (p - FNPred b radix precision q))%R with
(q - FNPred b radix precision q)%R; [ idtac | ring ].
unfold FtoRradix in |- *; apply FulpPred; auto.
case H'1; auto.
apply Rge_minus; apply Rle_ge; auto with real float.
case MaxMin with (3 := pGivesBound) (r := p) (p := q); auto with arith.
intros H'0 H'2; elim H'2; intros H'3 H'4; apply H'3; clear H'2; auto.
apply Rge_minus; apply Rle_ge; auto with real float.
apply isMax_inv1 with (1 := H'1).
Qed.
intros p q H'.
apply Rle_trans with (Fulp b radix precision q).
apply (ClosestUlp p q); auto.
replace (powerRZ radix (Fexp q)) with (FtoRradix (Float 1%nat (Fexp q))).
apply (FulpLe b radix); auto.
apply
RoundedModeBounded with (radix := radix) (P := Closest b radix) (r := p);
auto.
apply ClosestRoundedModeP with (precision := precision); auto.
unfold FtoRradix, FtoR in |- *; simpl in |- *.
ring.
Qed.
intros.
case (Zle_or_lt (Fexp p) (Fexp q)); auto; intros Z1.
absurd (powerRZ radix (Fexp p) <= powerRZ radix (Fexp q))%R.
2: apply Rle_powerRZ; auto with real arith.
apply Rgt_not_le.
red in |- *; apply Rlt_le_trans with (2%nat * powerRZ radix (Fexp q))%R.
apply Rltdouble; auto with real arith.
apply Rle_trans with (2%nat * Fabs q)%R.
apply Rmult_le_compat_l; auto with real arith.
replace 0%R with (INR 0); auto with real arith.
replace (powerRZ radix (Fexp q)) with (FtoRradix (Float 1%nat (Fexp q)));
auto.
apply (Fop.RleFexpFabs radix); auto with real zarith.
unfold FtoRradix, FtoR in |- *; simpl in |- *; ring.
rewrite (Fabs_correct radix); auto with arith.
replace (FtoR radix q) with (x - p)%R; auto.
apply ClosestExp; auto.
Qed.
intros p q H' H'0.
case (Rabs_pos (q - p)); intros H1.
Contradict H1; apply Rle_not_lt.
replace 0%R with (Rabs (p - p)); [ case H'0; auto | idtac ].
replace (p - p)%R with 0%R; [ apply Rabs_R0; auto | ring ].
apply Rplus_eq_reg_l with (r := (- p)%R).
apply trans_eq with 0%R; [ ring | idtac ].
apply trans_eq with (q - p)%R; [ idtac | ring ].
generalize H1; unfold Rabs in |- *; case (Rcase_abs (q - p)); auto.
intros r H0; replace 0%R with (-0)%R; [ rewrite H0 | idtac ]; ring.
Qed.
intros r1 r2 min max p q H' H'0 H'1 H'2 H'3.
case (Rle_or_lt r2 max); intros H'4.
2: apply (ClosestMonotone b radix) with (p := r1) (q := r2); auto.
2: apply Rle_lt_trans with (FtoRradix max); auto.
2: apply isMax_inv1 with (1 := H'0).
case H'4; clear H'4; intros H'4.
2: replace (FtoRradix q) with (FtoRradix max).
2: case ClosestMinOrMax with (1 := H'2); intros H'5.
2: replace (FtoRradix p) with (FtoRradix min).
2: apply Rle_trans with r1.
2: apply isMin_inv1 with (1 := H').
2: apply isMax_inv1 with (1 := H'0).
2: apply MinEq with (1 := H'); auto.
2: replace (FtoRradix p) with (FtoRradix max); auto with real.
2: apply MaxEq with (1 := H'0); auto.
2: apply ClosestIdem; auto.
2: case H'0; auto.
2: rewrite <- H'4; auto.
cut (min < r2)%R.
2: apply Rmult_lt_reg_l with (r := INR 2); auto with real.
2: replace (2%nat * min)%R with (min + min)%R;
[ idtac | simpl in |- *; ring ].
2: apply Rle_lt_trans with (2 := H'1).
2: apply Rplus_le_compat_l; auto with real.
2: apply Rle_trans with r1.
2: apply isMin_inv1 with (1 := H').
2: apply isMax_inv1 with (1 := H'0).
intros H'5.
replace (FtoRradix q) with (FtoRradix max).
case ClosestMinOrMax with (1 := H'2); intros H'6.
replace (FtoRradix p) with (FtoRradix min).
apply Rle_trans with r1.
apply isMin_inv1 with (1 := H').
apply isMax_inv1 with (1 := H'0).
apply MinEq with (1 := H'); auto.
replace (FtoRradix p) with (FtoRradix max); auto with real.
apply MaxEq with (1 := H'0); auto.
apply sym_eq.
apply (ClosestMaxEq b radix) with (r := r2) (min := min); auto.
apply isMinComp with (2 := H'0); auto.
apply isMaxComp with (1 := H'); auto.
