# Module Floats

Formalization of floating-point numbers, using the Flocq library.

Require Import Axioms.
Require Import Coqlib.
Require Import Integers.
Require Import Reals.
Require Import Fappli_IEEE.
Require Import Fappli_IEEE_bits.
Require Import Fcore.
Require Import Fcalc_round.
Require Import Fcalc_bracket.
Require Import Fprop_Sterbenz.
Require Import Program.
Require Import Omega.

Close Scope R_scope.

Definition float := binary64. (* the type of IEE754 doubles *)

Module Float.

Definition zero: float := B754_zero _ _ false. (* the float +0.0 *)

Definition eq_dec: forall (f1 f2: float), {f1 = f2} + {f1 <> f2}.
Proof.
Ltac try_not_eq := try solve [right; congruence].
destruct f1, f2;
try destruct b; try destruct b0;
try solve [left; auto]; try_not_eq;
destruct (positive_eq_dec m m0); try_not_eq;
destruct (Z_eq_dec e e1); try solve [right; intro H; inv H; congruence];
subst; left; rewrite (proof_irr e0 e2); auto.
Defined.

Arithmetic operations

Definition neg: float -> float := b64_opp. (* opposite (change sign) *)
Definition abs (x: float): float := (* absolute value (set sign to +) *)
match x with
| B754_nan => x
| B754_infinity _ => B754_infinity _ _ false
| B754_finite _ m e H => B754_finite _ _ false m e H
| B754_zero _ => B754_zero _ _ false
end.

Definition binary_normalize64 (m e:Z) (s:bool): float :=
binary_normalize 53 1024 eq_refl eq_refl mode_NE m e s.

Definition binary_normalize64_correct (m e:Z) (s:bool) :=
binary_normalize_correct 53 1024 eq_refl eq_refl mode_NE m e s.
Global Opaque binary_normalize64_correct.

Definition floatofbinary32 (f: binary32) : float := (* single precision embedding in double precision *)
match f with
| B754_nan => B754_nan _ _
| B754_infinity s => B754_infinity _ _ s
| B754_zero s => B754_zero _ _ s
| B754_finite s m e _ =>
binary_normalize64 (cond_Zopp s (Zpos m)) e s
end.

Definition binary32offloat (f: float) : binary32 := (* conversion to single precision *)
match f with
| B754_nan => B754_nan _ _
| B754_infinity s => B754_infinity _ _ s
| B754_zero s => B754_zero _ _ s
| B754_finite s m e _ =>
binary_normalize 24 128 eq_refl eq_refl mode_NE (cond_Zopp s (Zpos m)) e s
end.

Definition singleoffloat (f: float): float := (* conversion to single precision, embedded in double *)
floatofbinary32 (binary32offloat f).

Definition Zoffloat (f:float): option Z := (* conversion to Z *)
match f with
| B754_finite s m (Zpos e) _ => Some (cond_Zopp s (Zpos m) * Zpower_pos radix2 e)
| 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))
| B754_zero _ => Some 0
| _ => None
end.

Definition intoffloat (f:float): option int := (* conversion to signed 32-bit int *)
match Zoffloat f with
| Some n =>
if Zle_bool Int.min_signed n && Zle_bool n Int.max_signed then
Some (Int.repr n)
else
None
| None => None
end.

Definition intuoffloat (f:float): option int := (* conversion to unsigned 32-bit int *)
match Zoffloat f with
| Some n =>
if Zle_bool 0 n && Zle_bool n Int.max_unsigned then
Some (Int.repr n)
else
None
| None => None
end.

Definition floatofint (n:int): float := (* conversion from signed 32-bit int *)
binary_normalize64 (Int.signed n) 0 false.
Definition floatofintu (n:int): float:= (* conversion from unsigned 32-bit int *)
binary_normalize64 (Int.unsigned n) 0 false.
Definition add: float -> float -> float := b64_plus mode_NE. (* addition *)
Definition sub: float -> float -> float := b64_minus mode_NE. (* subtraction *)
Definition mul: float -> float -> float := b64_mult mode_NE. (* multiplication *)
Definition div: float -> float -> float := b64_div mode_NE. (* division *)

Definition order_float (f1 f2:float): option Datatypes.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.

Definition cmp (c:comparison) (f1 f2:float) : bool := (* comparison *)
match c with
| Ceq =>
match order_float f1 f2 with Some Eq => true | _ => false end
| Cne =>
match order_float f1 f2 with Some Eq => false | _ => true end
| Clt =>
match order_float f1 f2 with Some Lt => true | _ => false end
| Cle =>
match order_float f1 f2 with Some(Lt|Eq) => true | _ => false end
| Cgt =>
match order_float f1 f2 with Some Gt => true | _ => false end
| Cge =>
match order_float f1 f2 with Some(Gt|Eq) => true | _ => false end
end.

Conversions between floats and their concrete in-memory representation as a sequence of 64 bits (double precision) or 32 bits (single precision).

Definition bits_of_double (f: float): int64 := Int64.repr (bits_of_b64 f).
Definition double_of_bits (b: int64): float := b64_of_bits (Int64.unsigned b).

Definition bits_of_single (f: float) : int := Int.repr (bits_of_b32 (binary32offloat f)).
Definition single_of_bits (b: int): float := floatofbinary32 (b32_of_bits (Int.unsigned b)).

