(* *********************************************************************) (* *) (* The Compcert verified compiler *) (* *) (* Xavier Leroy, INRIA Paris-Rocquencourt *) (* Jacques-Henri Jourdan, INRIA Paris-Rocquencourt *) (* *) (* Copyright Institut National de Recherche en Informatique et en *) (* Automatique. All rights reserved. This file is distributed *) (* under the terms of the GNU General Public License as published by *) (* the Free Software Foundation, either version 2 of the License, or *) (* (at your option) any later version. This file is also distributed *) (* under the terms of the INRIA Non-Commercial License Agreement. *) (* *) (* *********************************************************************) (** Additional operations and proofs about IEEE-754 binary floating-point numbers, on top of the Flocq library. *) Require Import Psatz. Require Import Bool. Require Import Eqdep_dec. Require Import Fcore. Require Import Fcore_digits. Require Import Fcalc_digits. Require Import Fcalc_ops. Require Import Fcalc_round. Require Import Fcalc_bracket. Require Import Fprop_Sterbenz. Require Import Fappli_IEEE. Require Import Fappli_rnd_odd. Local Open Scope Z_scope. Section Extra_ops. (** [prec] is the number of bits of the mantissa including the implicit one. [emax] is the exponent of the infinities. Typically p=24 and emax = 128 in single precision. *) Variable prec emax : Z. Context (prec_gt_0_ : Prec_gt_0 prec). Let emin := (3 - emax - prec)%Z. Let fexp := FLT_exp emin prec. Hypothesis Hmax : (prec < emax)%Z. Let binary_float := binary_float prec emax. (** Remarks on [is_finite] *) Remark is_finite_not_is_nan: forall (f: binary_float), is_finite _ _ f = true -> is_nan _ _ f = false. Proof. destruct f; reflexivity || discriminate. Qed. Remark is_finite_strict_finite: forall (f: binary_float), is_finite_strict _ _ f = true -> is_finite _ _ f = true. Proof. destruct f; reflexivity || discriminate. Qed. (** Digression on FP numbers that cannot be [-0.0]. *) Definition is_finite_pos0 (f: binary_float) : bool := match f with | B754_zero s => negb s | B754_infinity _ => false | B754_nan _ _ => false | B754_finite _ _ _ _ => true end. Lemma Bsign_pos0: forall x, is_finite_pos0 x = true -> Bsign _ _ x = Rlt_bool (B2R _ _ x) 0%R. Proof. intros. destruct x as [ [] | | | [] ex mx Bx ]; try discriminate; simpl. - rewrite Rlt_bool_false; auto. lra. - rewrite Rlt_bool_true; auto. apply F2R_lt_0_compat. compute; auto. - rewrite Rlt_bool_false; auto. assert ((F2R (Float radix2 (Z.pos ex) mx) > 0)%R) by ( apply F2R_gt_0_compat; compute; auto ). lra. Qed. Theorem B2R_inj_pos0: forall x y, is_finite_pos0 x = true -> is_finite_pos0 y = true -> B2R _ _ x = B2R _ _ y -> x = y. Proof. intros. apply B2R_Bsign_inj. destruct x; reflexivity||discriminate. destruct y; reflexivity||discriminate. auto. rewrite ! Bsign_pos0 by auto. rewrite H1; auto. Qed. (** ** Decidable equality *) Definition Beq_dec: forall (f1 f2: binary_float), {f1 = f2} + {f1 <> f2}. Proof. assert (UIP_bool: forall (b1 b2: bool) (e e': b1 = b2), e = e'). { intros. apply UIP_dec. decide equality. } Ltac try_not_eq := try solve [right; congruence]. destruct f1 as [| |? []|], f2 as [| |? []|]; try destruct b; try destruct b0; try solve [left; auto]; try_not_eq. destruct (positive_eq_dec x x0); try_not_eq; subst; left; f_equal; f_equal; apply UIP_bool. destruct (positive_eq_dec x x0); try_not_eq; subst; left; f_equal; f_equal; apply UIP_bool. destruct (positive_eq_dec m m0); try_not_eq; destruct (Z_eq_dec e e1); try solve [right; intro H; inversion H; congruence]; subst; left; f_equal; apply UIP_bool. destruct (positive_eq_dec m m0); try_not_eq; destruct (Z_eq_dec e e1); try solve [right; intro H; inversion H; congruence]; subst; left; f_equal; apply UIP_bool. Defined. (** ** Comparison *) (** [Some c] means ordered as per [c]; [None] means unordered. *) Definition Bcompare (f1 f2: binary_float): option comparison := match f1, f2 with | B754_nan _ _,_ | _,B754_nan _ _ => None | B754_infinity true, B754_infinity true | B754_infinity false, B754_infinity false => Some Eq | B754_infinity true, _ => Some Lt | B754_infinity false, _ => Some Gt | _, B754_infinity true => Some Gt | _, B754_infinity false => Some Lt | B754_finite true _ _ _, B754_zero _ => Some Lt | B754_finite false _ _ _, B754_zero _ => Some Gt | B754_zero _, B754_finite true _ _ _ => Some Gt | B754_zero _, B754_finite false _ _ _ => Some Lt | B754_zero _, B754_zero _ => Some Eq | B754_finite s1 m1 e1 _, B754_finite s2 m2 e2 _ => match s1, s2 with | true, false => Some Lt | false, true => Some Gt | false, false => match Zcompare e1 e2 with | Lt => Some Lt | Gt => Some Gt | Eq => Some (Pcompare m1 m2 Eq) end | true, true => match Zcompare e1 e2 with | Lt => Some Gt | Gt => Some Lt | Eq => Some (CompOpp (Pcompare m1 m2 Eq)) end end end. Theorem Bcompare_finite_correct: forall f1 f2, is_finite _ _ f1 = true -> is_finite _ _ f2 = true -> Bcompare f1 f2 = Some (Rcompare (B2R _ _ f1) (B2R _ _ f2)). Proof. Ltac apply_Rcompare := match goal with | [ |- Some Lt = Some (Rcompare _ _) ] => f_equal; symmetry; apply Rcompare_Lt | [ |- Some Eq = Some (Rcompare _ _) ] => f_equal; symmetry; apply Rcompare_Eq | [ |- Some Gt = Some (Rcompare _ _) ] => f_equal; symmetry; apply Rcompare_Gt end. unfold Bcompare; intros. destruct f1, f2; try discriminate; unfold B2R, F2R, Fnum, Fexp, cond_Zopp; try (replace 0%R with (Z2R 0 * bpow radix2 e)%R by (simpl Z2R; ring); rewrite Rcompare_mult_r by (apply bpow_gt_0); rewrite Rcompare_Z2R). apply_Rcompare; reflexivity. destruct b0; reflexivity. destruct b; reflexivity. clear H H0. apply andb_prop in e0; destruct e0; apply (canonic_canonic_mantissa _ _ false) in H. apply andb_prop in e2; destruct e2; apply (canonic_canonic_mantissa _ _ false) in H1. pose proof (Zcompare_spec e e1); unfold canonic, Fexp in H1, H. assert (forall m1 m2 e1 e2, let x := (Z2R (Zpos m1) * bpow radix2 e1)%R in let y := (Z2R (Zpos m2) * bpow radix2 e2)%R in (canonic_exp radix2 fexp x < canonic_exp radix2 fexp y)%Z -> (x < y)%R). { intros; apply Rnot_le_lt; intro; apply (ln_beta_le radix2) in H5. unfold canonic_exp in H4. apply (fexp_monotone prec emax) in H5. unfold fexp, emin in H4. omega. apply Rmult_gt_0_compat; [apply (Z2R_lt 0); reflexivity|now apply bpow_gt_0]. } assert (forall m1 m2 e1 e2, (Z2R (- Zpos m1) * bpow radix2 e1 < Z2R (Zpos m2) * bpow radix2 e2)%R). { intros; apply (Rlt_trans _ 0%R). replace 0%R with (0*bpow radix2 e0)%R by ring; apply Rmult_lt_compat_r; [apply bpow_gt_0; reflexivity|now apply (Z2R_lt _ 0)]. apply Rmult_gt_0_compat; [apply (Z2R_lt 0); reflexivity|now apply bpow_gt_0]. } destruct b, b0; try (now apply_Rcompare; apply H5); inversion H3; try (apply_Rcompare; apply H4; rewrite H, H1 in H7; assumption); try (apply_Rcompare; do 2 rewrite Z2R_opp, Ropp_mult_distr_l_reverse; apply Ropp_lt_contravar; apply H4; rewrite H, H1 in H7; assumption); rewrite H7, Rcompare_mult_r, Rcompare_Z2R by (apply bpow_gt_0); reflexivity. Qed. Theorem Bcompare_swap: forall x y, Bcompare y x = match Bcompare x y with Some c => Some (CompOpp c) | None => None end. Proof. intros. destruct x as [ ? | [] | ? ? | [] mx ex Bx ]; destruct y as [ ? | [] | ? ? | [] my ey By ]; simpl; auto. - rewrite <- (Zcompare_antisym ex ey). destruct (ex ?