diff options
Diffstat (limited to 'flocq/Appli/Fappli_IEEE_bits.v')
-rw-r--r-- | flocq/Appli/Fappli_IEEE_bits.v | 546 |
1 files changed, 546 insertions, 0 deletions
diff --git a/flocq/Appli/Fappli_IEEE_bits.v b/flocq/Appli/Fappli_IEEE_bits.v new file mode 100644 index 00000000..06ed21e0 --- /dev/null +++ b/flocq/Appli/Fappli_IEEE_bits.v @@ -0,0 +1,546 @@ +(** +This file is part of the Flocq formalization of floating-point +arithmetic in Coq: http://flocq.gforge.inria.fr/ + +Copyright (C) 2011 Sylvie Boldo +#<br /># +Copyright (C) 2011 Guillaume Melquiond + +This library is free software; you can redistribute it and/or +modify it under the terms of the GNU Lesser General Public +License as published by the Free Software Foundation; either +version 3 of the License, or (at your option) any later version. + +This library is distributed in the hope that it will be useful, +but WITHOUT ANY WARRANTY; without even the implied warranty of +MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +COPYING file for more details. +*) + +(** * IEEE-754 encoding of binary floating-point data *) +Require Import Fcore. +Require Import Fcore_digits. +Require Import Fcalc_digits. +Require Import Fappli_IEEE. + +Section Binary_Bits. + +(** Number of bits for the fraction and exponent *) +Variable mw ew : Z. +Hypothesis Hmw : (0 < mw)%Z. +Hypothesis Hew : (0 < ew)%Z. + +Let emax := Zpower 2 (ew - 1). +Let prec := (mw + 1)%Z. +Let emin := (3 - emax - prec)%Z. +Let binary_float := binary_float prec emax. + +Let Hprec : (0 < prec)%Z. +unfold prec. +apply Zle_lt_succ. +now apply Zlt_le_weak. +Qed. + +Let Hm_gt_0 : (0 < 2^mw)%Z. +apply (Zpower_gt_0 radix2). +now apply Zlt_le_weak. +Qed. + +Let He_gt_0 : (0 < 2^ew)%Z. +apply (Zpower_gt_0 radix2). +now apply Zlt_le_weak. +Qed. + +Hypothesis Hmax : (prec < emax)%Z. + +Definition join_bits (s : bool) m e := + (((if s then Zpower 2 ew else 0) + e) * Zpower 2 mw + m)%Z. + +Definition split_bits x := + let mm := Zpower 2 mw in + let em := Zpower 2 ew in + (Zle_bool (mm * em) x, Zmod x mm, Zmod (Zdiv x mm) em)%Z. + +Theorem split_join_bits : + forall s m e, + (0 <= m < Zpower 2 mw)%Z -> + (0 <= e < Zpower 2 ew)%Z -> + split_bits (join_bits s m e) = (s, m, e). +Proof. +intros s m e Hm He. +unfold split_bits, join_bits. +apply f_equal2. +apply f_equal2. +(* *) +case s. +apply Zle_bool_true. +apply Zle_0_minus_le. +ring_simplify. +apply Zplus_le_0_compat. +apply Zmult_le_0_compat. +apply He. +now apply Zlt_le_weak. +apply Hm. +apply Zle_bool_false. +apply Zplus_lt_reg_l with (2^mw * (-e))%Z. +replace (2 ^ mw * - e + ((0 + e) * 2 ^ mw + m))%Z with (m * 1)%Z by ring. +rewrite <- Zmult_plus_distr_r. +apply Zlt_le_trans with (2^mw * 1)%Z. +now apply Zmult_lt_compat_r. +apply Zmult_le_compat_l. +clear -He. omega. +now apply Zlt_le_weak. +(* *) +rewrite Zplus_comm. +rewrite Z_mod_plus_full. +now apply Zmod_small. +(* *) +rewrite Z_div_plus_full_l. +rewrite Zdiv_small with (1 := Hm). +rewrite Zplus_0_r. +case s. +replace (2^ew + e)%Z with (e + 1 * 2^ew)%Z by ring. +rewrite