From ba7a3de74288f8721699291d8fc2464c452498e1 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 7 Jan 2022 15:02:58 +0100
Subject: progress in proofs

---
 kvx/FPDivision64.v | 183 +++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 183 insertions(+)
 create mode 100644 kvx/FPDivision64.v

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
new file mode 100644
index 00000000..33a980c6
--- /dev/null
+++ b/kvx/FPDivision64.v
@@ -0,0 +1,183 @@
+From Flocq Require Import Core Digits Operations Round Bracket Sterbenz
+                          Binary Round_odd Bits.
+Require Archi.
+Require Import Coqlib.
+Require Import Compopts.
+Require Import AST.
+Require Import Integers.
+Require Import Floats.
+Require Import Op.
+Require Import CminorSel.
+Require Import OpHelpers.
+Require Import ExtFloats.
+Require Import DecBoolOps.
+Require Import Chunks.
+Require Import Builtins.
+Require Import Values Globalenvs.
+Require Compopts.
+Require Import Psatz.
+Require Import IEEE754_extra.
+
+From Gappa Require Import Gappa_tactic.
+
+Definition approx_inv_longu b :=
+  let invb_s := ExtValues.invfs (Val.maketotal (Val.singleoflongu b)) in
+  let invb_d := Val.floatofsingle invb_s in
+  let b_d := Val.maketotal (Val.floatoflongu b) in
+  let one := Vfloat (ExtFloat.one) in
+  let alpha := ExtValues.fmsubf one invb_d b_d in
+  ExtValues.fmaddf invb_d alpha invb_d.
+
+Definition approx_inv_thresh := (1/70368744177664)%R.
+
+Lemma Rabs_relax:
+  forall b b' (INEQ : (b < b')%R) x,
+    (-b <= x <= b)%R -> (Rabs x < b')%R.
+Proof.
+  intros.
+  apply Rabs_lt.
+  lra.
+Qed.
+
+Theorem approx_inv_longu_correct :
+  forall b,
+    (1 <= Int64.unsigned b <= 9223372036854775808)%Z ->
+    exists (f : float),
+      (approx_inv_longu (Vlong b)) = Vfloat f /\
+      is_finite _ _ f = true /\ (Rabs((B2R _ _ f) - (1 / IZR (Int64.unsigned b))) <= approx_inv_thresh)%R.
+Proof.
+  intros b NONZ.
+  unfold approx_inv_longu.
+  cbn.
+  econstructor.
+  split.
+  reflexivity.
+  Local Transparent Float.neg Float.of_single Float32.of_longu Float32.div Float.of_longu Float32.of_int Float.of_int.
+  unfold Float.fma, Float.neg, Float.of_single, Float32.of_longu, ExtFloat32.inv, Float32.div, Float.of_longu, ExtFloat32.one, Float32.of_int, ExtFloat.one, Float.of_int.
+  set (re := (@eq_refl Datatypes.comparison Lt)).
+  change (Int.signed (Int.repr 1)) with 1%Z.
+  set (b' := Int64.unsigned b) in *.
+  assert(1 <= IZR b' <= 9223372036854775808)%R as RANGE.
+  { split; apply IZR_le; lia.
+  }
+
+  assert (-16777216 <= 1 <= 16777216)%Z as SILLY by lia.
+  destruct (BofZ_exact 24 128 re re 1 SILLY) as (C0R & C0F & _).
+  clear SILLY.
+  set (one_s := (BofZ 24 128 re re 1)) in *.
+  
+  pose proof (BofZ_correct 24 128 re re b') as C1.
+  cbn in C1.
+  rewrite Rlt_bool_true in C1; cycle 1.
+  { clear C1.
+    eapply (Rabs_relax (IZR 9223372036854775808)).
+    lra.
+    set (b'' := IZR b') in *.
+    gappa.
+  }
+  destruct C1 as (C1R & C1F & _).
+  set (b_s :=  (BofZ 24 128 re re b')) in *.
+
+  assert(1 <= B2R 24 128 b_s <= 9223372036854775808)%R as b_s_RANGE.
+  { rewrite C1R.
+    gappa.
+  }
+  assert(B2R 24 128 b_s <> 0)%R as b_s_NONZ by lra.
+  
+  pose proof (Bdiv_correct 24 128 re re Float32.binop_nan mode_NE one_s b_s b_s_NONZ) as C2.
+  rewrite Rlt_bool_true in C2; cycle 1.
+  { clear C2.
+    apply Rabs_relax with (b := 1%R).
+    { cbn; lra. }
+    rewrite C0R.
+    set (r_b_s := B2R 24 128 b_s) in *.
+    cbn.
+    gappa.
+  }
+  
+  destruct C2 as (C2R & C2F & _).
+  set (invb_s := (Bdiv 24 128 re re Float32.binop_nan mode_NE one_s b_s)) in *.
+  rewrite C0F in C2F.
+
+  assert ((1/9223372036854775808 <= B2R 24 128 invb_s <= 1)%R) as invb_s_RANGE.
+  { rewrite C2R.
+    set (r_b_s := B2R 24 128 b_s) in *.
+    rewrite C0R.
+    cbn.
+    gappa.
+  }
+  
+  pose proof (Bconv_correct 24 128 53 1024 re re Float.of_single_nan mode_NE invb_s C2F) as C3.
+  rewrite Rlt_bool_true in C3; cycle 1.
+  { clear C3.
+    set (r_invb_s := (B2R 24 128 invb_s)) in *.
+    apply Rabs_relax with (b := 1%R).
+    { replace 1%R with (bpow radix2 0)%R by reflexivity.
+      apply bpow_lt.
+      lia.
+    }
+    cbn.
+    gappa.
+  }
+  
+  destruct C3 as (C3R & C3F & _).
+  set (invb_d :=  (Bconv 24 128 53 1024 re re Float.of_single_nan mode_NE invb_s)) in *.
+  assert ((1/9223372036854775808 <= B2R 53 1024 invb_d <= 1)%R) as invb_d_RANGE.
+  { 
+    rewrite C3R.
+    set (r_invb_s := B2R 24 128 invb_s) in *.
+    cbn.
+    gappa.
+  }
+
+  pose proof (is_finite_Bopp 53 1024 Float.neg_nan invb_d) as opp_finite.
+  rewrite C3F in opp_finite.
+
+  pose proof (BofZ_correct 53 1024 re re 1) as C4.
+  rewrite Rlt_bool_true in C4; cycle 1.
+  { clear C4.
+    cbn.
+    eapply (Rabs_relax (IZR 9223372036854775808)).
+    lra.
+    set (b'' := IZR b') in *.
+    gappa.
+  }
+  destruct C4 as (C4R & C4F & _).
+  
+  pose proof (BofZ_correct 53 1024 re re b') as C5.
+  cbn in C5.
+  rewrite Rlt_bool_true in C5; cycle 1.
+  { clear C5.
+    eapply (Rabs_relax (IZR 9223372036854775808)).
+    lra.
+    set (b'' := IZR b') in *.
+    gappa.
+  }
+  destruct C5 as (C5R & C5F & _).
+  set (b_d :=  (BofZ 53 1024 re re b')) in *.
+  
+  assert(1 <= B2R 53 1024 b_d <= 9223372036854775808)%R as b_d_RANGE.
+  { rewrite C5R.
+    gappa.
+  }
+
+  pose proof (Bfma_correct 53 1024 re re Float.fma_nan mode_NE
+          (Bopp 53 1024 Float.neg_nan invb_d) (BofZ 53 1024 re re b')
+          (BofZ 53 1024 re re 1) opp_finite C5F C4F) as C6.
+  rewrite Rlt_bool_true in C6; cycle 1.
+  { clear C6.
+    rewrite C4R.
+    rewrite B2R_Bopp.
+    cbn.
+    eapply (Rabs_relax (IZR 9223372036854775808)).
+    { lra. }
+    fold invb_d.
+    fold b_d.
+    set (r_invb_d := B2R 53 1024 invb_d) in *.
+    set (r_b_d := B2R 53 1024 b_d) in *.
+    
+    unfold invb_d.
+    unfold invb_s.
+    gappa.
+  }
+Admitted.
-- 
cgit 


