path: root/flocq/Appli/Fappli_IEEE_bits.v

(**
This file is part of the Flocq formalization of floating-point
arithmetic in Coq: http://flocq.gforge.inria.fr/

Copyright (C) 2011-2013 Sylvie Boldo
#<br />#
Copyright (C) 2011-2013 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 sx (exist plx _) => join_bits sx (Zpos plx) (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 sx (exist plx _) => (sx, Zpos plx, 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 [plx Hplx]|sx mx ex Hx] ;
  try ( simpl ; apply split_join_bits ; split ; try apply Zle_refl ; try apply Zlt_pred ; trivial ; omega ).
simpl. apply split_join_bits; split; try (zify; omega).
destruct (digits2_Pnat_correct plx).
rewrite Zpower_nat_Z in H0.
eapply Zlt_le_trans. apply H0.
change 2%Z with (radix_val radix2). apply Zpower_le.
rewrite Z.ltb_lt in Hplx.
unfold prec in *. zify; 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.

Theorem bits_of_binary_float_range:
  forall x, (0 <= bits_of_binary_float x < 2^(mw+ew+1))%Z.
Proof.
  intros. 
Local Open Scope Z_scope.
  assert (J: forall s m e,
          0 <= m < 2^mw -> 0 <= e < 2^ew ->
          0 <= join_bits s m e < 2^(mw+ew+1)).
  {
    intros. unfold join_bits. 
    set (se := (if s then 2 ^ ew else 0) + e).
    assert (0 <= se < 2^(ew+1)).
    { rewrite (Zpower_plus radix2) by omega. change (radix2^1) with 2. simpl.
      unfold se. destruct s; omega. }
    assert (0 <= se * 2^mw <= (2^(ew+1) - 1) * 2^mw).
    { split. apply Zmult_gt_0_le_0_compat; omega.
      apply Zmult_le_compat_r; omega. }
    rewrite Z.mul_sub_distr_r in H2.
    replace (mw + ew + 1) with ((ew + 1) + mw) by omega. 
    rewrite (Zpower_plus radix2) by omega. simpl. omega.
  }
  assert (D: forall p n, Z.of_nat (S (digits2_Pnat p)) <= n ->
             0 <= Z.pos p < 2^n).
  {
    intros.  
    generalize (digits2_Pnat_correct p). simpl. rewrite ! Zpower_nat_Z. intros [A B].
    split. zify; omega. eapply Zlt_le_trans. eassumption. 
    apply (Zpower_le radix2); auto.
  } 
  destruct x; unfold bits_of_binary_float.
- apply J; omega.
- apply J; omega. 
- destruct n as [pl pl_range]. apply Z.ltb_lt in pl_range.
  apply J. apply D. unfold prec, Z_of_nat' in pl_range; omega. omega.
- unfold bounded in e0. apply Bool.andb_true_iff in e0; destruct e0 as [A B].
  apply Z.leb_le in B.
  unfold canonic_mantissa, FLT_exp in A. apply Zeq_bool_eq in A. 
  assert (G: Z.of_nat (S (digits2_Pnat m)) <= prec) by (zify; omega).
  assert (M: emin <= e) by (unfold emin; zify; omega). 
  generalize (Zle_bool_spec (2^mw) (Z.pos m)); intro SPEC; inversion SPEC.
  + apply J. 
    * split. omega. generalize (D _ _ G). unfold prec. 
      rewrite (Zpower_plus radix2) by omega.
      change (radix2^1) with 2. simpl radix_val. omega.
    * split. omega. unfold emin. replace (2^ew) with (2 * emax). omega.
      symmetry. replace ew with (1 + (ew - 1)) by omega. 
      apply (Zpower_plus radix2); omega.
  + apply J. zify; omega. omega. 
Local Close Scope Z_scope.
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 false xH (* dummy *)
    end
  else if Zeq_bool ex (Zpower 2 ew - 1) then
    match mx with
      | Z0 => F754_infinity sx
      | Zpos plx => F754_nan sx plx
      | Zneg _ => F754_nan false xH (* dummy *)
    end
  else
    match (mx + Zpower 2 mw)%Z with
    | Zpos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false xH (* 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.
simpl. intros. rewrite Z.ltb_lt. unfold prec. zify; omega.
case Zeq_bool_spec ; intros He2.
case_eq (x mod 2 ^ mw)%Z; try easy.
(* nan *)
intros plx Eqplx. apply Z.ltb_lt.
rewrite Z_of_nat_S_digits2_Pnat.
assert (forall a b, a <= b -> a < b+1)%Z by (intros; omega). apply H. clear H.
apply Zdigits_le_Zpower. simpl.
rewrite <- Eqplx. edestruct Z_mod_lt; eauto.
change 2%Z with (radix_val radix2).
apply Z.lt_gt, Zpower_gt_0. omega.
simpl. intros. rewrite Z.ltb_lt. unfold prec. zify; omega.
case_eq (x mod 2^mw + 2^mw)%Z ; try easy.
simpl. intros. rewrite Z.ltb_lt. unfold prec. zify; omega.
(* 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.
simpl. intros. rewrite Z.ltb_lt. unfold prec. zify; omega.
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 [plx Hplx]|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; auto.
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 ->
  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_eq mx; intros Hm.
now rewrite He2.
now rewrite He2.
intros. zify; omega.
(* 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.