= ey)%Z; auto. simpl. f_equal; f_equal. symmetry. apply Pcompare_antisym. - rewrite <- (Zcompare_antisym ex ey). destruct (ex ?= ey)%Z; auto. simpl. f_equal. symmetry. apply Pcompare_antisym. Qed. (** ** Absolute value *) Definition Babs abs_nan (x: binary_float) : binary_float := match x with | B754_nan sx plx => let '(sres, plres) := abs_nan sx plx in B754_nan _ _ sres plres | B754_infinity sx => B754_infinity _ _ false | B754_finite sx mx ex Hx => B754_finite _ _ false mx ex Hx | B754_zero sx => B754_zero _ _ false end. Theorem B2R_Babs : forall abs_nan x, B2R _ _ (Babs abs_nan x) = Rabs (B2R _ _ x). Proof. intros abs_nan [sx|sx|sx plx|sx mx ex Hx]; apply sym_eq ; try apply Rabs_R0. simpl. destruct abs_nan. simpl. apply Rabs_R0. simpl. rewrite <- F2R_abs. destruct sx; auto. Qed. Theorem is_finite_Babs : forall abs_nan x, is_finite _ _ (Babs abs_nan x) = is_finite _ _ x. Proof. intros abs_nan [| | |] ; try easy. intros s pl. simpl. now case abs_nan. Qed. Theorem sign_Babs: forall abs_nan x, is_nan _ _ x = false -> Bsign _ _ (Babs abs_nan x) = false. Proof. intros abs_nan [| | |]; reflexivity || discriminate. Qed. Theorem Babs_idempotent : forall abs_nan (x: binary_float), is_nan _ _ x = false -> Babs abs_nan (Babs abs_nan x) = Babs abs_nan x. Proof. now intros abs_nan [sx|sx|sx plx|sx mx ex Hx] ; auto. Qed. Theorem Babs_opp: forall abs_nan opp_nan x, is_nan _ _ x = false -> Babs abs_nan (Bopp _ _ opp_nan x) = Babs abs_nan x. Proof. intros abs_nan opp_nan [| | |]; reflexivity || discriminate. Qed. (** ** Conversion from an integer to a FP number *) (** Integers that can be represented exactly as FP numbers. *) Definition integer_representable (n: Z): Prop := Z.abs n <= 2^emax - 2^(emax - prec) /\ generic_format radix2 fexp (Z2R n). Let int_upper_bound_eq: 2^emax - 2^(emax - prec) = (2^prec - 1) * 2^(emax - prec). Proof. red in prec_gt_0_. ring_simplify. rewrite <- (Zpower_plus radix2) by omega. f_equal. f_equal. omega. Qed. Lemma integer_representable_n2p: forall n p, -2^prec < n < 2^prec -> 0 <= p -> p <= emax - prec -> integer_representable (n * 2^p). Proof. intros; split. - red in prec_gt_0_. replace (Z.abs (n * 2^p)) with (Z.abs n * 2^p). rewrite int_upper_bound_eq. apply Zmult_le_compat. zify; omega. apply (Zpower_le radix2); omega. zify; omega. apply (Zpower_ge_0 radix2). rewrite Z.abs_mul. f_equal. rewrite Z.abs_eq. auto. apply (Zpower_ge_0 radix2). - apply generic_format_FLT. exists (Float radix2 n p). unfold F2R; simpl. split. rewrite <- Z2R_Zpower by auto. apply Z2R_mult. split. zify; omega. unfold emin; red in prec_gt_0_; omega. Qed. Lemma integer_representable_2p: forall p, 0 <= p <= emax - 1 -> integer_representable (2^p). Proof. intros; split. - red in prec_gt_0_. rewrite Z.abs_eq by (apply (Zpower_ge_0 radix2)). apply Zle_trans with (2^(emax-1)). apply (Zpower_le radix2); omega. assert (2^emax = 2^(emax-1)*2). { change 2 with (2^1) at 3. rewrite <- (Zpower_plus radix2) by omega. f_equal. omega. } assert (2^(emax - prec) <= 2^(emax - 1)). { apply (Zpower_le radix2). omega. } omega. - red in prec_gt_0_. apply generic_format_FLT. exists (Float radix2 1 p). unfold F2R; simpl. split. rewrite Rmult_1_l. rewrite <- Z2R_Zpower. auto. omega. split. change 1 with (2^0). apply (Zpower_lt radix2). omega. auto. unfold emin; omega. Qed. Lemma integer_representable_opp: forall n, integer_representable n -> integer_representable (-n). Proof. intros n (A & B); split. rewrite Z.abs_opp. auto. rewrite Z2R_opp. apply generic_format_opp; auto. Qed. Lemma integer_representable_n2p_wide: forall n p, -2^prec <= n <= 2^prec -> 0 <= p -> p < emax - prec -> integer_representable (n * 2^p). Proof. intros. red in prec_gt_0_. destruct (Z.eq_dec n (2^prec)); [idtac | destruct (Z.eq_dec n (-2^prec))]. - rewrite e. rewrite <- (Zpower_plus radix2) by omega. apply integer_representable_2p. omega. - rewrite e. rewrite <- Zopp_mult_distr_l. apply integer_representable_opp. rewrite <- (Zpower_plus radix2) by omega. apply integer_representable_2p. omega. - apply integer_representable_n2p; omega. Qed. Lemma integer_representable_n: forall n, -2^prec <= n <= 2^prec -> integer_representable n. Proof. red in prec_gt_0_. intros. replace n with (n * 2^0) by (change (2^0) with 1; ring). apply integer_representable_n2p_wide. auto. omega. omega. Qed. Lemma round_int_no_overflow: forall n, Z.abs n <= 2^emax - 2^(emax-prec) -> (Rabs (round radix2 fexp (round_mode mode_NE) (Z2R n)) < bpow radix2 emax)%R. Proof. intros. red in prec_gt_0_. rewrite <- round_NE_abs. apply Rle_lt_trans with (Z2R (2^emax - 2^(emax-prec))). apply round_le_generic. apply fexp_correct; auto. apply valid_rnd_N. apply generic_format_FLT. exists (Float radix2 (2^prec-1) (emax-prec)). rewrite int_upper_bound_eq. unfold F2R; simpl. split. rewrite <- Z2R_Zpower by omega. rewrite <- Z2R_mult. auto. split. assert (0 < 2^prec) by (apply (Zpower_gt_0 radix2); omega). zify; omega. unfold emin; omega. rewrite <- Z2R_abs. apply Z2R_le. auto. rewrite <- Z2R_Zpower by omega. apply Z2R_lt. simpl. assert (0 < 2^(emax-prec)) by (apply (Zpower_gt_0 radix2); omega). omega. apply fexp_correct. auto. Qed. (** Conversion from an integer. Round to nearest. *) Definition BofZ (n: Z) : binary_float := binary_normalize prec emax prec_gt_0_ Hmax mode_NE n 0 false. Theorem BofZ_correct: forall n, if Rlt_bool (Rabs (round radix2 fexp (round_mode mode_NE) (Z2R n))) (bpow radix2 emax) then B2R prec emax (BofZ n) = round radix2 fexp (round_mode mode_NE) (Z2R n) /\ is_finite _ _ (BofZ n) = true /\ Bsign prec emax (BofZ n) = Zlt_bool n 0 else B2FF prec emax (BofZ n) = binary_overflow prec emax mode_NE (Zlt_bool n 0). Proof. intros. generalize (binary_normalize_correct prec emax prec_gt_0_ Hmax mode_NE n 0 false). fold emin; fold fexp; fold (BofZ n). replace (F2R {| Fnum := n; Fexp := 0 |}) with (Z2R n). destruct Rlt_bool. - intros (A & B & C). split; [|split]. + auto. + auto. + rewrite C. change 