From 4f8bd24d05bcea0a8b9793ee61450b1e0c7234c1 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 7 Jan 2022 15:17:12 +0100
Subject: this works

---
 kvx/FPDivision64.v | 31 ++++++++++++++++++++-----------
 1 file changed, 20 insertions(+), 11 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 33a980c6..424dfc77 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1,3 +1,12 @@
+(*
+This needs a special gappa script
+
+#!/bin/sh
+/home/monniaux/.opam/4.12.0+flambda/bin/gappa -Eprecision=100 "$@"
+
+in PATH before the normal gappa
+ *)
+
 From Flocq Require Import Core Digits Operations Round Bracket Sterbenz
                           Binary Round_odd Bits.
 Require Archi.
@@ -38,10 +47,10 @@ Proof.
   apply Rabs_lt.
   lra.
 Qed.
-
+    
 Theorem approx_inv_longu_correct :
   forall b,
-    (1 <= Int64.unsigned b <= 9223372036854775808)%Z ->
+    (1 <= Int64.unsigned b <= 18446744073709551616)%Z ->
     exists (f : float),
       (approx_inv_longu (Vlong b)) = Vfloat f /\
       is_finite _ _ f = true /\ (Rabs((B2R _ _ f) - (1 / IZR (Int64.unsigned b))) <= approx_inv_thresh)%R.
@@ -57,7 +66,7 @@ Proof.
   set (re := (@eq_refl Datatypes.comparison Lt)).
   change (Int.signed (Int.repr 1)) with 1%Z.
   set (b' := Int64.unsigned b) in *.
-  assert(1 <= IZR b' <= 9223372036854775808)%R as RANGE.
+  assert(1 <= IZR b' <= 18446744073709551616)%R as RANGE.
   { split; apply IZR_le; lia.
   }
 
@@ -70,7 +79,7 @@ Proof.
   cbn in C1.
   rewrite Rlt_bool_true in C1; cycle 1.
   { clear C1.
-    eapply (Rabs_relax (IZR 9223372036854775808)).
+    eapply (Rabs_relax (IZR 18446744073709551616)).
     lra.
     set (b'' := IZR b') in *.
     gappa.
@@ -78,7 +87,7 @@ Proof.
   destruct C1 as (C1R & C1F & _).
   set (b_s :=  (BofZ 24 128 re re b')) in *.
 
-  assert(1 <= B2R 24 128 b_s <= 9223372036854775808)%R as b_s_RANGE.
+  assert(1 <= B2R 24 128 b_s <= 18446744073709551616)%R as b_s_RANGE.
   { rewrite C1R.
     gappa.
   }
@@ -99,7 +108,7 @@ Proof.
   set (invb_s := (Bdiv 24 128 re re Float32.binop_nan mode_NE one_s b_s)) in *.
   rewrite C0F in C2F.
 
