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Commit 72dcf8b0 by Guillaume Melquiond

Added exponent parameter to rounding functions.

parent fb290c2c
 ... ... @@ -528,11 +528,11 @@ Qed. Theorem inbetween_float_rounding : forall rnd choice, ( forall x m l, inbetween_int m x l -> Zrnd rnd x = choice m l ) -> ( forall x m e l, inbetween_int m x l -> Zrnd rnd x e = choice m e l ) -> forall x m l, let e := canonic_exponent beta fexp x in inbetween_float m e x l -> rounding beta fexp rnd x = F2R (Float beta (choice m l) e). rounding beta fexp rnd x = F2R (Float beta (choice m e l) e). Proof. intros rnd choice Hc x m l e Hl. unfold rounding, F2R. simpl. ... ... @@ -549,9 +549,9 @@ Theorem inbetween_float_DN : inbetween_float m e x l -> rounding beta fexp ZrndDN x = F2R (Float beta m e). Proof. apply inbetween_float_rounding with (choice := fun m l => m). intros x m l Hl. apply Zfloor_imp. apply inbetween_float_rounding with (choice := fun m e l => m). intros x m e l Hl. refine (Zfloor_imp m _ _). apply inbetween_bounds with (2 := Hl). apply Z2R_lt. apply Zlt_succ. ... ... @@ -571,8 +571,8 @@ Theorem inbetween_float_UP : inbetween_float m e x l -> rounding beta fexp ZrndUP x = F2R (Float beta (cond_incr (round_UP l) m) e). Proof. apply inbetween_float_rounding with (choice := fun m l => cond_incr (round_UP l) m). intros x m l Hl. apply inbetween_float_rounding with (choice := fun m e l => cond_incr (round_UP l) m). intros x m e l Hl. assert (Hl': l = loc_Exact \/ (l <> loc_Exact /\ round_UP l = true)). case l ; try (now left) ; now right ; split. destruct Hl' as [Hl'|(Hl1, Hl2)]. ... ... @@ -580,7 +580,7 @@ destruct Hl' as [Hl'|(Hl1, Hl2)]. rewrite Hl'. destruct Hl ; try easy. rewrite H. apply Zceil_Z2R. exact (Zceil_Z2R _). (* not Exact *) rewrite Hl2. simpl. ... ... @@ -604,14 +604,14 @@ Theorem inbetween_float_NE : inbetween_float m e x l -> rounding beta fexp ZrndNE x = F2R (Float beta (cond_incr (round_NE (Zeven m) l) m) e). Proof. apply inbetween_float_rounding with (choice := fun m l => cond_incr (round_NE (Zeven m) l) m). intros x m l Hl. apply inbetween_float_rounding with (choice := fun m e l => cond_incr (round_NE (Zeven m) l) m). intros x m e l Hl. inversion_clear Hl as [Hx|l' Hx Hl']. (* Exact *) rewrite Hx. now rewrite Zrnd_Z2R. (* not Exact *) unfold Zrnd, ZrndNE, ZrndN, Znearest. unfold Zrnd, ZrndNE, ZrndN, Znearest, mkZrounding2. assert (Hm: Zfloor x = m). apply Zfloor_imp. exact (conj (Rlt_le _ _ (proj1 Hx)) (proj2 Hx)). ... ...