0%R with (Z2R 0). rewrite Rcompare_Z2R. unfold Zlt_bool. auto. - intros A; rewrite A. f_equal. change 0%R with (Z2R 0). generalize (Zlt_bool_spec n 0); intros SPEC; inversion SPEC. apply Rlt_bool_true; apply Z2R_lt; auto. apply Rlt_bool_false; apply Z2R_le; auto. - unfold F2R; simpl. ring. Qed. Theorem BofZ_finite: forall n, Z.abs n <= 2^emax - 2^(emax-prec) -> B2R _ _ (BofZ n) = round radix2 fexp (round_mode mode_NE) (Z2R n) /\ is_finite _ _ (BofZ n) = true /\ Bsign _ _ (BofZ n) = Zlt_bool n 0%Z. Proof. intros. generalize (BofZ_correct n). rewrite Rlt_bool_true. auto. apply round_int_no_overflow; auto. Qed. Theorem BofZ_representable: forall n, integer_representable n -> B2R _ _ (BofZ n) = Z2R n /\ is_finite _ _ (BofZ n) = true /\ Bsign _ _ (BofZ n) = (n B2R _ _ (BofZ n) = Z2R n /\ is_finite _ _ (BofZ n) = true /\ Bsign _ _ (BofZ n) = Zlt_bool n 0%Z. Proof. intros. apply BofZ_representable. apply integer_representable_n; auto. Qed. Lemma BofZ_finite_pos0: forall n, Z.abs n <= 2^emax - 2^(emax-prec) -> is_finite_pos0 (BofZ n) = true. Proof. intros. generalize (binary_normalize_correct prec emax prec_gt_0_ Hmax mode_NE n 0 false). fold emin; fold fexp; fold (BofZ n). replace (F2R {| Fnum := n; Fexp := 0 |}) with (Z2R n) by (unfold F2R; simpl; ring). rewrite Rlt_bool_true by (apply round_int_no_overflow; auto). intros (A & B & C). destruct (BofZ n); auto; try discriminate. simpl in *. rewrite C. change 0%R with (Z2R 0). rewrite Rcompare_Z2R. generalize (Zcompare_spec n 0); intros SPEC; inversion SPEC; auto. assert ((round radix2 fexp ZnearestE (Z2R n) <= -1)%R). { change (-1)%R with (Z2R (-1)). apply round_le_generic. apply fexp_correct. auto. apply valid_rnd_N. apply (integer_representable_opp 1). apply (integer_representable_2p 0). red in prec_gt_0_; omega. apply Z2R_le; omega. } lra. Qed. Lemma BofZ_finite_equal: forall x y, Z.abs x <= 2^emax - 2^(emax-prec) -> Z.abs y <= 2^emax - 2^(emax-prec) -> B2R _ _ (BofZ x) = B2R _ _ (BofZ y) -> BofZ x = BofZ y. Proof. intros. apply B2R_inj_pos0; auto; apply BofZ_finite_pos0; auto. Qed. (** Commutation properties with addition, subtraction, multiplication. *) Theorem BofZ_plus: forall nan p q, integer_representable p -> integer_representable q -> Bplus _ _ _ Hmax nan mode_NE (BofZ p) (BofZ q) = BofZ (p + q). Proof. intros. destruct (BofZ_representable p) as (A & B & C); auto. destruct (BofZ_representable q) as (D & E & F); auto. generalize (Bplus_correct _ _ _ Hmax nan mode_NE (BofZ p) (BofZ q) B E). fold emin; fold fexp. rewrite A, D. rewrite <- Z2R_plus. generalize (BofZ_correct (p + q)). destruct Rlt_bool. - intros (P & Q & R) (U & V & W). apply B2R_Bsign_inj; auto. rewrite P, U; auto. rewrite R, W, C, F. change 0%R with (Z2R 0). rewrite Rcompare_Z2R. unfold Zlt_bool at 3. generalize (Zcompare_spec (p + q) 0); intros SPEC; inversion SPEC; auto. assert (EITHER: 0 <= p \/ 0 <= q) by omega. destruct EITHER; [apply andb_false_intro1 | apply andb_false_intro2]; apply Zlt_bool_false; auto. - intros P (U & V). apply B2FF_inj. rewrite P, U, C. f_equal. rewrite C, F in V. generalize (Zlt_bool_spec p 0) (Zlt_bool_spec q 0). rewrite <- V. intros SPEC1 SPEC2; inversion SPEC1; inversion SPEC2; try congruence; symmetry. apply Zlt_bool_true; omega. apply Zlt_bool_false; omega. Qed. Theorem BofZ_minus: forall nan p q, integer_representable p -> integer_representable q -> Bminus _ _ _ Hmax nan mode_NE (BofZ p) (BofZ q) = BofZ (p - q). Proof. intros. destruct (BofZ_representable p) as (A & B & C); auto. destruct (BofZ_representable q) as (D & E & F); auto. generalize (Bminus_correct _ _ _ Hmax nan mode_NE (BofZ p) (BofZ q) B E). fold emin; fold fexp. rewrite A, D. rewrite <- Z2R_minus. generalize (BofZ_correct (p - q)). destruct Rlt_bool. - intros (P & Q & R) (U & V & W). apply B2R_Bsign_inj; auto. rewrite P, U; auto. rewrite R, W, C, F. change 0%R with (Z2R 0). rewrite Rcompare_Z2R. unfold Zlt_bool at 3. generalize (Zcompare_spec (p - q) 0); intros SPEC; inversion SPEC; auto. assert (EITHER: 0 <= p \/ q < 0) by omega. destruct EITHER; [apply andb_false_intro1 | apply andb_false_intro2]. rewrite Zlt_bool_false; auto. rewrite Zlt_bool_true; auto. - intros P (U & V). apply B2FF_inj. rewrite P, U, C. f_equal. rewrite C, F in V. generalize (Zlt_bool_spec p 0) (Zlt_bool_spec q 0). rewrite V. intros SPEC1 SPEC2; inversion SPEC1; inversion SPEC2; symmetry. rewrite <- H3 in H1; discriminate. apply Zlt_bool_true; omega. apply Zlt_bool_false; omega. rewrite <- H3 in H1; discriminate. Qed. Theorem BofZ_mult: forall nan p q, integer_representable p -> integer_representable q -> 0 < q -> Bmult _ _ _ Hmax nan mode_NE (BofZ p) (BofZ q) = BofZ (p * q). Proof. intros. assert (SIGN: xorb (p 2^prec <= Z.abs x -> 0 <= p <= emax - 1 -> Bmult _ _ _ Hmax nan mode_NE (BofZ x) (BofZ (2^p)) = BofZ (x * 2^p). Proof. intros. destruct (Z.eq_dec x 0). - subst x. apply BofZ_mult. apply integer_representable_n. generalize (Zpower_ge_0 radix2 prec). simpl; omega. apply integer_representable_2p. auto. apply (Zpower_gt_0 radix2). omega. - assert (Z2R x <> 0%R) by (apply (Z2R_neq _ _ n)). destruct (BofZ_finite x H) as (A & B & C). destruct (BofZ_representable (2^p)) as (D & E & F). apply integer_representable_2p. auto. assert (canonic_exp radix2 fexp (Z2R (x * 2^p)) = canonic_exp radix2 fexp (Z2R x) + p). { unfold canonic_exp, fexp. rewrite Z2R_mult. change (2^p) with (radix2^p). rewrite Z2R_Zpower by omega. rewrite ln_beta_mult_bpow by auto. assert (prec + 1 <= ln_beta radix2 (Z2R x)). { rewrite <- (ln_beta_abs radix2 (Z2R x)). rewrite <- (ln_beta_bpow radix2 prec). apply ln_beta_le. apply bpow_gt_0. rewrite <- Z2R_Zpower by (red in prec_gt_0_;omega). rewrite <- Z2R_abs. apply Z2R_le; auto. } unfold FLT_exp. unfold emin; red in prec_gt_0_; zify; omega. } assert (forall m, round radix2 fexp m (Z2R x) * Z2R (2^p) = round radix2 fexp m (Z2R (x * 2^p)))%R. { intros. unfold round, scaled_mantissa. rewrite H3. rewrite Z2R_mult. rewrite Z.opp_add_distr. rewrite bpow_plus. set (a := Z2R x); set (b := bpow radix2 (- canonic_exp radix2 fexp a)). replace (a * Z2R (2^p) * (b * bpow radix2 (-p)))%R with (a * b)%R. unfold F2R; simpl. rewrite Rmult_assoc. f_equal. rewrite bpow_plus. f_equal. apply (Z2R_Zpower radix2). omega. transitivity ((a * b) * (Z2R (2^p) * bpow radix2 (-p)))%R. rewrite (Z2R_Zpower radix2). rewrite <- bpow_plus. replace (p + -p) with 0 by omega. change (bpow radix2 0) with 1%R. ring. omega. ring. } assert (forall m x, round radix2 fexp (round_mode m) (round radix2 fexp (round_mode m) x) = round radix2 fexp (round_mode m) x). { intros. apply round_generic. apply