Definition from_words (hi lo: int) : float :=
double_of_bits
(Int64.or (Int64.shl (Int64.repr (Int.unsigned hi)) (Int64.repr 32))
(Int64.repr (Int.unsigned lo))).

Lemma from_words_eq:
forall lo hi,
from_words hi lo =
double_of_bits (Int64.repr (Int.unsigned hi * two_p 32 + Int.unsigned lo)).
Proof.
intros. unfold from_words. decEq.
apply Int64.eqm_samerepr. auto with ints.
change (Z_of_nat Int64.wordsize) with 64. omega.
generalize (Int.unsigned_range lo). intros [A B].
rewrite Int64.unsigned_repr. assumption.
assert (Int.modulus < Int64.max_unsigned). compute; auto. omega.
Qed.

Below are the only properties of floating-point arithmetic that we rely on in the compiler proof.

Some tactics *

Ltac compute_this val :=
let x := fresh in set val as x in *; vm_compute in x; subst x.

Ltac smart_omega :=
simpl radix_val in *; simpl Zpower in *;
compute_this Int.modulus; compute_this Int.half_modulus;
compute_this Int.max_unsigned;
compute_this Int.min_signed; compute_this Int.max_signed;
compute_this Int64.modulus; compute_this Int64.half_modulus;
compute_this Int64.max_unsigned;
compute_this (Zpower_pos 2 1024); compute_this (Zpower_pos 2 53); compute_this (Zpower_pos 2 52);
omega.

Proof.
intros.
destruct f1, f2; simpl; try reflexivity; try (destruct b, b0; reflexivity).
rewrite Zplus_comm; rewrite Zmin_comm; reflexivity.
Qed.

Theorem subf_addf_opp: forall f1 f2, sub f1 f2 = add f1 (neg f2).
Proof.
destruct f1, f2; reflexivity.
Qed.

Lemma floatofbinary32_exact :
forall f, is_finite_strict _ _ f = true ->
is_finite_strict _ _ (floatofbinary32 f) = true /\ B2R _ _ f = B2R _ _ (floatofbinary32 f).
Proof.
destruct f as [ | | |s m e]; try discriminate; intro.
pose proof (binary_normalize64_correct (cond_Zopp s (Zpos m)) e s).
match goal with [H0:if Rlt_bool (Rabs ?x) _ then _ else _ |- _ /\ ?y = _] => assert (x=y)%R end.
apply round_generic; [now apply valid_rnd_round_mode|].
apply (generic_inclusion_ln_beta _ (FLT_exp (3 - 128 - 24) 24)).
intro; eapply Zle_trans; [apply Zle_max_compat_l | apply Zle_max_compat_r]; omega.
apply generic_format_canonic; apply canonic_canonic_mantissa; apply (proj1 (andb_prop _ _ e0)).
rewrite H1, Rlt_bool_true in H0; intuition; unfold floatofbinary32, binary_normalize64.
match goal with [ |- _ _ _ ?x = true ] => destruct x end; try discriminate.
symmetry in H2; apply F2R_eq_0_reg in H2; destruct s; discriminate.
reflexivity.
eapply Rlt_trans.
unfold B2R; rewrite <- F2R_Zabs, abs_cond_Zopp; eapply bounded_lt_emax; now apply e0.
now apply bpow_lt.
Qed.

Lemma binary32offloatofbinary32 :
forall f, binary32offloat (floatofbinary32 f) = f.
Proof.
intro; pose proof (floatofbinary32_exact f); destruct f as [ | | |s m e]; try reflexivity.
specialize (H eq_refl); destruct H.
destruct (floatofbinary32 (B754_finite 24 128 s m e e0)) as [ | | |s1 m1 e1]; try discriminate.
unfold binary32offloat.
pose proof (binary_normalize_correct 24 128 eq_refl eq_refl mode_NE (cond_Zopp s1 (Zpos m1)) e1 s1).
unfold B2R at 2 in H0; cbv iota zeta beta in H0; rewrite <- H0, round_generic in H1.
rewrite Rlt_bool_true in H1.
apply B2R_inj; intuition; match goal with [|- _ _ _ ?f = true] => destruct f end; try discriminate.
symmetry in H2; apply F2R_eq_0_reg in H2; destruct s; discriminate.
reflexivity.
unfold B2R; rewrite <- F2R_Zabs, abs_cond_Zopp; eapply bounded_lt_emax; apply e0.
now apply valid_rnd_round_mode.
now apply generic_format_B2R.
Qed.

Theorem singleoffloat_idem:
forall f, singleoffloat (singleoffloat f) = singleoffloat f.
Proof.
intros; unfold singleoffloat; rewrite binary32offloatofbinary32; reflexivity.
Qed.

Properties of comparisons.

Theorem order_float_finite_correct:
forall f1 f2, is_finite _ _ f1 = true -> is_finite _ _ f2 = true ->
match order_float f1 f2 with
| Some c => Rcompare (B2R _ _ f1) (B2R _ _ f2) = c
| None => False
end.
Proof.
Ltac apply_Rcompare :=
match goal with
| [ |- Rcompare _ _ = Lt ] => apply Rcompare_Lt
| [ |- Rcompare _ _ = Eq ] => apply Rcompare_Eq
| [ |- Rcompare _ _ = Gt ] => apply Rcompare_Gt
end.
unfold order_float; 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 (FLT_exp (3-1024-53) 53) x < canonic_exp radix2 (FLT_exp (3-1024-53) 53) y -> (x < y)%R).
intros; apply Rnot_le_lt; intro; apply (ln_beta_le radix2) in H5.
apply (fexp_monotone 53 1024) in H5; unfold canonic_exp 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 cmp_swap:
forall c x y, Float.cmp (swap_comparison c) x y = Float.cmp c y x.
Proof.
destruct c, x, y; simpl; try destruct b; try destruct b0; try reflexivity;
rewrite <- (Zcompare_antisym e e1); destruct (e ?= e1); try reflexivity;
change Eq with (CompOpp Eq); rewrite <- (Pcompare_antisym m m0 Eq);
simpl; destruct ((m ?= m0)%positive Eq); try reflexivity.
Qed.