Z_mod_plus_full. +now apply Zmod_small. +now apply Zmod_small. +now apply Zgt_not_eq. +Qed. + +Theorem join_split_bits : + forall x, + (0 <= x < Zpower 2 (mw + ew + 1))%Z -> + let '(s, m, e) := split_bits x in + join_bits s m e = x. +Proof. +intros x Hx. +unfold split_bits, join_bits. +pattern x at 4 ; rewrite Z_div_mod_eq_full with x (2^mw)%Z. +apply (f_equal (fun v => (v + _)%Z)). +rewrite Zmult_comm. +apply f_equal. +pattern (x / (2^mw))%Z at 2 ; rewrite Z_div_mod_eq_full with (x / (2^mw))%Z (2^ew)%Z. +apply (f_equal (fun v => (v + _)%Z)). +replace (x / 2 ^ mw / 2 ^ ew)%Z with (if Zle_bool (2 ^ mw * 2 ^ ew) x then 1 else 0)%Z. +case Zle_bool. +now rewrite Zmult_1_r. +now rewrite Zmult_0_r. +rewrite Zdiv_Zdiv. +apply sym_eq. +case Zle_bool_spec ; intros Hs. +apply Zle_antisym. +cut (x / (2^mw * 2^ew) < 2)%Z. clear ; omega. +apply Zdiv_lt_upper_bound. +try apply Hx. (* 8.2/8.3 compatibility *) +now apply Zmult_lt_0_compat. +rewrite <- Zpower_exp ; try ( apply Zle_ge ; apply Zlt_le_weak ; assumption ). +change 2%Z at 1 with (Zpower 2 1). +rewrite <- Zpower_exp. +now rewrite Zplus_comm. +discriminate. +apply Zle_ge. +now apply Zplus_le_0_compat ; apply Zlt_le_weak. +apply Zdiv_le_lower_bound. +try apply Hx. (* 8.2/8.3 compatibility *) +now apply Zmult_lt_0_compat. +now rewrite Zmult_1_l. +apply Zdiv_small. +now split. +now apply Zlt_le_weak. +now apply Zlt_le_weak. +now apply Zgt_not_eq. +now apply Zgt_not_eq. +Qed. + +Theorem split_bits_inj : + forall x y, + (0 <= x < Zpower 2 (mw + ew + 1))%Z -> + (0 <= y < Zpower 2 (mw + ew + 1))%Z -> + split_bits x = split_bits y -> + x = y. +Proof. +intros x y Hx Hy. +generalize (join_split_bits x Hx) (join_split_bits y Hy). +destruct (split_bits x) as ((sx, mx), ex). +destruct (split_bits y) as ((sy, my), ey). +intros Jx Jy H. revert Jx Jy. +inversion_clear H. +intros Jx Jy. +now rewrite <- Jx. +Qed. + +Definition bits_of_binary_float (x : binary_float) := + match x with + | B754_zero sx => join_bits sx 0 0 + | B754_infinity sx => join_bits sx 0 (Zpower 2 ew - 1) + | B754_nan => join_bits false (Zpower 2 mw - 1) (Zpower 2 ew - 1) + | B754_finite sx mx ex _ => + if Zle_bool (Zpower 2 mw) (Zpos mx) then + join_bits sx (Zpos mx - Zpower 2 mw) (ex - emin + 1) + else + join_bits sx (Zpos mx) 0 + end. + +Definition split_bits_of_binary_float (x : binary_float) := + match x with + | B754_zero sx => (sx, 0, 0)%Z + | B754_infinity sx => (sx, 0, Zpower 2 ew - 1)%Z + | B754_nan => (false, Zpower 2 mw - 1, Zpower 2 ew - 1)%Z + | B754_finite sx mx ex _ => + if Zle_bool (Zpower 2 mw) (Zpos mx) then + (sx, Zpos mx - Zpower 2 mw, ex - emin + 1)%Z + else + (sx, Zpos mx, 0)%Z + end. + +Theorem split_bits_of_binary_float_correct : + forall x, + split_bits (bits_of_binary_float x) = split_bits_of_binary_float x. +Proof. +intros [sx|sx| |sx mx ex Hx] ; + try ( simpl ; apply split_join_bits ; split ; try apply Zle_refl ; try apply Zlt_pred ; trivial ; omega ). +unfold bits_of_binary_float, split_bits_of_binary_float. +assert (Hf: (emin <= ex /\ Zdigits radix2 (Zpos mx) <= prec)%Z). +destruct (andb_prop _ _ Hx) as (Hx', _). +unfold canonic_mantissa in Hx'. +rewrite Z_of_nat_S_digits2_Pnat in Hx'. +generalize (Zeq_bool_eq _ _ Hx'). +unfold FLT_exp. +change (Fcalc_digits.radix2) with radix2. +unfold emin. +clear ; zify ; omega. +destruct (Zle_bool_spec (2^mw) (Zpos mx)) as [H|H] ; + apply split_join_bits ; try now split. +(* *) +split. +clear -He_gt_0 H ; omega. +cut (Zpos mx < 2 * 2^mw)%Z. clear ; omega. +replace (2 * 2^mw)%Z with (2^prec)%Z. +apply (Zpower_gt_Zdigits radix2 _ (Zpos mx)). +apply Hf. +unfold prec. +rewrite Zplus_comm. +apply Zpower_exp ; apply Zle_ge. +discriminate. +now apply Zlt_le_weak. +(* *) +split. +generalize (proj1 Hf). +clear ; omega. +destruct (andb_prop _ _ Hx) as (_, Hx'). +unfold emin. +replace (2^ew)%Z with (2 * emax)%Z. +generalize (Zle_bool_imp_le _ _ Hx'). +clear ; omega. +apply sym_eq. +rewrite (Zsucc_pred ew). +unfold Zsucc. +rewrite Zplus_comm. +apply Zpower_exp ; apply Zle_ge. +discriminate. +now apply Zlt_0_le_0_pred. +Qed. + +Definition binary_float_of_bits_aux x := + let '(sx, mx, ex) := split_bits x in + if Zeq_bool ex 0 then + match mx with + | Z0 => F754_zero sx + | Zpos px => F754_finite sx px emin + | Zneg _ => F754_nan (* dummy *) + end + else if Zeq_bool ex (Zpower 2 ew - 1) then + if Zeq_bool mx 0 then F754_infinity sx else F754_nan + else + match (mx + Zpower 2 mw)%Z with + | Zpos px => F754_finite sx px (ex + emin - 1) + | _ => F754_nan (* dummy *) + end. + +Lemma binary_float_of_bits_aux_correct : + forall x, + valid_binary prec emax (binary_float_of_bits_aux x) = true. +Proof. +intros x. +unfold binary_float_of_bits_aux, split_bits. +case Zeq_bool_spec ; intros He1. +case_eq (x mod 2^mw)%Z ; try easy. +(* subnormal *) +intros px Hm. +assert (Zdigits radix2 (Zpos px) <= mw)%Z. +apply Zdigits_le_Zpower. +simpl. +rewrite <- Hm. +eapply Z_mod_lt. +now apply Zlt_gt. +apply bounded_canonic_lt_emax ; try assumption. +unfold canonic, canonic_exp. +fold emin. +rewrite ln_beta_F2R_Zdigits. 2: discriminate. +unfold Fexp, FLT_exp. +apply sym_eq. +apply Zmax_right. +clear -H Hprec. +unfold prec ; omega. +apply Rnot_le_lt. +intros H0. +refine (_ (ln_beta_le radix2 _ _ _ H0)). +rewrite ln_beta_bpow. +rewrite ln_beta_F2R_Zdigits. 2: discriminate. +unfold emin, prec. +apply Zlt_not_le. +cut (0 < emax)%Z. clear -H Hew ; omega. +apply (Zpower_gt_0 radix2). +clear -Hew ; omega. +apply bpow_gt_0. +case Zeq_bool_spec ; intros He2. +now case Zeq_bool. +case_eq (x mod 2^mw + 2^mw)%Z ; try easy. +(* normal *) +intros px Hm. +assert (prec = Zdigits radix2 (Zpos px)). +(* . *) +rewrite Zdigits_ln_beta. 2: discriminate. +apply sym_eq. +apply ln_beta_unique. +rewrite <- Z2R_abs. +unfold Zabs. +replace (prec - 1)%Z with mw by ( unfold prec ; ring ). +rewrite <- Z2R_Zpower with (1 := Zlt_le_weak _ _ Hmw). +rewrite <- Z2R_Zpower. 