-  assert ((1/9223372036854775808 <= B2R 24 128 invb_s <= 1)%R) as invb_s_RANGE.
+  assert ((1/18446744073709551616 <= B2R 24 128 invb_s <= 1)%R) as invb_s_RANGE.
   { rewrite C2R.
     set (r_b_s := B2R 24 128 b_s) in *.
     rewrite C0R.
@@ -122,7 +131,7 @@ Proof.
   
   destruct C3 as (C3R & C3F & _).
   set (invb_d :=  (Bconv 24 128 53 1024 re re Float.of_single_nan mode_NE invb_s)) in *.
-  assert ((1/9223372036854775808 <= B2R 53 1024 invb_d <= 1)%R) as invb_d_RANGE.
+  assert ((1/18446744073709551616 <= B2R 53 1024 invb_d <= 1)%R) as invb_d_RANGE.
   { 
     rewrite C3R.
     set (r_invb_s := B2R 24 128 invb_s) in *.
@@ -137,7 +146,7 @@ Proof.
   rewrite Rlt_bool_true in C4; cycle 1.
   { clear C4.
     cbn.
-    eapply (Rabs_relax (IZR 9223372036854775808)).
+    eapply (Rabs_relax (IZR 18446744073709551616)).
     lra.
     set (b'' := IZR b') in *.
     gappa.
@@ -148,7 +157,7 @@ Proof.
   cbn in C5.
   rewrite Rlt_bool_true in C5; cycle 1.
   { clear C5.
-    eapply (Rabs_relax (IZR 9223372036854775808)).
+    eapply (Rabs_relax (IZR 18446744073709551616)).
     lra.
     set (b'' := IZR b') in *.
     gappa.
@@ -156,7 +165,7 @@ Proof.
   destruct C5 as (C5R & C5F & _).
   set (b_d :=  (BofZ 53 1024 re re b')) in *.
   
-  assert(1 <= B2R 53 1024 b_d <= 9223372036854775808)%R as b_d_RANGE.
+  assert(1 <= B2R 53 1024 b_d <= 18446744073709551616)%R as b_d_RANGE.
   { rewrite C5R.
     gappa.
   }
@@ -169,7 +178,7 @@ Proof.
     rewrite C4R.
     rewrite B2R_Bopp.
     cbn.
-    eapply (Rabs_relax (IZR 9223372036854775808)).
+    eapply (Rabs_relax (IZR 18446744073709551616)).
     { lra. }
     fold invb_d.
     fold b_d.
-- 
cgit 


From af9f4682a8e7f06fdf9eba707d008bc21873d1ce Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 7 Jan 2022 15:27:56 +0100
Subject: progress in proof

---
 kvx/FPDivision64.v | 23 ++++++++++++++++++++---
 1 file changed, 20 insertions(+), 3 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 424dfc77..a44de06e 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -184,9 +184,26 @@ Proof.
     fold b_d.
     set (r_invb_d := B2R 53 1024 invb_d) in *.
     set (r_b_d := B2R 53 1024 b_d) in *.
-    
-    unfold invb_d.
-    unfold invb_s.
     gappa.
   }
+  fold b_d in C6.
+  destruct C6 as (C6R & C6F & _).
+
+  pose proof (Bfma_correct 53 1024 re re Float.fma_nan mode_NE
+       (Bfma 53 1024 re re Float.fma_nan mode_NE
+          (Bopp 53 1024 Float.neg_nan invb_d) b_d (BofZ 53 1024 re re 1))
+       invb_d invb_d C6F C3F C3F) as C7.
+  rewrite Rlt_bool_true in C7; cycle 1.
+  { clear C7.
+    rewrite C6R.
+    rewrite B2R_Bopp.
+    eapply (Rabs_relax (bpow radix2 64)).
+    { apply bpow_lt. lia. }
+    rewrite C4R.
+    cbn.
+    set (r_invb_d := B2R 53 1024 invb_d) in *.
+    set (r_b_d := B2R 53 1024 b_d) in *.
+    gappa.
+  }
+  
 Admitted.
-- 
cgit 


From b11fd5400cb042f4b0727e08d2fa53929092f74c Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 7 Jan 2022 15:32:51 +0100
Subject: progress

---
 kvx/FPDivision64.v | 3 +++
 1 file changed, 3 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index a44de06e..851a6cc4 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -205,5 +205,8 @@ Proof.
     set (r_b_d := B2R 53 1024 b_d) in *.
     gappa.
   }
+  destruct C7 as (C7R & C7F & _).
+
+  split. assumption.
   
 Admitted.
-- 
cgit 


From cd444fa2200427a1e792716ad5cbac9c1eac872e Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 7 Jan 2022 19:12:03 +0100
Subject: progress in proofs

---
 kvx/FPDivision64.v | 34 ++++++++++++++++++++++++++++++++++
 1 file changed, 34 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 851a6cc4..f61cc48c 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -208,5 +208,39 @@ Proof.
   destruct C7 as (C7R & C7F & _).
 