Theorem cmp_ne_eq:
forall f1 f2, cmp Cne f1 f2 = negb (cmp Ceq f1 f2).
Proof.
unfold cmp; intros; destruct (order_float f1 f2) as [ [] | ]; reflexivity.
Qed.

Theorem cmp_lt_eq_false:
forall f1 f2, cmp Clt f1 f2 = true -> cmp Ceq f1 f2 = true -> False.
Proof.
unfold cmp; intros; destruct (order_float f1 f2) as [ [] | ]; discriminate.
Qed.

Theorem cmp_le_lt_eq:
forall f1 f2, cmp Cle f1 f2 = cmp Clt f1 f2 || cmp Ceq f1 f2.
Proof.
unfold cmp; intros; destruct (order_float f1 f2) as [ [] | ]; reflexivity.
Qed.

Corollary cmp_gt_eq_false:
forall x y, cmp Cgt x y = true -> cmp Ceq x y = true -> False.
Proof.
intros; rewrite <- cmp_swap in H; rewrite <- cmp_swap in H0;
eapply cmp_lt_eq_false; now eauto.
Qed.

Corollary cmp_ge_gt_eq:
forall f1 f2, cmp Cge f1 f2 = cmp Cgt f1 f2 || cmp Ceq f1 f2.
Proof.
intros.
change Cge with (swap_comparison Cle); change Cgt with (swap_comparison Clt);
change Ceq with (swap_comparison Ceq).
repeat rewrite cmp_swap.
now apply cmp_le_lt_eq.
Qed.

Properties of conversions to/from in-memory representation. The double-precision conversions are bijective (one-to-one). The single-precision conversions lose precision exactly as described by singleoffloat rounding.

Theorem double_of_bits_of_double:
forall f, double_of_bits (bits_of_double f) = f.
Proof.
intros; unfold double_of_bits, bits_of_double, bits_of_b64, b64_of_bits.
rewrite Int64.unsigned_repr, binary_float_of_bits_of_binary_float; [reflexivity|].
destruct f.
simpl; try destruct b; vm_compute; split; congruence.
simpl; try destruct b; vm_compute; split; congruence.
simpl; vm_compute; split; congruence.
unfold bits_of_binary_float, join_bits.
destruct (andb_prop _ _ e0); apply Zle_bool_imp_le in H0; apply Zeq_bool_eq in H; unfold FLT_exp in H.
match goal with [H:Zmax ?x ?y = e|-_] => pose proof (Zle_max_l x y); pose proof (Zle_max_r x y) end.
rewrite H, Fcalc_digits.Z_of_nat_S_digits2_Pnat in *.
lapply (Fcalc_digits.Zpower_gt_Zdigits radix2 53 (Zpos m)). intro.
pose proof (Zle_bool_spec (2 ^ 52) (Zpos m)).
assert (Zpos m > 0); [vm_compute; exact eq_refl|].
compute_this (2^11); compute_this (2^(11-1)).
inversion H4; fold (2^52) in *; destruct H6; destruct b; now smart_omega.
Qed.

Theorem single_of_bits_of_single:
forall f, single_of_bits (bits_of_single f) = singleoffloat f.
Proof.
intros; unfold single_of_bits, bits_of_single, bits_of_b32, b32_of_bits.
rewrite Int.unsigned_repr, binary_float_of_bits_of_binary_float; [reflexivity|].
destruct (binary32offloat f).
simpl; try destruct b; vm_compute; split; congruence.
simpl; try destruct b; vm_compute; split; congruence.
simpl; vm_compute; split; congruence.
unfold bits_of_binary_float, join_bits.
destruct (andb_prop _ _ e0); apply Zle_bool_imp_le in H0; apply Zeq_bool_eq in H.
unfold FLT_exp in H.
match goal with [H:Zmax ?x ?y = e|-_] => pose proof (Zle_max_l x y); pose proof (Zle_max_r x y) end.
rewrite H, Fcalc_digits.Z_of_nat_S_digits2_Pnat in *.
lapply (Fcalc_digits.Zpower_gt_Zdigits radix2 24 (Zpos m)). intro.
pose proof (Zle_bool_spec (2 ^ 23) (Zpos m)).
compute_this (2^23); compute_this (2^24); compute_this (2^8); compute_this (2^(8-1)).
assert (Zpos m > 0); [exact eq_refl|].
inversion H4; destruct b; now smart_omega.
Qed.

Theorem bits_of_singleoffloat:
forall f, bits_of_single (singleoffloat f) = bits_of_single f.
Proof.
intro; unfold singleoffloat, bits_of_single; rewrite binary32offloatofbinary32; reflexivity.
Qed.