2: now apply Zlt_le_weak. +rewrite <- Hm. +split. +apply Z2R_le. +change (radix2^mw)%Z with (0 + 2^mw)%Z. +apply Zplus_le_compat_r. +eapply Z_mod_lt. +now apply Zlt_gt. +apply Z2R_lt. +unfold prec. +rewrite Zpower_exp. 2: now apply Zle_ge ; apply Zlt_le_weak. 2: discriminate. +rewrite <- Zplus_diag_eq_mult_2. +apply Zplus_lt_compat_r. +eapply Z_mod_lt. +now apply Zlt_gt. +(* . *) +apply bounded_canonic_lt_emax ; try assumption. +unfold canonic, canonic_exp. +rewrite ln_beta_F2R_Zdigits. 2: discriminate. +unfold Fexp, FLT_exp. +rewrite <- H. +set (ex := ((x / 2^mw) mod 2^ew)%Z). +replace (prec + (ex + emin - 1) - prec)%Z with (ex + emin - 1)%Z by ring. +apply sym_eq. +apply Zmax_left. +revert He1. +fold ex. +cut (0 <= ex)%Z. +unfold emin. +clear ; intros H1 H2 ; omega. +eapply Z_mod_lt. +apply Zlt_gt. +apply (Zpower_gt_0 radix2). +now apply Zlt_le_weak. +apply Rnot_le_lt. +intros H0. +refine (_ (ln_beta_le radix2 _ _ _ H0)). +rewrite ln_beta_bpow. +rewrite ln_beta_F2R_Zdigits. 2: discriminate. +rewrite <- H. +apply Zlt_not_le. +unfold emin. +apply Zplus_lt_reg_r with (emax - 1)%Z. +ring_simplify. +revert He2. +set (ex := ((x / 2^mw) mod 2^ew)%Z). +cut (ex < 2^ew)%Z. +replace (2^ew)%Z with (2 * emax)%Z. +clear ; intros H1 H2 ; omega. +replace ew with (1 + (ew - 1))%Z by ring. +rewrite Zpower_exp. +apply refl_equal. +discriminate. +clear -Hew ; omega. +eapply Z_mod_lt. +apply Zlt_gt. +apply (Zpower_gt_0 radix2). +now apply Zlt_le_weak. +apply bpow_gt_0. +Qed. + +Definition binary_float_of_bits x := + FF2B prec emax _ (binary_float_of_bits_aux_correct x). + +Theorem binary_float_of_bits_of_binary_float : + forall x, + binary_float_of_bits (bits_of_binary_float x) = x. +Proof. +intros x. +apply B2FF_inj. +unfold binary_float_of_bits. +rewrite B2FF_FF2B. +unfold binary_float_of_bits_aux. +rewrite split_bits_of_binary_float_correct. +destruct x as [sx|sx| |sx mx ex Bx]. +apply refl_equal. +(* *) +simpl. +rewrite Zeq_bool_false. +now rewrite Zeq_bool_true. +cut (1 < 2^ew)%Z. clear ; omega. +now apply (Zpower_gt_1 radix2). +(* *) +simpl. +rewrite Zeq_bool_false. +rewrite Zeq_bool_true. +rewrite Zeq_bool_false. +apply refl_equal. +cut (1 < 2^mw)%Z. clear ; omega. +now apply (Zpower_gt_1 radix2). +apply refl_equal. +cut (1 < 2^ew)%Z. clear ; omega. +now apply (Zpower_gt_1 radix2). +(* *) +unfold split_bits_of_binary_float. +case Zle_bool_spec ; intros Hm. +(* . *) +rewrite Zeq_bool_false. +rewrite Zeq_bool_false. +now ring_simplify (Zpos mx - 2 ^ mw + 2 ^ mw)%Z (ex - emin + 1 + emin - 1)%Z. +destruct (andb_prop _ _ Bx) as (_, H1). +generalize (Zle_bool_imp_le _ _ H1). +unfold emin. +replace (2^ew)%Z with (2 * emax)%Z. +clear ; omega. +replace ew with (1 + (ew - 1))%Z by ring. +rewrite Zpower_exp. +apply refl_equal. +discriminate. +clear -Hew ; omega. +destruct (andb_prop _ _ Bx) as (H1, _). +generalize (Zeq_bool_eq _ _ H1). +rewrite Z_of_nat_S_digits2_Pnat. +unfold FLT_exp, emin. +change Fcalc_digits.radix2 with radix2. +generalize (Zdigits radix2 (Zpos mx)). +clear. +intros ; zify ; omega. +(* . *) +rewrite Zeq_bool_true. 2: apply refl_equal. +simpl. +apply f_equal. +destruct (andb_prop _ _ Bx) as (H1, _). +generalize (Zeq_bool_eq _ _ H1). +rewrite Z_of_nat_S_digits2_Pnat. +unfold