   split. assumption.
+  rewrite C7R.
+  rewrite C6R.
+  rewrite C5R.
+  rewrite C4R.
+  rewrite B2R_Bopp.
+  rewrite C3R.
+  rewrite C2R.
+  rewrite C1R.
+  rewrite C0R.
+  cbn.
+  set(b1 := IZR b') in *.
+  replace (round radix2 (FLT_exp (-1074) 53) ZnearestE 1) with 1%R by gappa.
+  set (bd := round radix2 (FLT_exp (-1074) 53) ZnearestE b1).
+  set (x0 := round radix2 (FLT_exp (-1074) 53) ZnearestE
+         (round radix2 (FLT_exp (-149) 24) ZnearestE
+                    (1 / round radix2 (FLT_exp (-149) 24) ZnearestE b1))).
+  set (alpha0 := (- x0 * bd + 1)%R).
+  set (y1 := (round radix2 (FLT_exp (-1074) 53) ZnearestE alpha0 * x0 + x0)%R).
+  set (x1 := round radix2 (FLT_exp (-1074) 53) ZnearestE y1).
+  replace (x1 - 1/b1)%R with ((y1-1/b1)+(x1-y1))%R by ring.
+
+  assert(alpha0 = -((x0-1/bd)/(1/bd)))%R as alpha0_EQ.
+  { unfold alpha0.
+    field.
+    unfold bd.
+    gappa.
+  }
+  assert(y1-1/b1 = ((round radix2 (FLT_exp (-1074) 53) ZnearestE alpha0)
+                   - alpha0) * x0
+                   + alpha0*(x0-1/b1) - ((bd-b1)/b1) * x0)%R as y1_EQ.
+  { unfold y1, alpha0.
+    field.
+    lra.
+  }
   
 Admitted.
-- 
cgit 


From cf534a7a5144f4dadc0b98fc078cd2b11530650f Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 7 Jan 2022 23:02:33 +0100
Subject: qed for proof

---
 kvx/FPDivision64.v | 39 +++++++++++++++++++++++++++++++++++++--
 1 file changed, 37 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index f61cc48c..608b3710 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -242,5 +242,40 @@ Proof.
     field.
     lra.
   }
-  
-Admitted.
+  assert(Rabs alpha0 <= 257/2147483648)%R as alpha0_ABS.
+  { rewrite alpha0_EQ.
+    unfold x0, bd.
+    gappa.
+  }
+  assert (Rabs (x0 - 1 / b1) <= 3/33554432)%R as x0_delta_ABS.
+  { unfold x0.
+    gappa.
+  }
+  set (x0_delta := (x0 - 1 / b1)%R) in *.
+  assert (Rabs ((bd - b1) / b1) <= 1/9007199254740992)%R as bd_delta_ABS.
+  { unfold bd.
+    gappa.
+  }
+  set (bd_delta := ((bd - b1) / b1)%R) in *.
+  set (rnd_alpha0_delta := (round radix2 (FLT_exp (-1074) 53) ZnearestE alpha0 - alpha0)%R) in *.
+  assert (Rabs rnd_alpha0_delta <= 1/75557863725914323419136)%R as rnd_alpha0_delta_ABS.
+  { unfold rnd_alpha0_delta.
+    gappa.
+  }
+  assert (1/18446744073709551616 <= x0 <= 1)%R as x0_RANGE.
+  { unfold x0.
+    gappa.
+  }
+  assert (Rabs (y1 - 1 / b1) <= 49/4503599627370496)%R as y1_delta_ABS.
+  { rewrite y1_EQ.
+    gappa.
+  }
+  assert (Rabs(x1 - y1) <= 1/9007199254740992)%R as x1_delta_ABS.
+  { unfold x1.
+    gappa.
+  }
+  set (y1_delta := (y1 - 1 / b1)%R) in *.
+  set (x1_delta := (x1-y1)%R) in *.
+  unfold approx_inv_thresh.
+  gappa.
+Qed.
-- 
cgit 


From bfbd4b7235fe67cc3fa023bcdec88d89ca6f3c99 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 7 Jan 2022 23:05:19 +0100
Subject: nice intervals

---
 kvx/FPDivision64.v | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 608b3710..ffcc1ef9 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -50,7 +50,7 @@ Qed.
     
 Theorem approx_inv_longu_correct :
   forall b,
-    (1 <= Int64.unsigned b <= 18446744073709551616)%Z ->
+    (0 < Int64.unsigned b)%Z ->
     exists (f : float),
       (approx_inv_longu (Vlong b)) = Vfloat f /\
       is_finite _ _ f = true /\ (Rabs((B2R _ _ f) - (1 / IZR (Int64.unsigned b))) <= approx_inv_thresh)%R.
@@ -66,7 +66,9 @@ Proof.
   set (re := (@eq_refl Datatypes.comparison Lt)).
   change (Int.signed (Int.repr 1)) with 1%Z.
   set (b' := Int64.unsigned b) in *.
-  assert(1 <= IZR b' <= 18446744073709551616)%R as RANGE.
+  pose proof (Int64.unsigned_range b) as RANGE.
+  change Int64.modulus with 18446744073709551616%Z in RANGE.                                              
+  assert(1 <= IZR b' <= 18446744073709551616)%R as RANGE'.
   { split; apply IZR_le; lia.
   }
 
-- 
cgit 


From 41838c656dbbf817446af24b01eac8c071bafda7 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 7 Jan 2022 23:09:14 +0100
Subject: better bound

---
 kvx/FPDivision64.v | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index ffcc1ef9..ab593591 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -37,7 +37,8 @@ Definition approx_inv_longu b :=
   let alpha := ExtValues.fmsubf one invb_d b_d in
   ExtValues.fmaddf invb_d alpha invb_d.
 
-Definition approx_inv_thresh := (1/70368744177664)%R.
+Definition approx_inv_thresh := (25/2251799813685248)%R.
+(* 1.11022302462516e-14 *)
 
 Lemma Rabs_relax:
   forall b b' (INEQ : (b < b')%R) x,
-- 
cgit 


From 120446974d1a83e6f9ef99f049d597166be09271 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 10 Jan 2022 14:18:07 +0100
Subject: rough approx

---
 kvx/FPDivision64.v | 116 +++++++++++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 113 insertions(+), 3 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index ab593591..6c053d7c 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -37,9 +37,6 @@ Definition approx_inv_longu b :=
   let alpha := ExtValues.fmsubf one invb_d b_d in
   ExtValues.fmaddf invb_d alpha invb_d.
 