Theorem singleoffloat_of_bits:
forall b, singleoffloat (single_of_bits b) = single_of_bits b.
Proof.
intro; unfold singleoffloat, single_of_bits; rewrite binary32offloatofbinary32; reflexivity.
Qed.

Conversions between floats and unsigned ints can be defined in terms of conversions between floats and signed ints. (Most processors provide only the latter, forcing the compiler to emulate the former.)

Definition ox8000_0000 := Int.repr Int.half_modulus. (* 0x8000_0000 *)

Lemma round_exact:
forall n, -2^52 < n < 2^52 ->
round radix2 (FLT_exp (3 - 1024 - 53) 53)
(round_mode mode_NE) (Z2R n) = Z2R n.
Proof.
intros; rewrite round_generic; [reflexivity|now apply valid_rnd_round_mode|].
apply generic_format_FLT; exists (Float radix2 n 0).
unfold F2R, Fnum, Fexp, bpow; rewrite Rmult_1_r; intuition.
pose proof (Zabs_spec n); now smart_omega.
Qed.

Lemma binary_normalize64_exact:
forall n, -2^52 < n < 2^52 ->
B2R _ _ (binary_normalize64 n 0 false) = Z2R n /\
is_finite _ _ (binary_normalize64 n 0 false) = true.
Proof.
intros; pose proof (binary_normalize64_correct n 0 false).
unfold F2R, Fnum, Fexp, bpow in H0; rewrite Rmult_1_r, round_exact, Rlt_bool_true in H0; try assumption.
rewrite <- Z2R_abs; apply Z2R_lt; pose proof (Zabs_spec n); now smart_omega.
Qed.

Theorem floatofintu_floatofint_1:
forall x,
Int.ltu x ox8000_0000 = true ->
floatofintu x = floatofint x.
Proof.
unfold floatofintu, floatofint, Int.signed, Int.ltu; intro.
change (Int.unsigned ox8000_0000) with Int.half_modulus.
destruct (zlt (Int.unsigned x) Int.half_modulus); now intuition.
Qed.

Theorem floatofintu_floatofint_2:
forall x,
Int.ltu x ox8000_0000 = false ->
floatofintu x = add (floatofint (Int.sub x ox8000_0000))
(floatofintu ox8000_0000).
Proof.
unfold floatofintu, floatofint, Int.signed, Int.ltu, Int.sub; intros.
pose proof (Int.unsigned_range x).
compute_this (Int.unsigned ox8000_0000).
destruct (zlt (Int.unsigned x) 2147483648); try discriminate.
rewrite Int.unsigned_repr by smart_omega.
destruct (zlt ((Int.unsigned x) - 2147483648) Int.half_modulus).
match goal with [|- _ = Bplus _ _ _ _ _ ?x ?y] =>
pose proof (Bplus_correct 53 1024 eq_refl eq_refl mode_NE x y) end.
do 2 rewrite (fun x H => proj1 (binary_normalize64_exact x H)) in H1 by smart_omega.
do 2 rewrite (fun x H => proj2 (binary_normalize64_exact x H)) in H1 by smart_omega.
rewrite <- Z2R_plus, round_exact in H1 by smart_omega.
rewrite Rlt_bool_true in H1;
replace (Int.unsigned x - 2147483648 + 2147483648) with (Int.unsigned x) in * by ring.
apply B2R_inj.
destruct (binary_normalize64_exact (Int.unsigned x)); [now smart_omega|].
match goal with [|- _ _ _ ?f = _] => destruct f end; intuition.
exfalso; simpl in H2; change 0%R with (Z2R 0) in H2; apply eq_Z2R in H2; omega.
simpl Zcompare in *.
match goal with [|- _ _ _ ?f = _] => destruct f end; intuition.
exfalso; simpl in H0; change 0%R with (Z2R 0) in H0; apply eq_Z2R in H0; omega.
rewrite (fun x H => proj1 (binary_normalize64_exact x H)) by smart_omega; now intuition.
rewrite <- Z2R_Zpower, <- Z2R_abs by omega; apply Z2R_lt;
pose proof (Zabs_spec (Int.unsigned x)); now smart_omega.
exfalso; now smart_omega.
Qed.

Theorem Zoffloat_correct:
forall f,
match Zoffloat f with
| Some n =>
is_finite _ _ f = true /\
Z2R n = round radix2 (FIX_exp 0) (round_mode mode_ZR) (B2R _ _ f)
| None =>
is_finite _ _ f = false
end.
Proof.
destruct f; try now intuition.
simpl B2R. rewrite round_0. now intuition. now apply valid_rnd_round_mode.
destruct e. split. reflexivity.
rewrite round_generic. symmetry. now apply Rmult_1_r.
now apply valid_rnd_round_mode.
apply generic_format_FIX. exists (Float radix2 (cond_Zopp b (Zpos m)) 0). split; reflexivity.
split; [reflexivity|].
rewrite round_generic, Z2R_mult, Z2R_Zpower_pos, <- bpow_powerRZ;
[reflexivity|now apply valid_rnd_round_mode|apply generic_format_F2R; discriminate].
rewrite (inbetween_float_ZR_sign _ _ _ ((Zpos m) / Zpower_pos radix2 p)
pose proof (Rlt_bool_spec (Z2R (cond_Zopp b (Zpos m)) * / Z2R (Zpower_pos 2 p)) 0).
inversion H; rewrite <- (Rmult_0_l (bpow radix2 (Zneg p))) in H1.
apply Rmult_lt_reg_r in H1. apply (lt_Z2R _ 0) in H1.
destruct b; [split; [|ring_simplify];reflexivity|discriminate].
now apply bpow_gt_0.
apply Rmult_le_reg_r in H1. apply (le_Z2R 0) in H1.
destruct b; [destruct H1|split; [|ring_simplify]]; reflexivity.
now apply (bpow_gt_0 radix2 (Zneg p)).
unfold canonic_exp, FIX_exp; replace 0 with (Zneg p + Zpos p) by apply Zplus_opp_r.
apply (inbetween_float_new_location radix2 _ _ _ _ (Zpos p)); [reflexivity|].
apply inbetween_Exact; unfold B2R, F2R, Fnum, Fexp; destruct b.
rewrite Rabs_left; [simpl; ring_simplify; reflexivity|].
replace 0%R with (0*(bpow radix2 (Zneg p)))%R by ring; apply Rmult_gt_compat_r.
now apply bpow_gt_0.
apply (Z2R_lt _ 0); reflexivity.
apply Rabs_right; replace 0%R with (0*(bpow radix2 (Zneg p)))%R by ring; apply Rgt_ge.
apply Rmult_gt_compat_r; [now apply bpow_gt_0|apply (Z2R_lt 0); reflexivity].
Qed.