valid_rnd_round_mode. apply generic_format_round. apply fexp_correct; auto. apply valid_rnd_round_mode. } assert (xorb (x 1 -> prec' > 1 -> prec' >= prec + 2 -> emin' <= emin - 2 -> round radix2 fexp (Znearest choice) (round radix2 (FLT_exp emin' prec') Zrnd_odd x) = round radix2 fexp (Znearest choice) x. Proof. intros. apply round_odd_prop. auto. apply fexp_correct; auto. apply exists_NE_FLT. right; omega. apply FLT_exp_valid. red; omega. apply exists_NE_FLT. right; omega. unfold fexp, FLT_exp; intros. zify; omega. Qed. Corollary round_odd_fix: forall x p choice, prec > 1 -> 0 <= p -> (bpow radix2 (prec + p + 1) <= Rabs x)%R -> round radix2 fexp (Znearest choice) (round radix2 (FIX_exp p) Zrnd_odd x) = round radix2 fexp (Znearest choice) x. Proof. intros. destruct (Req_EM_T x 0%R). - subst x. rewrite round_0. auto. apply valid_rnd_odd. - set (prec' := ln_beta radix2 x - p). set (emin' := emin - 2). assert (PREC: ln_beta radix2 (bpow radix2 (prec + p + 1)) <= ln_beta radix2 x). { rewrite <- (ln_beta_abs radix2 x). apply ln_beta_le; auto. apply bpow_gt_0. } rewrite ln_beta_bpow in PREC. assert (CANON: canonic_exp radix2 (FLT_exp emin' prec') x = canonic_exp radix2 (FIX_exp p) x). { unfold canonic_exp, FLT_exp, FIX_exp. replace (ln_beta radix2 x - prec') with p by (unfold prec'; omega). apply Z.max_l. unfold emin', emin. red in prec_gt_0_; omega. } assert (RND: round radix2 (FIX_exp p) Zrnd_odd x = round radix2 (FLT_exp emin' prec') Zrnd_odd x). { unfold round, scaled_mantissa. rewrite CANON. auto. } rewrite RND. apply round_odd_flt. auto. unfold prec'. red in prec_gt_0_; omega. unfold prec'. omega. unfold emin'. omega. Qed. Definition int_round_odd (x: Z) (p: Z) := (if Z.eqb (x mod 2^p) 0 || Z.odd (x / 2^p) then x / 2^p else x / 2^p + 1) * 2^p. Lemma Zrnd_odd_int: forall n p, 0 <= p -> Zrnd_odd (Z2R n * bpow radix2 (-p)) * 2^p = int_round_odd n p. Proof. intros. assert (0 < 2^p) by (apply (Zpower_gt_0 radix2); omega). assert (n = (n / 2^p) * 2^p + n mod 2^p) by (rewrite Zmult_comm; apply Z.div_mod; omega). assert (0 <= n mod 2^p < 2^p) by (apply Z_mod_lt; omega). unfold int_round_odd. set (q := n / 2^p) in *; set (r := n mod 2^p) in *. f_equal. pose proof (bpow_gt_0 radix2 (-p)). assert (bpow radix2 p * bpow radix2 (-p) = 1)%R. { rewrite <- bpow_plus. replace (p + -p) with 0 by omega. auto. } assert (Z2R n * bpow radix2 (-p) = Z2R q + Z2R r * bpow radix2 (-p))%R. { rewrite H1. rewrite Z2R_plus, Z2R_mult. change (Z2R (2^p)) with (Z2R (radix2^p)). rewrite Z2R_Zpower by omega. ring_simplify. rewrite Rmult_assoc. rewrite H4. ring. } assert (0 <= Z2R r < bpow radix2 p)%R. { split. change 0%R with (Z2R 0). apply Z2R_le; omega. rewrite <- Z2R_Zpower by omega. apply Z2R_lt; tauto. } assert (0 <= Z2R r * bpow radix2 (-p) < 1)%R. { generalize (bpow_gt_0 radix2 (-p)). intros. split. apply Rmult_le_pos; lra. rewrite <- H4. apply Rmult_lt_compat_r. auto. tauto. } assert (Zfloor (Z2R n * bpow radix2 (-p)) = q). { apply Zfloor_imp. rewrite H5. rewrite Z2R_plus. change (Z2R 1) with 1%R. lra. } unfold Zrnd_odd. destruct Req_EM_T. - assert (Z2R r * bpow radix2 (-p) = 0)%R. { rewrite H8 in e. rewrite e in H5. lra. } apply Rmult_integral in H9. destruct H9; [ | lra ]. apply (eq_Z2R r 0) in H9. apply <- Z.eqb_eq in H9. rewrite H9. assumption. - assert (Z2R r * bpow radix2 (-p) <> 0)%R. { rewrite H8 in n0. lra. } destruct (Z.eqb r 0) eqn:RZ. apply Z.eqb_eq in RZ. rewrite RZ in H9. change (Z2R 0) with 0%R in H9. rewrite Rmult_0_l in H9. congruence. rewrite Zceil_floor_neq by lra. rewrite H8. change Zeven with Z.even. rewrite Zodd_even_bool. destruct (Z.even q); auto. Qed. Lemma int_round_odd_le: forall p x y, 0 <= p -> x <= y -> int_round_odd x p <= int_round_odd y p. Proof. intros. assert (Zrnd_odd (Z2R x * bpow radix2 (-p)) <= Zrnd_odd (Z2R y * bpow radix2 (-p))). { apply Zrnd_le. apply valid_rnd_odd. apply Rmult_le_compat_r. apply bpow_ge_0. apply Z2R_le; auto. } rewrite <- ! Zrnd_odd_int by auto. apply Zmult_le_compat_r. auto. apply (Zpower_ge_0 radix2). Qed. Lemma int_round_odd_exact: forall p x, 0 <= p -> (2^p | x) -> int_round_odd x p = x. Proof. intros. unfold int_round_odd. apply Znumtheory.Zdivide_mod in H0. rewrite H0. simpl. rewrite Zmult_comm. symmetry. apply Z_div_exact_2. apply Zlt_gt. apply (Zpower_gt_0 radix2). auto. auto. Qed. Theorem BofZ_round_odd: forall x p, prec > 1 -> Z.abs x <= 2^emax - 2^(emax-prec) -> 0 <= p <= emax - prec -> 2^(prec + p + 1) <= Z.abs x -> BofZ x = BofZ (int_round_odd x p). Proof. intros x p PREC XRANGE PRANGE XGE. assert (DIV: (2^p | 2^emax - 2^(emax - prec))). { rewrite int_upper_bound_eq. apply Z.divide_mul_r. exists (2^(emax - prec - p)). red in prec_gt_0_. rewrite <- (Zpower_plus radix2) by omega. f_equal; omega. } assert (YRANGE: Z.abs (int_round_odd x p) <= 2^emax - 2^(emax-prec)). { apply Z.abs_le. split. replace (-(2^emax - 2^(emax-prec))) with (int_round_odd (-(2^emax - 2^(emax-prec))) p). apply int_round_odd_le; zify; omega. apply int_round_odd_exact. omega. apply Z.divide_opp_r. auto. replace (2^emax - 2^(emax-prec)) with (int_round_odd (2^emax - 2^(emax-prec)) p). apply int_round_odd_le; zify; omega. apply int_round_odd_exact. omega. auto. } destruct (BofZ_finite x XRANGE) as (X1 & X2 & X3). destruct (BofZ_finite (int_round_odd x p) YRANGE) as (Y1 & Y2 & Y3). apply BofZ_finite_equal; auto. rewrite X1, Y1. assert (Z2R (int_round_odd x p) = round radix2 (FIX_exp p) Zrnd_odd (Z2R x)). { unfold round, scaled_mantissa, canonic_exp, FIX_exp. rewrite <- Zrnd_odd_int by omega. unfold F2R; simpl. rewrite Z2R_mult. f_equal. apply (Z2R_Zpower radix2). omega. } rewrite H. symmetry. apply round_odd_fix. auto. omega. rewrite <- Z2R_Zpower. rewrite <- Z2R_abs. apply Z2R_le; auto. red in prec_gt_0_; omega. Qed. Lemma int_round_odd_shifts: forall x p, 0 <= p -> int_round_odd x p = Z.shiftl (if Z.eqb (x mod 2^p) 0 then Z.shiftr x p else Z.lor (Z.shiftr x p) 1) p. Proof. intros. unfold int_round_odd. rewrite Z.shiftl_mul_pow2 by auto. f_equal. rewrite Z.shiftr_div_pow2 by auto. destruct (x mod 2^p =? 0) eqn:E. auto. assert (forall n, (if Z.odd n then n else n + 1) = Z.lor n 1). { destruct n; simpl; auto. destruct p0; auto. destruct p0; auto. induction p0; auto. } simpl. apply H0. Qed. Lemma int_round_odd_bits: forall x y p, 0 <= p -> (forall i, 0 <= i < p -> Z.testbit y i = false) -> Z.testbit y p = (if Z.eqb (x mod 2^p) 0 then Z.testbit x p else true) -> (forall i, p < i -> Z.testbit y i = Z.testbit x i) -> int_round_odd x p = y. Proof. intros until p; intros PPOS BELOW AT ABOVE. rewrite int_round_odd_shifts by auto. apply Z.bits_inj'. intros. generalize (Zcompare_spec n p); intros SPEC; inversion SPEC. - rewrite BELOW by auto. apply Z.shiftl_spec_low; auto. - subst n. rewrite AT. rewrite Z.shiftl_spec_high by omega. replace (p - p) with 0 by omega. destruct (x mod 2^p =? 