 ... ... @@ -294,9 +294,9 @@ Qed. Section Fcore_generic_rounding_pos. Record Zrounding := mkZrounding { Zrnd : R -> Z ; Zrnd_monotone : forall x y, (x <= y)%R -> (Zrnd x <= Zrnd y)%Z ; Zrnd_Z2R : forall n, Zrnd (Z2R n) = n Zrnd : R -> Z -> Z ; Zrnd_monotone : forall x y e, (x <= y)%R -> (Zrnd x e <= Zrnd y e)%Z ; Zrnd_Z2R : forall n e, Zrnd (Z2R n) e = n }. Variable rnd : Zrounding. ... ... @@ -305,18 +305,18 @@ Let Zrnd_monotone := Zrnd_monotone rnd. Let Zrnd_Z2R := Zrnd_Z2R rnd. Theorem Zrnd_DN_or_UP : forall x, Zrnd x = Zfloor x \/ Zrnd x = Zceil x. forall x e, Zrnd x e = Zfloor x \/ Zrnd x e = Zceil x. Proof. intros x. destruct (Zle_or_lt (Zrnd x) (Zfloor x)) as [Hx|Hx]. intros x e. destruct (Zle_or_lt (Zrnd x e) (Zfloor x)) as [Hx|Hx]. left. apply Zle_antisym with (1 := Hx). rewrite <- (Zrnd_Z2R (Zfloor x)). rewrite <- (Zrnd_Z2R (Zfloor x) e). apply Zrnd_monotone. apply Zfloor_lb. right. apply Zle_antisym. rewrite <- (Zrnd_Z2R (Zceil x)). rewrite <- (Zrnd_Z2R (Zceil x) e). apply Zrnd_monotone. apply Zceil_ub. rewrite Zceil_floor_neq. ... ... @@ -328,7 +328,7 @@ apply Zlt_irrefl with (1 := Hx). Qed. Definition rounding x := F2R (Float beta (Zrnd (scaled_mantissa x)) (canonic_exponent x)). F2R (Float beta (Zrnd (scaled_mantissa x) (canonic_exponent x)) (canonic_exponent x)). Theorem rounding_monotone_pos : forall x y, (0 < x)%R -> (x <= y)%R -> (rounding x <= rounding y)%R. ... ... @@ -365,14 +365,14 @@ apply Zrnd_monotone. apply Rmult_le_compat_r. apply bpow_ge_0. exact Hxy. apply Rle_trans with (F2R (Float beta (Zrnd (bpow (ey - 1) * bpow (- fexp ey))%R) (fexp ey))). apply Rle_trans with (F2R (Float beta (Zrnd (bpow (ey - 1) * bpow (- fexp ey)) (fexp ey)) (fexp ey))). rewrite <- bpow_add. rewrite <- (Z2R_Zpower beta (ey - 1 + -fexp ey)). 2: omega. rewrite Zrnd_Z2R. destruct (Zle_or_lt ex (fexp ex)) as [Hx1|Hx1]. apply Rle_trans with (F2R (Float beta 1 (fexp ex))). apply F2R_le_compat. rewrite <- (Zrnd_Z2R 1). rewrite <- (Zrnd_Z2R 1 (fexp ex)). apply Zrnd_monotone. apply Rlt_le. exact (proj2 (mantissa_small_pos _ _ Hex Hx1)). ... ... @@ -381,7 +381,7 @@ rewrite Z2R_Zpower. 2: omega. rewrite <- bpow_add, Rmult_1_l. apply -> bpow_le. omega. apply Rle_trans with (F2R (Float beta (Zrnd (bpow ex * bpow (- fexp ex))%R) (fexp ex))). apply Rle_trans with (F2R (Float beta (Zrnd (bpow ex * bpow (- fexp ex)) (fexp ex)) (fexp ex))). apply F2R_le_compat. apply Zrnd_monotone. apply Rmult_le_compat_r. ... ... @@ -441,7 +441,7 @@ intros x ex He Hx. unfold rounding, scaled_mantissa. rewrite (canonic_exponent_fexp_pos _ _ Hx). unfold F2R. simpl. destruct (Zrnd_DN_or_UP (x * bpow (- fexp ex))) as [Hr|Hr] ; rewrite Hr. destruct (Zrnd_DN_or_UP (x * bpow (- fexp ex)) (fexp ex)) as [Hr|Hr] ; rewrite Hr. (* DN *) split. replace (ex - 1)%Z with (ex - 1 + - fexp ex + fexp ex)%Z by ring. ... ... @@ -500,7 +500,7 @@ intros x ex He Hx. unfold rounding, scaled_mantissa. rewrite (canonic_exponent_fexp_pos _ _ Hx). unfold F2R. simpl. destruct (Zrnd_DN_or_UP (x * bpow (-fexp ex))) as [Hr|Hr] ; rewrite Hr. destruct (Zrnd_DN_or_UP (x * bpow (-fexp ex)) (fexp ex)) as [Hr|Hr] ; rewrite Hr. (* DN *) left. apply Rmult_eq_0_compat_r. ... ... @@ -556,21 +556,23 @@ Section Zrounding_opp. Variable rnd : Zrounding. Definition Zrnd_opp x := Zopp (Zrnd rnd (-x)). Definition Zrnd_opp x e := Zopp (Zrnd rnd (-x) e). Lemma Zrnd_opp_le : forall x y, (x <= y)%R -> (Zrnd_opp x <= Zrnd_opp