-Definition approx_inv_thresh := (25/2251799813685248)%R.
-(* 1.11022302462516e-14 *)
-
 Lemma Rabs_relax:
   forall b b' (INEQ : (b < b')%R) x,
     (-b <= x <= b)%R -> (Rabs x < b')%R.
@@ -48,6 +45,9 @@ Proof.
   apply Rabs_lt.
   lra.
 Qed.
+
+Definition approx_inv_thresh := (25/2251799813685248)%R.
+(* 1.11022302462516e-14 *)
     
 Theorem approx_inv_longu_correct :
   forall b,
@@ -282,3 +282,113 @@ Proof.
   unfold approx_inv_thresh.
   gappa.
 Qed.
+
+Definition rough_approx_inv_longu b :=
+  let invb_s := ExtValues.invfs (Val.maketotal (Val.singleoflongu b)) in
+  Val.floatofsingle invb_s.
+
+Definition rough_approx_inv_thresh := (3/33554432)%R.
+(* 8.94069671630859e-8 *)
+    
+Theorem rough_approx_inv_longu_correct :
+  forall b,
+    (0 < Int64.unsigned b)%Z ->
+    exists (f : float),
+      (rough_approx_inv_longu (Vlong b)) = Vfloat f /\
+      is_finite _ _ f = true /\ (Rabs((B2R _ _ f) - (1 / IZR (Int64.unsigned b))) <= rough_approx_inv_thresh)%R.
+Proof.
+  intros b NONZ.
+  unfold rough_approx_inv_longu.
+  cbn.
+  econstructor.
+  split.
+  reflexivity.
+  Local Transparent Float.neg Float.of_single Float32.of_longu Float32.div Float.of_longu Float32.of_int Float.of_int.
+  unfold Float.fma, Float.neg, Float.of_single, Float32.of_longu, ExtFloat32.inv, Float32.div, Float.of_longu, ExtFloat32.one, Float32.of_int, ExtFloat.one, Float.of_int.
+  set (re := (@eq_refl Datatypes.comparison Lt)).
+  change (Int.signed (Int.repr 1)) with 1%Z.
+  set (b' := Int64.unsigned b) in *.
+  pose proof (Int64.unsigned_range b) as RANGE.
+  change Int64.modulus with 18446744073709551616%Z in RANGE.                                              
+  assert(1 <= IZR b' <= 18446744073709551616)%R as RANGE'.
+  { split; apply IZR_le; lia.
+  }
+
+  assert (-16777216 <= 1 <= 16777216)%Z as SILLY by lia.
+  destruct (BofZ_exact 24 128 re re 1 SILLY) as (C0R & C0F & _).
+  clear SILLY.
+  set (one_s := (BofZ 24 128 re re 1)) in *.
+  
+  pose proof (BofZ_correct 24 128 re re b') as C1.
+  cbn in C1.
+  rewrite Rlt_bool_true in C1; cycle 1.
+  { clear C1.
+    eapply (Rabs_relax (IZR 18446744073709551616)).
+    lra.
+    set (b'' := IZR b') in *.
+    gappa.
+  }
+  destruct C1 as (C1R & C1F & _).
+  set (b_s :=  (BofZ 24 128 re re b')) in *.
+
+  assert(1 <= B2R 24 128 b_s <= 18446744073709551616)%R as b_s_RANGE.
+  { rewrite C1R.
+    gappa.
+  }
+  assert(B2R 24 128 b_s <> 0)%R as b_s_NONZ by lra.
+  
+  pose proof (Bdiv_correct 24 128 re re Float32.binop_nan mode_NE one_s b_s b_s_NONZ) as C2.
+  rewrite Rlt_bool_true in C2; cycle 1.
+  { clear C2.
+    apply Rabs_relax with (b := 1%R).
+    { cbn; lra. }
+    rewrite C0R.
+    set (r_b_s := B2R 24 128 b_s) in *.
+    cbn.
+    gappa.
+  }
+  
+  destruct C2 as (C2R & C2F & _).
+  set (invb_s := (Bdiv 24 128 re re Float32.binop_nan mode_NE one_s b_s)) in *.
+  rewrite C0F in C2F.
+
+  assert ((1/18446744073709551616 <= B2R 24 128 invb_s <= 1)%R) as invb_s_RANGE.
+  { rewrite C2R.
+    set (r_b_s := B2R 24 128 b_s) in *.
+    rewrite C0R.
+    cbn.
+    gappa.
+  }
+  
+  pose proof (Bconv_correct 24 128 53 1024 re re Float.of_single_nan mode_NE invb_s C2F) as C3.
+  rewrite Rlt_bool_true in C3; cycle 1.
+  { clear C3.
+    set (r_invb_s := (B2R 24 128 invb_s)) in *.
+    apply Rabs_relax with (b := 1%R).
+    { replace 1%R with (bpow radix2 0)%R by reflexivity.
+      apply bpow_lt.
+      lia.
+    }
+    cbn.
+    gappa.
+  }
+  
+  destruct C3 as (C3R & C3F & _).
+  set (invb_d :=  (Bconv 24 128 53 1024 re re Float.of_single_nan mode_NE invb_s)) in *.
+  assert ((1/18446744073709551616 <= B2R 53 1024 invb_d <= 1)%R) as invb_d_RANGE.
+  { 
+    rewrite C3R.
+    set (r_invb_s := B2R 24 128 invb_s) in *.
+    cbn.
+    gappa.
+  }
+
+  split. assumption.
+  unfold rough_approx_inv_thresh.
+  rewrite C3R.
+  rewrite C2R.
+  rewrite C1R.
+  rewrite C0R.
+  cbn.
+  gappa.
+Qed.
-- 
cgit 