Theorem intoffloat_correct:
forall f,
match intoffloat f with
| Some n =>
is_finite _ _ f = true /\
Z2R (Int.signed n) = round radix2 (FIX_exp 0) (round_mode mode_ZR) (B2R _ _ f)
| None =>
is_finite _ _ f = false \/
(B2R _ _ f <= Z2R (Zpred Int.min_signed)\/
Z2R (Zsucc Int.max_signed) <= B2R _ _ f)%R
end.
Proof.
intro; pose proof (Zoffloat_correct f); unfold intoffloat; destruct (Zoffloat f).
pose proof (Zle_bool_spec Int.min_signed z); pose proof (Zle_bool_spec z Int.max_signed).
compute_this Int.min_signed; compute_this Int.max_signed; destruct H.
inversion H0; [inversion H1|].
rewrite <- (Int.signed_repr z) in H2 by smart_omega; split; assumption.
right; right; eapply Rle_trans; [apply Z2R_le; apply Zlt_le_succ; now apply H6|].
rewrite H2, round_ZR_pos.
unfold round, scaled_mantissa, canonic_exp, FIX_exp, F2R, Fnum, Fexp; simpl bpow.
do 2 rewrite Rmult_1_r; now apply Zfloor_lb.
apply Rnot_lt_le; intro; apply Rlt_le in H7; apply (round_le radix2 (FIX_exp 0) (round_mode mode_ZR)) in H7;
rewrite <- H2, round_0 in H7; [apply (le_Z2R _ 0) in H7; now smart_omega|now apply valid_rnd_round_mode].
right; left; eapply Rle_trans; [|apply (Z2R_le z); simpl; omega].
rewrite H2, round_ZR_neg.
unfold round, scaled_mantissa, canonic_exp, FIX_exp, F2R, Fnum, Fexp; simpl bpow.
do 2 rewrite Rmult_1_r; now apply Zceil_ub.
apply Rnot_lt_le; intro; apply Rlt_le in H5; apply (round_le radix2 (FIX_exp 0) (round_mode mode_ZR)) in H5.
rewrite <- H2, round_0 in H5; [apply (le_Z2R 0) in H5; omega|now apply valid_rnd_round_mode].
left; assumption.
Qed.

Theorem intuoffloat_correct:
forall f,
match intuoffloat f with
| Some n =>
is_finite _ _ f = true /\
Z2R (Int.unsigned n) = round radix2 (FIX_exp 0) (round_mode mode_ZR) (B2R _ _ f)
| None =>
is_finite _ _ f = false \/
(B2R _ _ f <= -1 \/
Z2R (Zsucc Int.max_unsigned) <= B2R _ _ f)%R
end.
Proof.
intro; pose proof (Zoffloat_correct f); unfold intuoffloat; destruct (Zoffloat f).
pose proof (Zle_bool_spec 0 z); pose proof (Zle_bool_spec z Int.max_unsigned).
compute_this Int.max_unsigned; destruct H.
inversion H0. inversion H1.
rewrite <- (Int.unsigned_repr z) in H2 by smart_omega; split; assumption.
right; right; eapply Rle_trans; [apply Z2R_le; apply Zlt_le_succ; now apply H6|].
rewrite H2, round_ZR_pos.
unfold round, scaled_mantissa, canonic_exp, FIX_exp, F2R, Fnum, Fexp; simpl bpow;
do 2 rewrite Rmult_1_r; now apply Zfloor_lb.
apply Rnot_lt_le; intro; apply Rlt_le in H7; eapply (round_le radix2 (FIX_exp 0) (round_mode mode_ZR)) in H7;
rewrite <- H2, round_0 in H7; [apply (le_Z2R _ 0) in H7; now smart_omega|now apply valid_rnd_round_mode].
right; left; eapply Rle_trans; [|change (-1)%R with (Z2R (-1)); apply (Z2R_le z); omega].
rewrite H2, round_ZR_neg; unfold round, scaled_mantissa, canonic_exp, FIX_exp, F2R, Fnum, Fexp; simpl bpow.
do 2 rewrite Rmult_1_r; now apply Zceil_ub.
apply Rnot_lt_le; intro; apply Rlt_le in H5; apply (round_le radix2 (FIX_exp 0) (round_mode mode_ZR)) in H5;
rewrite <- H2, round_0 in H5; [apply (le_Z2R 0) in H5; omega|now apply valid_rnd_round_mode].
left; assumption.
Qed.