Qed.
intros x z y H' H'0 H'1.
case MinEx with (r := (/ 2%nat * x)%R) (3 := pGivesBound); auto with arith.
intros min isMin.
case MaxEx with (r := (/ 2%nat * x)%R) (3 := pGivesBound); auto with arith.
intros max isMax.
case (Rle_or_lt y max); intros Rl1.
case Rl1; clear Rl1; intros Rl1.
replace (FtoRradix z) with (FtoRradix max).
apply isMax_inv1 with (1 := isMax).
apply sym_eq.
unfold FtoRradix in |- *;
apply ClosestMaxEq with (b := b) (r := y) (min := min);
auto.
apply isMinComp with (r1 := (/ 2%nat * x)%R) (max := max); auto.
apply Rle_lt_trans with (2 := H'1); auto.
apply isMin_inv1 with (1 := isMin).
apply isMaxComp with (r1 := (/ 2%nat * x)%R) (min := min); auto.
apply Rle_lt_trans with (2 := H'1); auto.
apply isMin_inv1 with (1 := isMin).
replace (FtoR radix min + FtoR radix max)%R with (FtoRradix x).
apply Rmult_lt_reg_l with (r := (/ 2%nat)%R); auto with real.
rewrite <- Rmult_assoc; rewrite Rinv_l; try rewrite Rmult_1_l; auto with real.
unfold FtoRradix in |- *; apply (div2IsBetween b radix precision); auto.
cut (Closest b radix max z); [ intros C0 | idtac ].
replace (FtoRradix z) with (FtoRradix max); auto.
rewrite <- Rl1; auto.
apply Rlt_le; auto.
apply ClosestIdem; auto.
case isMax; auto.
apply (ClosestCompatible b radix y max z z); auto.
case H'0; auto.
apply Rle_trans with (FtoRradix max); auto.
apply isMax_inv1 with (1 := isMax).
apply (ClosestMonotone b radix (FtoRradix max) y); auto.
apply (RoundedModeProjectorIdem b radix (Closest b radix)); auto.
apply ClosestRoundedModeP with (precision := precision); auto.
case isMax; auto.
Qed.
intros p q x H H0 H1.
apply Rle_trans with (Fulp b radix precision p * / 2%nat)%R.
rewrite H1.
replace (Rabs (x - p)) with (2%nat * Rabs (x - p) * / 2%nat)%R;
[ idtac | field; auto with real ].
apply Rmult_le_compat_r; auto with real.
apply ClosestUlp; auto.
apply Rmult_le_compat_r.
apply Rlt_le.
apply Rinv_0_lt_compat; auto with real.
unfold FtoRradix in |- *; apply FulpLe; auto.
Qed.
intros p q x H H0 H1 H2; exists (Fnormalize radix b precision q).
cut (Fcanonic radix b (Fnormalize radix b precision q));
[ intros C1 | apply FnormalizeCanonic; auto with arith ].
split.
apply FcanonicBound with (radix := radix); auto.
split.
apply (FnormalizeCorrect radix); auto.
case C1; intros C2.
apply Z.le_trans with (Fexp p - precision)%Z; auto with zarith.
apply Zplus_le_reg_l with (Z_of_nat precision).
replace (precision + (Fexp p - precision))%Z with (Fexp p); [ idtac | ring ].
replace (precision + Fexp (Fnormalize radix b precision q))%Z with
(Z.succ (Z.pred precision + Fexp (Fnormalize radix b precision q)));
[ idtac | unfold Z.pred, Z.succ in |- *; ring ].
apply Zlt_le_succ.
apply Zlt_powerRZ with (IZR radix); auto with real zarith.
rewrite powerRZ_add; auto with real zarith.
apply
Rle_lt_trans
with
(Z.abs (Fnum (Fnormalize radix b precision q)) *
powerRZ radix (Fexp (Fnormalize radix b precision q)))%R.
apply Rmult_le_compat_r; auto with real zarith.
replace (Z.pred precision) with
(Z_of_nat (pred (digit radix (Fnum (Fnormalize radix b precision q))))).
rewrite <- Zpower_nat_Z_powerRZ.
apply Rle_IZR; apply digitLess; auto with real zarith.
change (~ is_Fzero (Fnormalize radix b precision q)) in |- *;
apply (FnormalNotZero radix b); auto with float.
change
(Z_of_nat (pred (Fdigit radix (Fnormalize radix b precision q))) =
Z.pred precision) in |- *.
rewrite FnormalPrecision with (precision := precision) (4 := C2);
auto with zarith arith.
apply inj_pred; auto with arith.
change (Fabs (Fnormalize radix b precision q) < powerRZ radix (Fexp p))%R
in |- *.
rewrite (Fabs_correct radix); auto; rewrite (FnormalizeCorrect radix); auto.