y)%Z. forall x y e, (x <= y)%R -> (Zrnd_opp x e <= Zrnd_opp y e)%Z. Proof. intros x y Hxy. intros x y e Hxy. unfold Zrnd_opp. generalize (Zrnd_monotone rnd _ _ (Ropp_le_contravar _ _ Hxy)). omega. apply Zopp_le_cancel. rewrite 2!Zopp_involutive. apply Zrnd_monotone. now apply Ropp_le_contravar. Qed. Lemma Zrnd_opp_Z2R : forall n, Zrnd_opp (Z2R n) = n. forall n e, Zrnd_opp (Z2R n) e = n. Proof. intros n. intros n e. unfold Zrnd_opp. rewrite <- opp_Z2R, Zrnd_Z2R. apply Zopp_involutive. ... ... @@ -592,8 +594,11 @@ Qed. End Zrounding_opp. Definition ZrndDN := mkZrounding Zfloor Zfloor_le Zfloor_Z2R. Definition ZrndUP := mkZrounding Zceil Zceil_le Zceil_Z2R. Definition mkZrounding2 rnd (mono : forall x y, (x <= y)%R -> (rnd x <= rnd y)%Z) (z2r : forall n, rnd (Z2R n) = n) := mkZrounding (fun x _ => rnd x) (fun x y _ => mono x y) (fun n _ => z2r n). Definition ZrndDN := mkZrounding2 Zfloor Zfloor_le Zfloor_Z2R. Definition ZrndUP := mkZrounding2 Zceil Zceil_le Zceil_Z2R. Theorem rounding_DN_or_UP : forall rnd x, ... ... @@ -602,7 +607,7 @@ Proof. intros rnd x. unfold rounding. unfold Zrnd at 2 4. simpl. destruct (Zrnd_DN_or_UP rnd (scaled_mantissa x)) as [Hx|Hx]. destruct (Zrnd_DN_or_UP rnd (scaled_mantissa x) (canonic_exponent x)) as [Hx|Hx]. left. now rewrite Hx. right. now rewrite Hx. Qed. ... ... @@ -629,7 +634,7 @@ now apply Ropp_le_contravar. (* . 0 <= y *) apply Rle_trans with R0. apply F2R_le_0_compat. simpl. rewrite <- (Zrnd_Z2R rnd 0). rewrite <- (Zrnd_Z2R rnd 0 (canonic_exponent x)). apply Zrnd_monotone. simpl. rewrite <- (Rmult_0_l (bpow (- fexp (projT1 (ln_beta beta x))))). ... ... @@ -637,7 +642,7 @@ apply Rmult_le_compat_r. apply bpow_ge_0. now apply Rlt_le. apply F2R_ge_0_compat. simpl. rewrite <- (Zrnd_Z2R rnd 0). rewrite <- (Zrnd_Z2R rnd 0 (canonic_exponent y)). apply Zrnd_monotone. apply Rmult_le_pos. exact Hy. ... ... @@ -647,7 +652,7 @@ rewrite Hx. rewrite rounding_0. apply F2R_ge_0_compat. simpl. rewrite <- (Zrnd_Z2R rnd 0). rewrite <- (Zrnd_Z2R rnd 0 (canonic_exponent y)). apply Zrnd_monotone. apply Rmult_le_pos. now rewrite <- Hx. ... ... @@ -1135,7 +1140,7 @@ rewrite opp_Z2R. apply Rplus_comm. Qed. Definition ZrndN := mkZrounding Znearest Znearest_monotone Znearest_Z2R. Definition ZrndN := mkZrounding2 Znearest Znearest_monotone Znearest_Z2R. Theorem Znearest_N_strict : forall x, ... ...
 ... ... @@ -77,10 +77,10 @@ apply Hex. apply Hey. (* *) assert (Hr: ((F2R (Float beta (- (Ztrunc (scaled_mantissa beta (FLX_exp prec) x) * Ztrunc (scaled_mantissa beta (FLX_exp prec) y)) + Zrnd rnd (scaled_mantissa beta (FLX_exp prec) (x * y)) * Ztrunc (scaled_mantissa beta (FLX_exp prec) y)) + Zrnd rnd (scaled_mantissa beta (FLX_exp prec) (x * y)) (canonic_exponent beta (FLX_exp prec) (x * y)) * radix_val beta ^ (cexp (x * y)%R - (cexp x + cexp y))) (cexp x + cexp y))) = f - x * y)%R). rewrite Hx at 6. rewrite Hy at 6. rewrite Hx at 7. rewrite Hy at 7. rewrite <- mult_F2R. simpl. unfold f, rounding, Rminus. ... ...
 ... ... @@ -30,7 +30,7 @@ rewrite Z2R_Zpower. 2: omega. rewrite <- bpow_add. apply (f_equal (fun v => Z2R m * bpow v)%R). ring. exists ((Zrnd rnd (Z2R m * bpow (e - e'))) * Zpower (radix_val beta) (e' - e))%Z. exists ((Zrnd rnd (Z2R m * bpow (e - e')) e') * Zpower (radix_val beta) (e' - e))%Z. unfold F2R. simpl. rewrite mult_Z2R. rewrite Z2R_Zpower. 2: omega. ... ...
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