From f411b8eb07064070440169269d9f402b3cdbb009 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 10 Jan 2022 14:55:20 +0100
Subject: state theorem

---
 kvx/FPDivision64.v | 15 +++++++++++++++
 1 file changed, 15 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 6c053d7c..9baa3f94 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -392,3 +392,18 @@ Proof.
   cbn.
   gappa.
 Qed.
+
+Definition rough_approx_div_longu a b :=
+  Val.maketotal (Val.longuoffloat
+    (Val.mulf (Val.maketotal (Val.floatoflongu a)) (rough_approx_inv_longu b))).
+
+Definition rough_approx_div_thresh := 1649267441663%Z.
+
+Theorem rough_approx_div_longu_correct :
+  forall a b,
+    (0 < Int64.unsigned b)%Z ->
+    exists (q : int64),
+      (rough_approx_div_longu (Vlong a) (Vlong b)) = Vlong q /\
+        (Z.abs ((Int64.unsigned a) - (Int64.unsigned q) * (Int64.unsigned b)) <= rough_approx_div_thresh)%Z.
+Proof.
+Admitted.
-- 
cgit 


From c31acf1e0448b9c6d0a6b175d0ac605a9b1b82ba Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 10 Jan 2022 15:17:10 +0100
Subject: progress in proofs

---
 kvx/FPDivision64.v | 25 +++++++++++++++++++++++++
 1 file changed, 25 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 9baa3f94..e025fa94 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -406,4 +406,29 @@ Theorem rough_approx_div_longu_correct :
       (rough_approx_div_longu (Vlong a) (Vlong b)) = Vlong q /\
         (Z.abs ((Int64.unsigned a) - (Int64.unsigned q) * (Int64.unsigned b)) <= rough_approx_div_thresh)%Z.
 Proof.
+  intros a b b_RANGE.
+  unfold rough_approx_div_longu.
+  destruct (rough_approx_inv_longu_correct b b_RANGE) as (f & C1R & C1F & C1D).
+  rewrite C1R.
+  cbn.
+  Local Transparent Float.to_longu Float.mul.
+  unfold Float.to_longu, Float.mul, Float.of_longu.
+  set (re := (@eq_refl Datatypes.comparison Lt)).
+
+  pose proof (Int64.unsigned_range a) as a_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in a_RANGE.
+  set (a' := Int64.unsigned a) in *.
+  assert (0 <= IZR a' <= 18446744073709551615)%R as IZR_a_RANGE.
+  { split ; apply IZR_le; lia.
+  }
+
+  pose proof (BofZ_correct 53 1024 re re a') as C2.
+  rewrite Rlt_bool_true in C2; cycle 1.
+  { clear C2.
+    apply Rabs_relax with (b := bpow radix2 64).
+    { apply bpow_lt. lia. }
+    cbn.
+    gappa.
+  }
+  destruct C2 as (C2R & C2F & _).
 Admitted.
-- 
cgit 


From b7875749ae445c43130fabd9eb9099eddea48318 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 10 Jan 2022 15:31:39 +0100
Subject: proof progress

---
 kvx/FPDivision64.v | 26 ++++++++++++++++++++++++--
 1 file changed, 24 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index e025fa94..9463313a 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -406,11 +406,12 @@ Theorem rough_approx_div_longu_correct :
       (rough_approx_div_longu (Vlong a) (Vlong b)) = Vlong q /\
         (Z.abs ((Int64.unsigned a) - (Int64.unsigned q) * (Int64.unsigned b)) <= rough_approx_div_thresh)%Z.
 Proof.
-  intros a b b_RANGE.
+  intros a b b_NONZ.
   unfold rough_approx_div_longu.
-  destruct (rough_approx_inv_longu_correct b b_RANGE) as (f & C1R & C1F & C1D).
+  destruct (rough_approx_inv_longu_correct b b_NONZ) as (f & C1R & C1F & C1D).
   rewrite C1R.
   cbn.
+
   Local Transparent Float.to_longu Float.mul.
   unfold Float.to_longu, Float.mul, Float.of_longu.
   set (re := (@eq_refl Datatypes.comparison Lt)).
@@ -422,6 +423,13 @@ Proof.
   { split ; apply IZR_le; lia.
   }
 
+  pose proof (Int64.unsigned_range b) as b_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in b_RANGE.
+  set (b' := Int64.unsigned b) in *.
+  assert (1 <= IZR b' <= 18446744073709551615)%R as IZR_b_RANGE.
+  { split ; apply IZR_le; lia.
+  }
+
   pose proof (BofZ_correct 53 1024 re re a') as C2.
   rewrite Rlt_bool_true in C2; cycle 1.
   { clear C2.
@@ -431,4 +439,18 @@ Proof.
     gappa.
   }
   destruct C2 as (C2R & C2F & _).
+
+  pose proof (Bmult_correct  53 1024 re re Float.binop_nan mode_NE
+                             (BofZ 53 1024 re re a') f) as C3.
+  rewrite Rlt_bool_true in C3; cycle 1.
+  { clear C3.
+    rewrite C2R.
+    replace (B2R 53 1024 f)%R with ((B2R 53 1024 f - 1 / IZR b') + 1 / IZR b')%R by ring.
+    unfold rough_approx_inv_thresh in C1D.
+    apply Rabs_relax with (b := bpow radix2 65).
+    { apply bpow_lt ; lia. }
+    cbn.
+    gappa.
+  }
+  
 Admitted.
-- 
cgit 


From f42abd286e90cad92174cd77b2dd068ed8dd11ae Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 10 Jan 2022 15:49:39 +0100
Subject: wrong operator

---
 kvx/FPDivision64.v | 14 ++++++++++++--
 1 file changed, 12 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 9463313a..9cde7d97 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -394,7 +394,7 @@ Proof.
 Qed.
 