apply Rle_lt_trans with (Float 1%nat (Fexp p) * / 2%nat)%R.
apply ClosestErrorBound with (x := x); auto.
unfold FtoRradix in |- *; unfold FtoR in |- *; simpl in |- *.
pattern (powerRZ radix (Fexp p)) at 2 in |- *;
replace (powerRZ radix (Fexp p)) with (powerRZ radix (Fexp p) * 1)%R;
[ idtac | ring ].
replace (1 * powerRZ radix (Fexp p))%R with (powerRZ radix (Fexp p));
[ apply Rmult_lt_compat_l | ring ].
apply powerRZ_lt; auto with arith real.
pattern 1%R at 3 in |- *; replace 1%R with (/ 1)%R.
apply Rinv_1_lt_contravar; auto with real.
replace 2%R with (INR 2); auto with real arith.
apply Z.le_trans with (- dExp b)%Z; auto with float zarith.
case C2.
intros H3 (H4, H5); rewrite H4; auto with zarith.
Qed.
intros x p H H'.
apply Rle_trans with (/ 2%nat * Fulp b radix precision p)%R.
replace (Rabs (x - FtoRradix p)) with
(/ 2%nat * (2%nat * Rabs (x - FtoRradix p)))%R.
apply Rmult_le_compat_l; auto with real.
apply ClosestUlp; auto.
rewrite <- Rmult_assoc; rewrite Rinv_l; simpl in |- *; auto with real.
apply
Rle_trans with (/ 2%nat * (Rabs p * (radix * / Zpos (vNum b))))%R;
[ apply Rmult_le_compat_l | right; ring; ring ].
apply Rlt_le; apply Rinv_0_lt_compat; auto with real arith.
unfold Fulp in |- *.
replace (Fexp (Fnormalize radix b precision p)) with
(Fexp (Fnormalize radix b precision p) + precision + - precision)%Z;
[ idtac | ring ].
rewrite powerRZ_add; auto with real zarith.
apply Rle_trans with (Rabs p * radix * powerRZ radix (- precision))%R;
[ apply Rmult_le_compat_r | right ]; auto with real zarith.
2: rewrite pGivesBound; simpl in |- *.
2: rewrite powerRZ_Zopp; auto with real zarith; rewrite Zpower_nat_Z_powerRZ;
auto with real zarith; ring.
replace (FtoRradix p) with (FtoRradix (Fnormalize radix b precision p));
[ idtac | apply (FnormalizeCorrect radix) ]; auto.
rewrite <- (Fabs_correct radix); unfold FtoR in |- *; simpl in |- *;
auto with arith.
rewrite powerRZ_add; auto with real zarith.
replace
(Z.abs (Fnum (Fnormalize radix b precision p)) *
powerRZ radix (Fexp (Fnormalize radix b precision p)) * radix)%R with
(powerRZ radix (Fexp (Fnormalize radix b precision p)) *
(Z.abs (Fnum (Fnormalize radix b precision p)) * radix))%R;
[ idtac | ring ].
apply Rmult_le_compat_l; auto with arith real.
rewrite <- Zpower_nat_Z_powerRZ; auto with real zarith.
rewrite <- Rmult_IZR; apply Rle_IZR.
rewrite <- pGivesBound; pattern radix at 2 in |- *;
rewrite <- (Z.abs_eq radix); auto with zarith.
rewrite <- Zabs_Zmult.
rewrite Zmult_comm; elim H'; auto.
Qed.
intros x p H.
cut (Fcanonic radix b (Fnormalize radix b precision p));
[ intros tmp; Casec tmp; intros Fs | idtac ].
3: apply FnormalizeCanonic; auto with arith.
3: apply
RoundedModeBounded with (radix := radix) (P := Closest b radix) (r := x);
auto.
3: apply ClosestRoundedModeP with (precision := precision); auto.
apply
Rle_trans with (Rabs p * (/ 2%nat * (radix * / Zpos (vNum b))))%R;
[ idtac | apply RmaxLess1 ].
apply ClosestErrorBoundNormal_aux; auto.
apply Rle_trans with (/ 2%nat * Fulp b radix precision p)%R.
replace (Rabs (x - FtoRradix p)) with
(/ 2%nat * (2%nat * Rabs (x - FtoRradix p)))%R.
apply Rmult_le_compat_l; auto with real.
apply ClosestUlp; auto.
rewrite <- Rmult_assoc; rewrite Rinv_l; simpl in |- *; auto with real.
elim Fs; intros H1 H2; elim H2; intros; clear H2.
unfold Fulp in |- *; rewrite H0; apply RmaxLess2.
Qed.

Theorem ClosestErrorBoundNormal :
forall (x : R) (p : float),
Closest b radix x p ->
Fnormal radix b (Fnormalize radix b precision p) ->
(Rabs (x - p) <= Rabs p * (/ 2%nat * powerRZ radix (Z.succ (- precision))))%R.