FLT_exp, emin, prec. +change Fcalc_digits.radix2 with radix2. +generalize (Zdigits_le_Zpower radix2 _ (Zpos mx) Hm). +generalize (Zdigits radix2 (Zpos mx)). +clear. +intros ; zify ; omega. +Qed. + +Theorem bits_of_binary_float_of_bits : + forall x, + (0 <= x < 2^(mw+ew+1))%Z -> + binary_float_of_bits x <> B754_nan prec emax -> + bits_of_binary_float (binary_float_of_bits x) = x. +Proof. +intros x Hx. +unfold binary_float_of_bits, bits_of_binary_float. +set (Cx := binary_float_of_bits_aux_correct x). +clearbody Cx. +rewrite match_FF2B. +revert Cx. +generalize (join_split_bits x Hx). +unfold binary_float_of_bits_aux. +case_eq (split_bits x). +intros (sx, mx) ex Sx. +assert (Bm: (0 <= mx < 2^mw)%Z). +inversion_clear Sx. +apply Z_mod_lt. +now apply Zlt_gt. +case Zeq_bool_spec ; intros He1. +(* subnormal *) +case_eq mx. +intros Hm Jx _ _. +now rewrite He1 in Jx. +intros px Hm Jx _ _. +rewrite Zle_bool_false. +now rewrite <- He1. +now rewrite <- Hm. +intros px Hm _ _ _. +apply False_ind. +apply Zle_not_lt with (1 := proj1 Bm). +now rewrite Hm. +case Zeq_bool_spec ; intros He2. +(* infinity/nan *) +case Zeq_bool_spec ; intros Hm. +now rewrite Hm, He2. +intros _ Cx Nx. +now elim Nx. +(* normal *) +case_eq (mx + 2 ^ mw)%Z. +intros Hm. +apply False_ind. +clear -Bm Hm ; omega. +intros p Hm Jx Cx _. +rewrite <- Hm. +rewrite Zle_bool_true. +now ring_simplify (mx + 2^mw - 2^mw)%Z (ex + emin - 1 - emin + 1)%Z. +now apply (Zplus_le_compat_r 0). +intros p Hm. +apply False_ind. +clear -Bm Hm ; zify ; omega. +Qed. + +End Binary_Bits. + +(** Specialization for IEEE single precision operations *) +Section B32_Bits. + +Definition binary32 := binary_float 24 128. + +Let Hprec : (0 < 24)%Z. +apply refl_equal. +Qed. + +Let Hprec_emax : (24 < 128)%Z. +apply refl_equal. +Qed. + +Definition b32_opp := Bopp 24 128. +Definition b32_plus := Bplus _ _ Hprec Hprec_emax. +Definition b32_minus := Bminus _ _ Hprec Hprec_emax. +Definition b32_mult := Bmult _ _ Hprec Hprec_emax. +Definition b32_div := Bdiv _ _ Hprec Hprec_emax. +Definition b32_sqrt := Bsqrt _ _ Hprec Hprec_emax. + +Definition b32_of_bits : Z -> binary32 := binary_float_of_bits 23 8 (refl_equal _) (refl_equal _) (refl_equal _). +Definition bits_of_b32 : binary32 -> Z := bits_of_binary_float 23 8. + +End B32_Bits. + +(** Specialization for IEEE double precision operations *) +Section B64_Bits. + +Definition binary64 := binary_float 53 1024. + +Let Hprec : (0 < 53)%Z. +apply refl_equal. +Qed. + +Let Hprec_emax : (53 < 1024)%Z. +apply refl_equal. +Qed. + +Definition b64_opp := Bopp 53 1024. +Definition b64_plus := Bplus _ _ Hprec Hprec_emax. +Definition b64_minus := Bminus _ _ Hprec Hprec_emax. +Definition b64_mult := Bmult _ _ Hprec Hprec_emax. +Definition b64_div := Bdiv _ _ Hprec Hprec_emax. +Definition b64_sqrt := Bsqrt _ _ Hprec Hprec_emax. + +Definition b64_of_bits : Z -> binary64 := binary_float_of_bits 52 11 (refl_equal _) (refl_equal _) (refl_equal _). +Definition bits_of_b64 : binary64 -> Z := bits_of_binary_float 52 11. + +End B64_Bits. |