Lemma intuoffloat_interval:
forall f n,
intuoffloat f = Some n ->
(-1 < B2R _ _ f < Z2R (Zsucc Int.max_unsigned))%R.
Proof.
intro; pose proof (intuoffloat_correct f); destruct (intuoffloat f); try discriminate; destruct H.
destruct f; try discriminate; intros.
simpl B2R; change 0%R with (Z2R 0); change (-1)%R with (Z2R (-1)); split; apply Z2R_lt; reflexivity.
pose proof (Int.unsigned_range i).
unfold round, scaled_mantissa, B2R, F2R, Fnum, Fexp in H0 |- *; simpl bpow in H0; do 2 rewrite Rmult_1_r in H0;
apply eq_Z2R in H0.
split; apply Rnot_le_lt; intro.
rewrite Ztrunc_ceil in H0;
[apply Zceil_le in H3; change (-1)%R with (Z2R (-1)) in H3; rewrite Zceil_Z2R in H3; omega|].
eapply Rle_trans; [now apply H3|apply (Z2R_le (-1) 0); discriminate].
rewrite Ztrunc_floor in H0; [apply Zfloor_le in H3; rewrite Zfloor_Z2R in H3; now smart_omega|].
eapply Rle_trans; [|now apply H3]; apply (Z2R_le 0); discriminate.
Qed.

Theorem intuoffloat_intoffloat_1:
forall x n,
cmp Clt x (floatofintu ox8000_0000) = true ->
intuoffloat x = Some n ->
intoffloat x = Some n.
Proof.
intros; unfold cmp in H; pose proof (order_float_finite_correct x (floatofintu ox8000_0000)).
destruct (order_float x (floatofintu ox8000_0000)); try destruct c; try discriminate.
pose proof (intuoffloat_correct x); rewrite H0 in H2; destruct H2.
specialize (H1 H2 eq_refl); pose proof (intoffloat_correct x); destruct (intoffloat x).
f_equal; rewrite <- (proj2 H4) in H3; apply eq_Z2R in H3.
pose proof (eq_refl (Int.repr (Int.unsigned n))); rewrite H3 in H5 at 1.
rewrite Int.repr_signed, Int.repr_unsigned in H5; assumption.
destruct H4; [rewrite H2 in H4; discriminate|].
apply intuoffloat_interval in H0; exfalso; destruct H0, H4.
eapply Rlt_le_trans in H0; [|now apply H4]; apply (lt_Z2R (-1)) in H0; discriminate.
apply Rcompare_Lt_inv in H1; eapply Rle_lt_trans in H1; [|now apply H4].
unfold floatofintu in H1; rewrite (fun x H => proj1 (binary_normalize64_exact x H)) in H1;
[apply lt_Z2R in H1; discriminate|split; reflexivity].
Qed.

Lemma 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 ring; do 2 rewrite Z2R_minus.
split;
[apply Rplus_le_compat_r; now apply Zfloor_lb|
apply Rplus_lt_compat_r; rewrite Z2R_plus; now apply Zfloor_ub].
Qed.

Theorem intuoffloat_intoffloat_2:
forall x n,
cmp Clt x (floatofintu ox8000_0000) = false ->
intuoffloat x = Some n ->
intoffloat (sub x (floatofintu ox8000_0000)) = Some (Int.sub n ox8000_0000).
Proof.
assert (B2R _ _ (floatofintu ox8000_0000) = Z2R (Int.unsigned ox8000_0000)).
apply (fun x H => proj1 (binary_normalize64_exact x H)); split; reflexivity.
intros; unfold cmp in H0; pose proof (order_float_finite_correct x (floatofintu ox8000_0000)).
destruct (order_float x (floatofintu ox8000_0000)); try destruct c; try discriminate;
pose proof (intuoffloat_correct x); rewrite H1 in H3; destruct H3; specialize (H2 H3 eq_refl).
apply Rcompare_Eq_inv in H2; apply B2R_inj in H2.
subst x; vm_compute in H1; injection H1; intro; subst n; vm_compute; reflexivity.
destruct x; try discriminate H3;
[rewrite H in H2; simpl B2R in H2; apply (eq_Z2R 0) in H2; discriminate|reflexivity].
reflexivity.
rewrite H in H2; apply Rcompare_Gt_inv in H2; pose proof (intuoffloat_interval _ _ H1).
unfold sub, b64_minus.
exploit (Bminus_correct 53 1024 eq_refl eq_refl mode_NE x (floatofintu ox8000_0000)); [assumption|reflexivity|]; intro.
rewrite H, round_generic in H6.
match goal with [H6:if Rlt_bool ?x ?y then _ else _|-_] =>
pose proof (Rlt_bool_spec x y); destruct (Rlt_bool x y) end.
destruct H6.
match goal with [|- _ ?y = _] => pose proof (intoffloat_correct y); destruct (intoffloat y) end.
destruct H9.
f_equal; rewrite <- (Int.repr_signed i); unfold Int.sub; f_equal; apply eq_Z2R.
rewrite Z2R_minus, H10, H4.
unfold round, scaled_mantissa, F2R, Fexp, Fnum, round_mode; simpl bpow; repeat rewrite Rmult_1_r;
rewrite <- Z2R_minus; f_equal.
rewrite (Ztrunc_floor (B2R _ _ x)), <- Zfloor_minus, <- Ztrunc_floor;
[f_equal; assumption|apply Rle_0_minus; left; assumption|].
left; eapply Rlt_trans; [|now apply H2]; apply (Z2R_lt 0); reflexivity.
exfalso; simpl Zcompare in H6, H8; rewrite H6, H8 in H9.
destruct H9 as [|[]]; [discriminate|..].
eapply Rle_trans in H9; [|apply Rle_0_minus; left; assumption]; apply (le_Z2R 0) in H9; apply H9; reflexivity.
eapply Rle_lt_trans in H9; [|apply Rplus_lt_compat_r; now apply (proj2 H5)].
rewrite <- Z2R_opp, <- Z2R_plus in H9; apply lt_Z2R in H9; discriminate.
exfalso; inversion H7; rewrite Rabs_right in H8.
eapply Rle_lt_trans in H8. apply Rle_not_lt in H8; [assumption|apply (bpow_le _ 31); discriminate].
change (bpow radix2 31) with (Z2R(Zsucc Int.max_unsigned - Int.unsigned ox8000_0000)); rewrite Z2R_minus.
apply Rplus_lt_compat_r; exact (proj2 H5).
apply Rle_ge; apply Rle_0_minus; left; assumption.
now apply valid_rnd_round_mode.
apply Fprop_Sterbenz.sterbenz_aux; [now apply fexp_monotone|now apply generic_format_B2R| |].
rewrite <- H; now apply generic_format_B2R.
destruct H5; split; left; assumption.
now destruct H2.
Qed.