 Definition rough_approx_div_longu a b :=
-  Val.maketotal (Val.longuoffloat
+  Val.maketotal (Val.longuoffloat_ne
     (Val.mulf (Val.maketotal (Val.floatoflongu a)) (rough_approx_inv_longu b))).
 
 Definition rough_approx_div_thresh := 1649267441663%Z.
@@ -413,7 +413,7 @@ Proof.
   cbn.
 
   Local Transparent Float.to_longu Float.mul.
-  unfold Float.to_longu, Float.mul, Float.of_longu.
+  unfold Float.to_longu_ne, Float.mul, Float.of_longu.
   set (re := (@eq_refl Datatypes.comparison Lt)).
 
   pose proof (Int64.unsigned_range a) as a_RANGE.
@@ -452,5 +452,15 @@ Proof.
     cbn.
     gappa.
   }
+  rewrite C1F in C3.
+  rewrite C2F in C3.
+  cbn in C3.
+  destruct C3 as (C3R & C3F & _).
+
+  pose proof (ZofB_ne_range_correct 53 1024
+               (Bmult 53 1024 re re Float.binop_nan mode_NE
+                      (BofZ 53 1024 re re a') f) 0 Int64.max_unsigned) as C4.
+  rewrite C3F in C4.
+  rewrite C3R in C4.
   
 Admitted.
-- 
cgit 


From a0303be2ae9b7b53f012a74c3e42020acb225fb9 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 10 Jan 2022 17:22:18 +0100
Subject: signs

---
 kvx/FPDivision64.v | 36 +++++++++++++++++++++++++++++++-----
 1 file changed, 31 insertions(+), 5 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 9cde7d97..1dbf2782 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -295,7 +295,9 @@ Theorem rough_approx_inv_longu_correct :
     (0 < Int64.unsigned b)%Z ->
     exists (f : float),
       (rough_approx_inv_longu (Vlong b)) = Vfloat f /\
-      is_finite _ _ f = true /\ (Rabs((B2R _ _ f) - (1 / IZR (Int64.unsigned b))) <= rough_approx_inv_thresh)%R.
+        is_finite _ _ f = true /\
+        (Rabs((B2R _ _ f) - (1 / IZR (Int64.unsigned b))) <= rough_approx_inv_thresh)%R (* /\
+        Bsign _ _ f = false. *)
 Proof.
   intros b NONZ.
   unfold rough_approx_inv_longu.
@@ -328,7 +330,8 @@ Proof.
     set (b'' := IZR b') in *.
     gappa.
   }
-  destruct C1 as (C1R & C1F & _).
+  rewrite (Zlt_bool_false b' 0) in C1 by lia.
+  destruct C1 as (C1R & C1F & C1S).
   set (b_s :=  (BofZ 24 128 re re b')) in *.
 
   assert(1 <= B2R 24 128 b_s <= 18446744073709551616)%R as b_s_RANGE.
@@ -347,10 +350,19 @@ Proof.
     cbn.
     gappa.
   }
-  
-  destruct C2 as (C2R & C2F & _).
+  destruct C2 as (C2R & C2F & C2Sz).
+  rewrite C1S in C2Sz.
+  change (xorb _ _) with false in C2Sz.
   set (invb_s := (Bdiv 24 128 re re Float32.binop_nan mode_NE one_s b_s)) in *.
   rewrite C0F in C2F.
+  assert (is_nan 24 128 invb_s = false) as NAN.
+  { apply is_finite_not_is_nan.
+    assumption.
+  }
+  pose proof (C2Sz NAN) as C2S.
+  clear C2Sz.
+  
+  apply is_finite_not_is_nan in C2S.
 
   assert ((1/18446744073709551616 <= B2R 24 128 invb_s <= 1)%R) as invb_s_RANGE.
   { rewrite C2R.
@@ -462,5 +474,19 @@ Proof.
                       (BofZ 53 1024 re re a') f) 0 Int64.max_unsigned) as C4.
   rewrite C3F in C4.
   rewrite C3R in C4.
-  
+  rewrite C2R in C4.
+
+  assert(0 <=
+      (round radix2 (FLT_exp (-1074) 53) ZnearestE
+              (round radix2 (FLT_exp (3 - 1024 - 53) 53) 
+                 (round_mode mode_NE) (IZR a') * B2R 53 1024 f))
+         <= IZR Int64.max_unsigned)%R as q_RANGE.
+  { clear C4.
+    change (IZR Int64.max_unsigned) with 18446744073709551615%R.
+    cbn.
+    replace (B2R 53 1024 f)%R with ((B2R 53 1024 f - 1 / IZR b') + 1 / IZR b')%R by ring.
+    unfold rough_approx_inv_thresh in C1D.
+    set (delta :=  (B2R 53 1024 f - 1 / IZR b')%R) in *.
+    gappa.
+  }
 Admitted.
-- 
cgit 