0). + rewrite Z.shiftr_spec by omega. f_equal; omega. + rewrite Z.lor_spec. apply orb_true_r. - rewrite ABOVE by auto. rewrite Z.shiftl_spec_high by omega. destruct (x mod 2^p =? 0). rewrite Z.shiftr_spec by omega. f_equal; omega. rewrite Z.lor_spec, Z.shiftr_spec by omega. change 1 with (Z.ones 1). rewrite Z.ones_spec_high by omega. rewrite orb_false_r. f_equal; omega. Qed. (** ** Conversion from a FP number to an integer *) (** Always rounds toward zero. *) Definition ZofB (f: binary_float): option Z := match f with | B754_finite s m (Zpos e) _ => Some (cond_Zopp s (Zpos m) * Zpower_pos radix2 e)%Z | B754_finite s m 0 _ => Some (cond_Zopp s (Zpos m)) | B754_finite s m (Zneg e) _ => Some (cond_Zopp s (Zpos m / Zpower_pos radix2 e))%Z | B754_zero _ => Some 0%Z | _ => None end. Theorem ZofB_correct: forall f, ZofB f = if is_finite _ _ f then Some (Ztrunc (B2R _ _ f)) else None. Proof. destruct f; simpl; auto. - f_equal. symmetry. apply (Ztrunc_Z2R 0). - destruct e; f_equal. + unfold F2R; simpl. rewrite Rmult_1_r. rewrite Ztrunc_Z2R. auto. + unfold F2R; simpl. rewrite <- Z2R_mult. rewrite Ztrunc_Z2R. auto. + unfold F2R; simpl. rewrite Z2R_cond_Zopp. rewrite <- cond_Ropp_mult_l. assert (EQ: forall x, Ztrunc (cond_Ropp b x) = cond_Zopp b (Ztrunc x)). { intros. destruct b; simpl; auto. apply Ztrunc_opp. } rewrite EQ. f_equal. generalize (Zpower_pos_gt_0 2 p (refl_equal _)); intros. rewrite Ztrunc_floor. symmetry. apply Zfloor_div. omega. apply Rmult_le_pos. apply (Z2R_le 0). compute; congruence. apply Rlt_le. apply Rinv_0_lt_compat. apply (Z2R_lt 0). auto. Qed. (** Interval properties. *) Remark Ztrunc_range_pos: forall x, 0 < Ztrunc x -> (Z2R (Ztrunc x) <= x < Z2R (Ztrunc x + 1)%Z)%R. Proof. intros. rewrite Ztrunc_floor. split. apply Zfloor_lb. rewrite Z2R_plus. apply Zfloor_ub. generalize (Rle_bool_spec 0%R x). intros RLE; inversion RLE; subst; clear RLE. auto. rewrite Ztrunc_ceil in H by lra. unfold Zceil in H. assert (-x < 0)%R. { apply Rlt_le_trans with (Z2R (Zfloor (-x)) + 1)%R. apply Zfloor_ub. change 0%R with (Z2R 0). change 1%R with (Z2R 1). rewrite <- Z2R_plus. apply Z2R_le. omega. } lra. Qed. Remark Ztrunc_range_zero: forall x, Ztrunc x = 0 -> (-1 < x < 1)%R. Proof. intros; generalize (Rle_bool_spec 0%R x). intros RLE; inversion RLE; subst; clear RLE. - rewrite Ztrunc_floor in H by auto. split. + apply Rlt_le_trans with 0%R; auto. rewrite <- Ropp_0. apply Ropp_lt_contravar. apply Rlt_0_1. + replace 1%R with (Z2R (Zfloor x) + 1)%R. apply Zfloor_ub. rewrite H. simpl. apply Rplus_0_l. - rewrite Ztrunc_ceil in H by (apply Rlt_le; auto). split. + apply Ropp_lt_cancel. rewrite Ropp_involutive. replace 1%R with (Z2R (Zfloor (-x)) + 1)%R. apply Zfloor_ub. unfold Zceil in H. replace (Zfloor (-x)) with 0 by omega. simpl. apply Rplus_0_l. + apply Rlt_le_trans with 0%R; auto. apply Rle_0_1. Qed. Theorem ZofB_range_pos: forall f n, ZofB f = Some n -> 0 < n -> (Z2R n <= B2R _ _ f < Z2R (n + 1)%Z)%R. Proof. intros. rewrite ZofB_correct in H. destruct (is_finite prec emax f) eqn:FIN; inversion H. apply Ztrunc_range_pos. congruence. Qed. Theorem ZofB_range_neg: forall f n, ZofB f = Some n -> n < 0 -> (Z2R (n - 1)%Z < B2R _ _ f <= Z2R n)%R. Proof. intros. rewrite ZofB_correct in H. destruct (is_finite prec emax f) eqn:FIN; inversion H. set (x := B2R prec emax f) in *. set (y := (-x)%R). assert (A: (Z2R (Ztrunc y) <= y < Z2R (Ztrunc y + 1)%Z)%R). { apply Ztrunc_range_pos. unfold y. rewrite Ztrunc_opp. omega. } destruct A as [B C]. unfold y in B, C. rewrite Ztrunc_opp in B, C. replace (- Ztrunc x + 1) with (- (Ztrunc x - 1)) in C by omega. rewrite Z2R_opp in B, C. lra. Qed. Theorem ZofB_range_zero: forall f, ZofB f = Some 0 -> (-1 < B2R _ _ f < 1)%R. Proof. intros. rewrite ZofB_correct in H. destruct (is_finite prec emax f) eqn:FIN; inversion H. apply Ztrunc_range_zero. auto. Qed. Theorem ZofB_range_nonneg: forall f n, ZofB f = Some n -> 0 <= n -> (-1 < B2R _ _ f < Z2R (n + 1)%Z)%R. Proof. intros. destruct (Z.eq_dec n 0). - subst n. apply ZofB_range_zero. auto. - destruct (ZofB_range_pos f n) as (A & B). auto. omega. split; auto. apply Rlt_le_trans with (Z2R 0). simpl; lra. apply Rle_trans with (Z2R n); auto. apply Z2R_le; auto. Qed. (** For representable integers, [ZofB] is left inverse of [BofZ]. *) Theorem ZofBofZ_exact: forall n, integer_representable n -> ZofB (BofZ n) = Some n. Proof. intros. destruct (BofZ_representable n H) as (A & B & C). rewrite ZofB_correct. rewrite A, B. f_equal. apply Ztrunc_Z2R. Qed. (** Compatibility with subtraction *) Remark Zfloor_minus: forall x n, Zfloor (x - Z2R n) = Zfloor x - n. Proof. intros. apply Zfloor_imp. replace (Zfloor x - n + 1) with ((Zfloor x + 1) - n) by omega. rewrite ! Z2R_minus. unfold Rminus. split. apply Rplus_le_compat_r. apply Zfloor_lb. apply Rplus_lt_compat_r. rewrite Z2R_plus. apply Zfloor_ub. Qed. Theorem ZofB_minus: forall minus_nan m f p q, ZofB f = Some p -> 0 <= p < 2*q -> q <= 2^prec -> (Z2R q <= B2R _ _ f)%R -> ZofB (Bminus _ _ _ Hmax minus_nan m f (BofZ q)) = Some (p - q). Proof. intros. assert (Q: -2^prec <= q <= 2^prec). { split; auto. generalize (Zpower_ge_0 radix2 prec); simpl; omega. } assert (RANGE: (-1 < B2R _ _ f < Z2R (p + 1)%Z)%R) by (apply ZofB_range_nonneg; auto; omega). rewrite ZofB_correct in H. destruct (is_finite prec emax f) eqn:FIN; try discriminate. assert (PQ2: (Z2R (p + 1) <= Z2R q * 2)%R). { change 2%R with (Z2R 2). rewrite <- Z2R_mult. apply Z2R_le. omega. } assert (EXACT: round radix2 fexp (round_mode m) (B2R _ _ f - Z2R q)%R = (B2R _ _ f - Z2R q)%R). { apply round_generic. apply valid_rnd_round_mode. apply sterbenz_aux. apply FLT_exp_monotone. apply generic_format_B2R. apply integer_representable_n. auto. lra. } destruct (BofZ_exact q Q) as (A & B & C). generalize (Bminus_correct _ _ _ Hmax minus_nan m f (BofZ q) FIN B). rewrite Rlt_bool_true. - fold emin; fold fexp. intros (D & E & F). rewrite ZofB_correct. rewrite E. rewrite D. rewrite A. rewrite EXACT. inversion H. f_equal. rewrite ! Ztrunc_floor. apply Zfloor_minus. lra. lra. - rewrite A. fold emin; fold fexp. rewrite EXACT. apply Rle_lt_trans with (bpow radix2 prec). apply Rle_trans with (Z2R q). apply Rabs_le. lra. rewrite <- Z2R_Zpower. apply Z2R_le; auto. red in prec_gt_0_; omega. apply bpow_lt. auto. Qed. (** A variant of [ZofB] that bounds the range of representable integers. *) Definition ZofB_range (f: binary_float) (zmin