Conversions from ints to floats can be defined as bitwise manipulations over the in-memory representation. This is what the PowerPC port does. The trick is that from_words 0x4330_0000 x is the float 2^52 + floatofintu x.

Definition ox4330_0000 := Int.repr 1127219200. (* 0x4330_0000 *)

Lemma split_bits_or:
forall x,
split_bits 52 11
(Int64.unsigned
(Int64.or
(Int64.shl (Int64.repr (Int.unsigned ox4330_0000)) (Int64.repr 32))
(Int64.repr (Int.unsigned x)))) = (false, Int.unsigned x, 1075).
Proof.
intro; pose proof (Int.unsigned_range x); unfold split_bits; f_equal; f_equal.
apply Zle_bool_false.
match goal with [|-_ ?x < _] =>
pose proof (Int64.sign_bit_of_Z x);
destruct (zlt (Int64.unsigned x) Int64.half_modulus)
end.
assumption.
unfold Int64.or, Int64.bitwise_binop in H0.
rewrite Int64.unsigned_repr, Int64.bits_of_Z_of_bits in H0.
rewrite orb_false_intro in H0; [discriminate|reflexivity|].
rewrite Int64.sign_bit_of_Z.
match goal with [|- ((if ?c then _ else _) = _)] => destruct c end.
reflexivity.
rewrite Int64.unsigned_repr in z0; [exfalso|]; now smart_omega.
vm_compute; split; congruence.
now apply Int64.Z_of_bits_range_2.
change (2^52) with (two_p 52).
rewrite <- Int64.zero_ext_mod, Int64.zero_ext_and by (vm_compute; intuition).
rewrite Int64.and_commut, Int64.and_or_distrib.
match goal with [|- _ (_ ?o _) = _] => change o with Int64.zero end.
rewrite Int64.or_commut, Int64.or_zero, Int64.and_commut.
rewrite <- Int64.zero_ext_and, Int64.zero_ext_mod by (compute; intuition).
rewrite Int64.unsigned_repr; [apply Zmod_small; compute_this (two_p 52)|]; now smart_omega.
match goal with [|- ?x mod _ = _] => rewrite <- (Int64.unsigned_repr x) end.
change (2^52) with (two_p (Int64.unsigned (Int64.repr 52))). rewrite <- Int64.shru_div_two_p.
change (2^11) with (two_p 11). rewrite <- Int64.or_shru.
replace (Int64.shru (Int64.repr (Int.unsigned x)) (Int64.repr 52)) with Int64.zero;
[vm_compute; reflexivity|].
rewrite Int64.shru_div_two_p, Int64.unsigned_repr by smart_omega.
unfold Int64.zero; f_equal; rewrite Zdiv_small; [reflexivity|].
simpl; compute_this (two_power_pos 52); now smart_omega.
apply Zdiv_interval_2; try smart_omega; now apply Int64.unsigned_range_2.
Qed.

Lemma from_words_value:
forall x,
B2R _ _ (from_words ox4330_0000 x) =
(bpow radix2 52 + Z2R (Int.unsigned x))%R /\
is_finite _ _ (from_words ox4330_0000 x) = true.
Proof.
intros; unfold from_words, double_of_bits, b64_of_bits, binary_float_of_bits.
rewrite B2R_FF2B; unfold is_finite; rewrite match_FF2B;
unfold binary_float_of_bits_aux; rewrite split_bits_or; simpl; pose proof (Int.unsigned_range x).
destruct (Int.unsigned x + Zpower_pos 2 52) as []_eqn.
exfalso; now smart_omega.
simpl; rewrite <- Heqz; unfold F2R; simpl.
rewrite <- (Z2R_plus 4503599627370496), Rmult_1_r.
split; [f_equal; compute_this (Zpower_pos 2 52); ring | reflexivity].
assert (Zneg p < 0) by reflexivity.
exfalso; now smart_omega.
Qed.