zmax: Z): option Z := match ZofB f with | None => None | Some z => if Zle_bool zmin z && Zle_bool z zmax then Some z else None end. Theorem ZofB_range_correct: forall f min max, let n := Ztrunc (B2R _ _ f) in ZofB_range f min max = if is_finite _ _ f && Zle_bool min n && Zle_bool n max then Some n else None. Proof. intros. unfold ZofB_range. rewrite ZofB_correct. fold n. destruct (is_finite prec emax f); auto. Qed. Lemma ZofB_range_inversion: forall f min max n, ZofB_range f min max = Some n -> min <= n /\ n <= max /\ ZofB f = Some n. Proof. intros. rewrite ZofB_range_correct in H. rewrite ZofB_correct. destruct (is_finite prec emax f); try discriminate. set (n1 := Ztrunc (B2R _ _ f)) in *. destruct (min <=? n1) eqn:MIN; try discriminate. destruct (n1 <=? max) eqn:MAX; try discriminate. simpl in H. inversion H. subst n. split. apply Zle_bool_imp_le; auto. split. apply Zle_bool_imp_le; auto. auto. Qed. Theorem ZofB_range_minus: forall minus_nan m f p q, ZofB_range f 0 (2 * q - 1) = Some p -> q <= 2^prec -> (Z2R q <= B2R _ _ f)%R -> ZofB_range (Bminus _ _ _ Hmax minus_nan m f (BofZ q)) (-q) (q - 1) = Some (p - q). Proof. intros. destruct (ZofB_range_inversion _ _ _ _ H) as (A & B & C). set (f' := Bminus prec emax prec_gt_0_ Hmax minus_nan m f (BofZ q)). assert (D: ZofB f' = Some (p - q)). { apply ZofB_minus. auto. omega. auto. auto. } unfold ZofB_range. rewrite D. rewrite Zle_bool_true by omega. rewrite Zle_bool_true by omega. auto. Qed. (** ** Algebraic identities *) (** Commutativity of addition and multiplication *) Theorem Bplus_commut: forall plus_nan mode (x y: binary_float), plus_nan x y = plus_nan y x -> Bplus _ _ _ Hmax plus_nan mode x y = Bplus _ _ _ Hmax plus_nan mode y x. Proof. intros until y; intros NAN. pose proof (Bplus_correct _ _ _ Hmax plus_nan mode x y). pose proof (Bplus_correct _ _ _ Hmax plus_nan mode y x). unfold Bplus in *; destruct x; destruct y; auto. - rewrite (eqb_sym b0 b). destruct (eqb b b0) eqn:EQB; auto. f_equal; apply eqb_prop; auto. - rewrite NAN; auto. - rewrite (eqb_sym b0 b). destruct (eqb b b0) eqn:EQB. f_equal; apply eqb_prop; auto. rewrite NAN; auto. - rewrite NAN; auto. - rewrite NAN; auto. - rewrite NAN; auto. - rewrite NAN; auto. - rewrite NAN; auto. - rewrite NAN; auto. - generalize (H (refl_equal _) (refl_equal _)); clear H. generalize (H0 (refl_equal _) (refl_equal _)); clear H0. fold emin. fold fexp. set (x := B754_finite prec emax b0 m0 e1 e2). set (rx := B2R _ _ x). set (y := B754_finite prec emax b m e e0). set (ry := B2R _ _ y). rewrite (Rplus_comm ry rx). destruct Rlt_bool. + intros (A1 & A2 & A3) (B1 & B2 & B3). apply B2R_Bsign_inj; auto. rewrite <- B1 in A1. auto. rewrite Z.add_comm. rewrite Z.min_comm. auto. + intros (A1 & A2) (B1 & B2). apply B2FF_inj. rewrite B2 in B1. rewrite <- B1 in A1. auto. Qed. Theorem Bmult_commut: forall mult_nan mode (x y: binary_float), mult_nan x y = mult_nan y x -> Bmult _ _ _ Hmax mult_nan mode x y = Bmult _ _ _ Hmax mult_nan mode y x. Proof. intros until y; intros NAN. pose proof (Bmult_correct _ _ _ Hmax mult_nan mode x y). pose proof (Bmult_correct _ _ _ Hmax mult_nan mode y x). unfold Bmult in *; destruct x; destruct y; auto. - rewrite (xorb_comm b0 b); auto. - rewrite NAN; auto. - rewrite NAN; auto. - rewrite (xorb_comm b0 b); auto. - rewrite NAN; auto. - rewrite (xorb_comm b0 b); auto. - rewrite NAN; auto. - rewrite (xorb_comm b0 b); auto. - rewrite NAN; auto. - rewrite NAN; auto. - rewrite NAN; auto. - rewrite NAN; auto. - rewrite (xorb_comm b0 b); auto. - rewrite (xorb_comm b0 b); auto. - rewrite NAN; auto. - revert H H0. fold emin. fold fexp. set (x := B754_finite prec emax b0 m0 e1 e2). set (rx := B2R _ _ x). set (y := B754_finite prec emax b m e e0). set (ry := B2R _ _ y). rewrite (Rmult_comm ry rx). destruct Rlt_bool. + intros (A1 & A2 & A3) (B1 & B2 & B3). apply B2R_Bsign_inj; auto. rewrite <- B1 in A1. auto. rewrite ! Bsign_FF2B. f_equal. f_equal. apply xorb_comm. apply Pos.mul_comm. apply Z.add_comm. + intros A B. apply B2FF_inj. etransitivity. eapply A. rewrite xorb_comm. auto. Qed. (** Multiplication by 2 is diagonal addition. *) Theorem Bmult2_Bplus: forall plus_nan mult_nan mode (f: binary_float), (forall (x y: binary_float), is_nan _ _ x = true -> is_finite _ _ y = true -> plus_nan x x = mult_nan x y) -> Bplus _ _ _ Hmax plus_nan mode f f = Bmult _ _ _ Hmax mult_nan mode f (BofZ 2%Z). Proof. intros until f; intros NAN. destruct (BofZ_representable 2) as (A & B & C). apply (integer_representable_2p 1). red in prec_gt_0_; omega. pose proof (Bmult_correct _ _ _ Hmax mult_nan mode f (BofZ 2%Z)). fold emin in H. rewrite A, B, C in H. rewrite xorb_false_r in H. destruct (is_finite _ _ f) eqn:FIN. - pose proof (Bplus_correct _ _ _ Hmax plus_nan mode f f FIN FIN). fold emin in H0. assert (EQ: (B2R prec emax f * Z2R 2%Z = B2R prec emax f + B2R prec emax f)%R). { change (Z2R 2%Z) with 2%R. ring. } rewrite <- EQ in H0. destruct Rlt_bool. + destruct H0 as (P & Q & R). destruct H as (S & T & U). apply B2R_Bsign_inj; auto. rewrite P, S. auto. rewrite R, U. replace 0%R with (0 * Z2R 2%Z)%R by ring. rewrite Rcompare_mult_r. rewrite andb_diag, orb_diag. destruct f; try discriminate; simpl. rewrite Rcompare_Eq by auto. destruct mode; auto. replace 0%R with (@F2R radix2 {| Fnum := 0%Z; Fexp := e |}). rewrite Rcompare_F2R. destruct b; auto. unfold F2R. simpl. ring. change 0%R with (Z2R 0%Z). apply Z2R_lt. omega. destruct (Bmult prec emax prec_gt_0_ Hmax mult_nan mode f (BofZ 2)); reflexivity || discriminate. + destruct H0 as (P & Q). apply B2FF_inj. rewrite P, H. auto. - destruct f; try discriminate. + simpl Bplus. rewrite eqb_true. destruct (BofZ 2) eqn:B2; try discriminate; simpl in *. assert ((0 = 2)%Z) by (apply eq_Z2R; auto). discriminate. subst b0. rewrite xorb_false_r. auto. auto. + unfold Bplus, Bmult. rewrite <- NAN by auto. auto. Qed. (** Divisions that can be turned into multiplications by an inverse *) Definition Bexact_inverse_mantissa := Z.iter (prec - 1) xO xH. Remark Bexact_inverse_mantissa_value: Zpos Bexact_inverse_mantissa = 2 ^ (prec - 1). Proof. assert (REC: forall n, Z.pos (nat_iter n xO xH) = 2 ^ (Z.of_nat n)). { induction n. reflexivity. simpl nat_iter. transitivity (2 * Z.pos (nat_iter n xO xH)). reflexivity. rewrite inj_S. rewrite IHn. unfold Z.succ. rewrite Zpower_plus by omega. change (2 ^ 1) with 2. ring. } red in prec_gt_0_. unfold Bexact_inverse_mantissa. rewrite iter_nat_of_Z by omega. rewrite REC. rewrite Zabs2Nat.id_abs. rewrite Z.abs_eq by omega. auto. Qed. Remark Bexact_inverse_mantissa_digits2_Pnat: digits2_Pnat Bexact_inverse_mantissa = Z.to_nat (prec - 1). Proof. assert (DIGITS: forall n, digits2_Pnat (nat_iter n xO xH) = n). { induction n; simpl. auto. congruence. } red in prec_gt_0_. unfold Bexact_inverse_mantissa. rewrite iter_nat_of_Z by omega. rewrite DIGITS. apply Zabs2Nat.abs_nat_nonneg. omega. Qed. Remark bounded_Bexact_inverse: forall e, emin <= e <= emax - prec <-> bounded prec emax Bexact_inverse_mantissa e = true. Proof. intros. unfold bounded, canonic_mantissa. rewrite andb_true_iff. rewrite <- Zeq_is_eq_bool. rewrite <- Zle_is_le_bool. rewrite Bexact_inverse_mantissa_digits2_Pnat. rewrite inj_S. red in prec_gt_0_. rewrite Z2Nat.id by omega. split. - intros; split. unfold FLT_exp. unfold emin in H. zify; omega. omega. - intros [A B]. unfold FLT_exp in A. unfold emin. zify; omega. Qed. Program Definition Bexact_inverse (f: binary_float) : option binary_float := match f with | B754_finite s m e B => if positive_eq_dec m Bexact_inverse_mantissa then let e' := -e - (prec - 1) * 2 in if Z_le_dec emin e' then if Z_le_dec e' emax then Some(B754_finite _ _ s m e' _) else None else None else None | _ => None end. Next Obligation. rewrite <- bounded_Bexact_inverse in B. rewrite <- bounded_Bexact_inverse. unfold emin in *. omega. Qed. Lemma Bexact_inverse_correct: forall f f', Bexact_inverse f = Some f' -> is_finite_strict _ _ f = true /\ is_finite_strict _ _ f' = true /\ B2R _ _ f' = (/ B2R _ _ f)%R /\ B2R _ _ f <> 0%R /\ Bsign _ _ f' = Bsign _ _ f. Proof with (try discriminate). intros f f' EI. unfold Bexact_inverse in EI. destruct f... destruct (Pos.eq_dec m Bexact_inverse_mantissa)... set (e' := -e - (prec - 1) * 2) in *. destruct (Z_le_dec emin e')... destruct (Z_le_dec e' emax)... inversion EI; clear EI; subst f' m. split. auto. split. auto. split. unfold B2R. rewrite Bexact_inverse_mantissa_value. unfold F2R; simpl. rewrite Z2R_cond_Zopp. rewrite <- ! cond_Ropp_mult_l. red in prec_gt_0_. replace (Z2R (2 ^ (prec - 1))) with (bpow radix2 (prec - 1)) by (symmetry; apply (Z2R_Zpower radix2); omega). rewrite <- ! bpow_plus. replace (prec - 1 + e') with (- (prec - 1 + e)) by (unfold e'; omega). rewrite bpow_opp. unfold cond_Ropp; destruct b; auto. rewrite Ropp_inv_permute. auto. apply Rgt_not_eq. apply bpow_gt_0. split. simpl. red; intros. apply F2R_eq_0_reg in H. destruct b; simpl in H; discriminate. auto. Qed. Theorem Bdiv_mult_inverse: forall div_nan mult_nan mode x y z, (forall (x y z: binary_float), is_nan _ _ x = true -> is_finite _ _ y = true -> is_finite _ _ z = true -> div_nan x y = mult_nan x z) -> Bexact_inverse y = Some z -> Bdiv _ _ _ Hmax div_nan mode x y = Bmult _ _ _ Hmax mult_nan mode x z. Proof. intros until z; intros NAN; intros. destruct (Bexact_inverse_correct _ _ H) as (A & B & C & D & E). pose proof (Bmult_correct _ _ _ Hmax mult_nan mode x z). fold emin in H0. fold fexp in H0. pose proof (Bdiv_correct _ _ _ Hmax div_nan mode x y D). fold emin in H1. fold fexp in H1. unfold Rdiv in H1. rewrite <- C in H1. destruct (is_finite _ _ x) eqn:FINX. - destruct Rlt_bool. + destruct H0 as (P & Q & R). destruct H1 as (S & T & U). apply B2R_Bsign_inj; auto. rewrite Q. simpl. apply is_finite_strict_finite; auto. rewrite P, S. auto. rewrite R, U, E. auto. apply is_finite_not_is_nan; auto. apply is_finite_not_is_nan. rewrite Q. simpl. apply is_finite_strict_finite; auto. + apply B2FF_inj. rewrite H0, H1. rewrite E. auto. - destruct y; try discriminate. destruct z; try discriminate. destruct x; try discriminate; simpl. + simpl in E; congruence. + erewrite NAN; eauto. Qed. End Extra_ops. (** ** Conversions between two FP formats *) Section Conversions. Variable prec1 emax1 prec2 emax2 : Z. Context (prec1_gt_0_ : Prec_gt_0 prec1) (prec2_gt_0_ : Prec_gt_0 prec2). Let emin1 := (3 - emax1 - prec1)%Z. Let fexp1 := FLT_exp emin1 prec1. Let emin2 := (3 - emax2 - prec2)%Z. Let fexp2 := FLT_exp emin2 prec2. Hypothesis Hmax1 : (prec1 < emax1)%Z. Hypothesis Hmax2 : (prec2 < emax2)%Z. Let binary_float1 := binary_float prec1 emax1. Let binary_float2 := binary_float prec2 emax2. Definition Bconv (conv_nan: bool -> nan_pl prec1 -> bool * nan_pl prec2) (md: mode) (f: binary_float1) : binary_float2 := match f with | B754_nan s pl => let '(s, pl) := conv_nan s pl in B754_nan _ _ s pl | B754_infinity s => B754_infinity _ _ s | B754_zero s => B754_zero _ _ s | B754_finite s m e _ => binary_normalize _ _ _ Hmax2 md (cond_Zopp s (Zpos m)) e s end. Theorem Bconv_correct: forall conv_nan m f, is_finite _ _ f = true -> if Rlt_bool (Rabs (round radix2 fexp2 (round_mode m) (B2R _ _ f))) (bpow radix2 emax2) then B2R _ _ (Bconv conv_nan m f) = round radix2 fexp2 (round_mode m) (B2R _ _ f) /\ is_finite _ _ (Bconv conv_nan m f) = true /\ Bsign _ _ (Bconv conv_nan m f) = Bsign _ _ f else B2FF _ _ (Bconv conv_nan m f) = binary_overflow prec2 emax2 m (Bsign _ _ f). Proof. intros. destruct f; try discriminate. - simpl. rewrite round_0. rewrite Rabs_R0. rewrite Rlt_bool_true. auto. apply bpow_gt_0. apply valid_rnd_round_mode. - generalize (binary_normalize_correct _ _ _ Hmax2 m (cond_Zopp b (Zpos m0)) e b). fold emin2; fold fexp2. simpl. destruct Rlt_bool. + intros (A & B & C). split. auto. split. auto. rewrite C. destruct b; simpl. rewrite Rcompare_Lt. auto. apply F2R_lt_0_compat. simpl. compute; auto. rewrite Rcompare_Gt. auto. apply F2R_gt_0_compat. simpl. compute; auto. + intros A. rewrite A. f_equal. destruct b. apply Rlt_bool_true. apply F2R_lt_0_compat. simpl. compute; auto. apply Rlt_bool_false. apply Rlt_le. apply Rgt_lt. apply F2R_gt_0_compat. simpl. compute; auto. Qed. (** Converting a finite FP number to higher or equal precision preserves its value. *) Theorem Bconv_widen_exact: (prec2 >= prec1)%Z -> (emax2 >= emax1)%Z -> forall conv_nan m f, is_finite _ _ f = true -> B2R _ _ (Bconv conv_nan m f) = B2R _ _ f /\ is_finite _ _ (Bconv conv_nan m f) = true /\ Bsign _ _ (Bconv conv_nan m f) = Bsign _ _ f. Proof. intros PREC EMAX; intros. generalize (Bconv_correct conv_nan m f H). assert (LT: (Rabs (B2R _ _ f) < bpow radix2 emax2)%R). { destruct f; try discriminate; simpl. rewrite Rabs_R0. apply bpow_gt_0. apply Rlt_le_trans with (bpow radix2 emax1). rewrite F2R_cond_Zopp. rewrite abs_cond_Ropp. rewrite <- F2R_Zabs. simpl Z.abs. eapply bounded_lt_emax; eauto. apply bpow_le. omega. } assert (EQ: round radix2 fexp2 (round_mode m) (B2R prec1 emax1 f) = B2R prec1 emax1 f). { apply round_generic. apply valid_rnd_round_mode. eapply generic_inclusion_le. 5: apply generic_format_B2R. apply fexp_correct; auto. apply fexp_correct; auto. instantiate (1 := emax2). intros. unfold fexp2, FLT_exp. unfold emin2. zify; omega. apply Rlt_le; auto. } rewrite