Theorem floatofintu_from_words:
forall x,
floatofintu x =
sub (from_words ox4330_0000 x) (from_words ox4330_0000 Int.zero).
Proof.
intros; destruct (Int.eq_dec x Int.zero); [subst; vm_compute; reflexivity|].
assert (Int.unsigned x <> 0).
intro; destruct n; rewrite <- (Int.repr_unsigned x), H; reflexivity.
pose proof (Int.unsigned_range x).
pose proof (binary_normalize64_exact (Int.unsigned x)). destruct H1; [smart_omega|].
unfold floatofintu, sub, b64_minus.
match goal with [|- _ = Bminus _ _ _ _ _ ?x ?y] =>
pose proof (Bminus_correct 53 1024 eq_refl eq_refl mode_NE x y) end.
apply (fun f x y => f x y) in H3; try apply (fun x => proj2 (from_words_value x)).
do 2 rewrite (fun x => proj1 (from_words_value x)) in H3.
replace (bpow radix2 52 + Z2R (Int.unsigned x) -
(bpow radix2 52 + Z2R (Int.unsigned Int.zero)))%R with (Z2R (Int.unsigned x)) in H3 by (simpl; ring).
rewrite round_exact in H3 by smart_omega.
match goal with [H3:if Rlt_bool ?x ?y then _ else _ |- _] =>
pose proof (Rlt_bool_spec x y); destruct (Rlt_bool x y) end; destruct H3.
simpl Zcompare in *; apply B2R_inj;
try match goal with [H':B2R _ _ ?f = _ , H'':is_finite _ _ ?f = true |- is_finite_strict _ _ ?f = true] =>
destruct f; [
simpl in H'; change 0%R with (Z2R 0) in H'; apply eq_Z2R in H'; now destruct (H (eq_sym H')) |
discriminate H'' | discriminate H'' | reflexivity
]
end.
rewrite H3; assumption.
inversion H4; change (bpow radix2 1024) with (Z2R (radix2 ^ 1024)) in H6; rewrite <- Z2R_abs in H6.
apply le_Z2R in H6; pose proof (Zabs_spec (Int.unsigned x));
exfalso; now smart_omega.
Qed.

Lemma ox8000_0000_signed_unsigned:
forall x,
Int.unsigned (Int.add x ox8000_0000) = Int.signed x + Int.half_modulus.
Proof.
intro; unfold Int.signed, Int.add; pose proof (Int.unsigned_range x).
destruct (zlt (Int.unsigned x) Int.half_modulus).
rewrite Int.unsigned_repr; compute_this (Int.unsigned ox8000_0000); now smart_omega.
rewrite (Int.eqm_samerepr _ (Int.unsigned x + -2147483648)).
rewrite Int.unsigned_repr; now smart_omega.
apply Int.eqm_add; [now apply Int.eqm_refl|exists 1;reflexivity].
Qed.

Theorem floatofint_from_words:
forall x,
floatofint x =
sub (from_words ox4330_0000 (Int.add x ox8000_0000))
(from_words ox4330_0000 ox8000_0000).
Proof.
intros; destruct (Int.eq_dec x Int.zero); [subst; vm_compute; reflexivity|].
assert (Int.signed x <> 0).
intro; destruct n; rewrite <- (Int.repr_signed x), H; reflexivity.
pose proof (Int.signed_range x).
pose proof (binary_normalize64_exact (Int.signed x)); destruct H1; [now smart_omega|].
unfold floatofint, sub, b64_minus.
match goal with [|- _ = Bminus _ _ _ _ _ ?x ?y] =>
pose proof (Bminus_correct 53 1024 eq_refl eq_refl mode_NE x y) end.
apply (fun f x y => f x y) in H3; try apply (fun x => proj2 (from_words_value x)).
do 2 rewrite (fun x => proj1 (from_words_value x)) in H3.
(bpow radix2 52 + Z2R (Int.unsigned ox8000_0000)))%R with (Z2R (Int.signed x)) in H3
by (rewrite ox8000_0000_signed_unsigned; rewrite Z2R_plus; simpl; ring).
rewrite round_exact in H3 by smart_omega.
match goal with [H3:if Rlt_bool ?x ?y then _ else _ |- _] =>
pose proof (Rlt_bool_spec x y); destruct (Rlt_bool x y) end; destruct H3.
simpl Zcompare in *; apply B2R_inj;
try match goal with [H':B2R _ _ ?f = _ , H'':is_finite _ _ ?f = true |- is_finite_strict _ _ ?f = true] =>
destruct f; [
simpl in H'; change 0%R with (Z2R 0) in H'; apply eq_Z2R in H'; now destruct (H (eq_sym H')) |
discriminate H'' | discriminate H'' | reflexivity
]
end.
rewrite H3; assumption.
inversion H4; unfold bpow in H6; rewrite <- Z2R_abs in H6;
apply le_Z2R in H6; pose proof (Zabs_spec (Int.signed x)); exfalso; now smart_omega.
Qed.

Global Opaque
zero eq_dec neg abs singleoffloat intoffloat intuoffloat floatofint floatofintu
add sub mul div cmp bits_of_double double_of_bits bits_of_single single_of_bits from_words.

End Float.