EQ. rewrite Rlt_bool_true by auto. auto. Qed. (** Conversion from integers and change of format *) Theorem Bconv_BofZ: forall conv_nan n, integer_representable prec1 emax1 n -> Bconv conv_nan mode_NE (BofZ prec1 emax1 _ Hmax1 n) = BofZ prec2 emax2 _ Hmax2 n. Proof. intros. destruct (BofZ_representable _ _ _ Hmax1 n H) as (A & B & C). set (f := BofZ prec1 emax1 prec1_gt_0_ Hmax1 n) in *. generalize (Bconv_correct conv_nan mode_NE f B). unfold BofZ. generalize (binary_normalize_correct _ _ _ Hmax2 mode_NE n 0 false). fold emin2; fold fexp2. rewrite A. replace (F2R {| Fnum := n; Fexp := 0 |}) with (Z2R n). destruct Rlt_bool. - intros (P & Q & R) (D & E & F). apply B2R_Bsign_inj; auto. congruence. rewrite F, C, R. change 0%R with (Z2R 0). rewrite Rcompare_Z2R. unfold Zlt_bool. auto. - intros P Q. apply B2FF_inj. rewrite P, Q. rewrite C. f_equal. change 0%R with (Z2R 0). generalize (Zlt_bool_spec n 0); intros LT; inversion LT. rewrite Rlt_bool_true; auto. apply Z2R_lt; auto. rewrite Rlt_bool_false; auto. apply Z2R_le; auto. - unfold F2R; simpl. rewrite Rmult_1_r. auto. Qed. (** Change of format (to higher precision) and conversion to integer. *) Theorem ZofB_Bconv: prec2 >= prec1 -> emax2 >= emax1 -> forall conv_nan m f n, ZofB _ _ f = Some n -> ZofB _ _ (Bconv conv_nan m f) = Some n. Proof. intros. rewrite ZofB_correct in H1. destruct (is_finite _ _ f) eqn:FIN; inversion H1. destruct (Bconv_widen_exact H H0 conv_nan m f) as (A & B & C). auto. rewrite ZofB_correct. rewrite B. rewrite A. auto. Qed. Theorem ZofB_range_Bconv: forall min1 max1 min2 max2, prec2 >= prec1 -> emax2 >= emax1 -> min2 <= min1 -> max1 <= max2 -> forall conv_nan m f n, ZofB_range _ _ f min1 max1 = Some n -> ZofB_range _ _ (Bconv conv_nan m f) min2 max2 = Some n. Proof. intros. destruct (ZofB_range_inversion _ _ _ _ _ _ H3) as (A & B & C). unfold ZofB_range. erewrite ZofB_Bconv by eauto. rewrite ! Zle_bool_true by omega. auto. Qed. (** Change of format (to higher precision) and comparison. *) Theorem Bcompare_Bconv_widen: prec2 >= prec1 -> emax2 >= emax1 -> forall conv_nan m x y, Bcompare _ _ (Bconv conv_nan m x) (Bconv conv_nan m y) = Bcompare _ _ x y. Proof. intros. destruct (is_finite _ _ x && is_finite _ _ y) eqn:FIN. - apply andb_true_iff in FIN. destruct FIN. destruct (Bconv_widen_exact H H0 conv_nan m x H1) as (A & B & C). destruct (Bconv_widen_exact H H0 conv_nan m y H2) as (D & E & F). rewrite ! Bcompare_finite_correct by auto. rewrite A, D. auto. - generalize (Bconv_widen_exact H H0 conv_nan m x) (Bconv_widen_exact H H0 conv_nan m y); intros P Q. destruct x, y; try discriminate; simpl in P, Q; simpl; repeat (match goal with |- context [conv_nan ?b ?pl] => destruct (conv_nan b pl) end); auto. destruct Q as (D & E & F); auto. destruct (binary_normalize prec2 emax2 prec2_gt_0_ Hmax2 m (cond_Zopp b0 (Z.pos m0)) e b0); discriminate || reflexivity. destruct P as (A & B & C); auto. destruct (binary_normalize prec2 emax2 prec2_gt_0_ Hmax2 m (cond_Zopp b (Z.pos m0)) e b); try discriminate; simpl. destruct b; auto. destruct b, b1; auto. destruct P as (A & B & C); auto. destruct (binary_normalize prec2 emax2 prec2_gt_0_ Hmax2 m (cond_Zopp b (Z.pos m0)) e b); try discriminate; simpl. destruct b; auto. destruct b, b2; auto. Qed. End Conversions. Section Compose_Conversions. Variable prec1 emax1 prec2 emax2 : Z. Context (prec1_gt_0_ : Prec_gt_0 prec1) (prec2_gt_0_ : Prec_gt_0 prec2). Let emin1 := (3 - emax1 - prec1)%Z. Let fexp1 := FLT_exp emin1 prec1. Let emin2 := (3 - emax2 - prec2)%Z. Let fexp2 := FLT_exp emin2 prec2. Hypothesis Hmax1 : (prec1 < emax1)%Z. Hypothesis Hmax2 : (prec2 < emax2)%Z. Let binary_float1 := binary_float prec1 emax1. Let binary_float2 := binary_float prec2 emax2. (** Converting to a higher precision then down to the original format is the identity. *) Theorem Bconv_narrow_widen: prec2 >= prec1 -> emax2 >= emax1 -> forall narrow_nan widen_nan m f, is_nan _ _ f = false -> Bconv prec2 emax2 prec1 emax1 _ Hmax1 narrow_nan m (Bconv prec1 emax1 prec2 emax2 _ Hmax2 widen_nan m f) = f. Proof. intros. destruct (is_finite _ _ f) eqn:FIN. - assert (EQ: round radix2 fexp1 (round_mode m) (B2R prec1 emax1 f) = B2R prec1 emax1 f). { apply round_generic. apply valid_rnd_round_mode. apply generic_format_B2R. } generalize (Bconv_widen_exact _ _ _ _ _ _ Hmax2 H H0 widen_nan m f FIN). set (f' := Bconv prec1 emax1 prec2 emax2 _ Hmax2 widen_nan m f). intros (A & B & C). generalize (Bconv_correct _ _ _ _ _ Hmax1 narrow_nan m f' B). fold emin1. fold fexp1. rewrite A, C, EQ. rewrite Rlt_bool_true. intros (D & E & F). apply B2R_Bsign_inj; auto. destruct f; try discriminate; simpl. rewrite Rabs_R0. apply bpow_gt_0. rewrite F2R_cond_Zopp. rewrite abs_cond_Ropp. rewrite <- F2R_Zabs. simpl Z.abs. eapply bounded_lt_emax; eauto. - destruct f; try discriminate. simpl. auto. Qed. End Compose_Conversions. (** Specialization to binary32 and binary64 formats. *) Require Import Fappli_IEEE_bits. Section B3264. Let prec32 : (0 < 24)%Z. apply refl_equal. Qed. Let emax32 : (24 < 128)%Z. apply refl_equal. Qed. Let prec64 : (0 < 53)%Z. apply refl_equal. Qed. Let emax64 : (53 < 1024)%Z. apply refl_equal. Qed. Definition b32_abs : (bool -> nan_pl 24 -> bool * nan_pl 24) -> binary32 -> binary32 := Babs 24 128. Definition b32_eq_dec : forall (f1 f2: binary32), {f1=f2} + {f1<>f2} := Beq_dec 24 128. Definition b32_compare : binary32 -> binary32 -> option comparison := Bcompare 24 128. Definition b32_of_Z : Z -> binary32 := BofZ 24 128 prec32 emax32. Definition b32_to_Z : binary32 -> option Z := ZofB 24 128. Definition b32_to_Z_range : binary32 -> Z -> Z -> option Z := ZofB_range 24 128. Definition b32_exact_inverse : binary32 -> option binary32 := Bexact_inverse 24 128 prec32. Definition b64_abs : (bool -> nan_pl 53 -> bool * nan_pl 53) -> binary64 -> binary64 := Babs 53 1024. Definition b64_eq_dec : forall (f1 f2: binary64), {f1=f2} + {f1<>f2} := Beq_dec 53 1024. Definition b64_compare : binary64 -> binary64 -> option comparison := Bcompare 53 1024. Definition b64_of_Z : Z -> binary64 := BofZ 53 1024 prec64 emax64. Definition b64_to_Z : binary64 -> option Z := ZofB 53 1024. Definition b64_to_Z_range : binary64 -> Z -> Z -> option Z := ZofB_range 53 1024. Definition b64_exact_inverse : binary64 -> option binary64 := Bexact_inverse 53 1024 prec64. Definition b64_of_b32 : (bool -> nan_pl 24 -> bool * nan_pl 53) -> mode -> binary32 -> binary64 := Bconv 24 128 53 1024 prec32 prec64. Definition b32_of_b64 : (bool -> nan_pl 53 -> bool * nan_pl 24) -> mode -> binary64 -> binary32 := Bconv 53 1024 24 128 prec64 prec32. End B3264.