From e1e99fcf500c8c9822a017258acb9c63fa40229f Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 10 Jan 2022 18:23:42 +0100
Subject: some more proof

---
 kvx/FPDivision64.v | 54 +++++++++++++++++++++++++++++++++++++++++++++++-------
 1 file changed, 47 insertions(+), 7 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 1dbf2782..4d9f0537 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -296,8 +296,8 @@ Theorem rough_approx_inv_longu_correct :
     exists (f : float),
       (rough_approx_inv_longu (Vlong b)) = Vfloat f /\
         is_finite _ _ f = true /\
-        (Rabs((B2R _ _ f) - (1 / IZR (Int64.unsigned b))) <= rough_approx_inv_thresh)%R (* /\
-        Bsign _ _ f = false. *)
+        (Rabs((B2R _ _ f) - (1 / IZR (Int64.unsigned b))) <= rough_approx_inv_thresh)%R /\
+        Bsign _ _ f = false.
 Proof.
   intros b NONZ.
   unfold rough_approx_inv_longu.
@@ -362,8 +362,6 @@ Proof.
   pose proof (C2Sz NAN) as C2S.
   clear C2Sz.
   
-  apply is_finite_not_is_nan in C2S.
-
   assert ((1/18446744073709551616 <= B2R 24 128 invb_s <= 1)%R) as invb_s_RANGE.
   { rewrite C2R.
     set (r_b_s := B2R 24 128 b_s) in *.
@@ -385,7 +383,7 @@ Proof.
     gappa.
   }
   
-  destruct C3 as (C3R & C3F & _).
+  destruct C3 as (C3R & C3F & C3S).
   set (invb_d :=  (Bconv 24 128 53 1024 re re Float.of_single_nan mode_NE invb_s)) in *.
   assert ((1/18446744073709551616 <= B2R 53 1024 invb_d <= 1)%R) as invb_d_RANGE.
   { 
@@ -394,8 +392,9 @@ Proof.
     cbn.
     gappa.
   }
+  rewrite C2S in C3S.
 
-  split. assumption.
+  split. assumption. split. 2: assumption.
   unfold rough_approx_inv_thresh.
   rewrite C3R.
   rewrite C2R.
@@ -411,6 +410,22 @@ Definition rough_approx_div_longu a b :=
 
 Definition rough_approx_div_thresh := 1649267441663%Z.
 
+Lemma Bsign_false_nonneg:
+  forall prec emax f,
+    (Bsign prec emax f) = false -> (0 <= (B2R prec emax f))%R.
+Proof.
+  intros until f. intro SIGN.
+  destruct f.
+  1, 2, 3: cbn; lra.
+  cbn.
+  apply F2R_ge_0.
+  cbn.
+  cbn in SIGN.
+  rewrite SIGN.
+  cbn.
+  lia.
+Qed.
+
 Theorem rough_approx_div_longu_correct :
   forall a b,
     (0 < Int64.unsigned b)%Z ->
@@ -420,7 +435,7 @@ Theorem rough_approx_div_longu_correct :
 Proof.
   intros a b b_NONZ.
   unfold rough_approx_div_longu.
-  destruct (rough_approx_inv_longu_correct b b_NONZ) as (f & C1R & C1F & C1D).
+  destruct (rough_approx_inv_longu_correct b b_NONZ) as (f & C1R & C1F & C1D & C1S).
   rewrite C1R.
   cbn.
 
@@ -435,6 +450,12 @@ Proof.
   { split ; apply IZR_le; lia.
   }
 
+  set (a_d := (round radix2 (FLT_exp (-1074) 53) ZnearestE (IZR a'))).
+  assert (0 <= a_d <= 18446744073709551616)%R as a_d_RANGE.
+  { unfold a_d.
+    gappa.
+  }
+
   pose proof (Int64.unsigned_range b) as b_RANGE.
   change Int64.modulus with 18446744073709551616%Z in b_RANGE.
   set (b' := Int64.unsigned b) in *.
@@ -442,6 +463,19 @@ Proof.
   { split ; apply IZR_le; lia.
   }
 
+  set (f_r := (B2R _ _ f)) in *.
+  assert (0 <= f_r <= 1)%R as f_RANGE.
+  { split.
+    { apply Bsign_false_nonneg. trivial .}
+    unfold rough_approx_inv_thresh in C1D.
+    replace f_r with ((f_r - 1 / IZR b') + 1/ IZR b')%R by (field; lra).
+    set (delta := (f_r - 1 / IZR b')%R) in *.
+    destruct (Z_le_gt_dec b' 1) as [LE | GT].
+    { admit. }
+    assert (2 <= IZR b')%R as NOT1.
+    { apply IZR_le. lia. }
+    gappa.
+  }
   pose proof (BofZ_correct 53 1024 re re a') as C2.
   rewrite Rlt_bool_true in C2; cycle 1.
   { clear C2.
@@ -482,7 +516,13 @@ Proof.
                  (round_mode mode_NE) (IZR a') * B2R 53 1024 f))
          <= IZR Int64.max_unsigned)%R as q_RANGE.
   { clear C4.
+    cbn.
     change (IZR Int64.max_unsigned) with 18446744073709551615%R.
+    fold a_d.
+    gappa.
+    set (y := ((IZR a') * B2R 53 1024 f)%R).
+    assert ((round radix2 (FLT_exp (-1074) 53) ZnearestE (IZR a') * B2R 53 1024 f) >= 0)%R.
+   
     cbn.
     replace (B2R 53 1024 f)%R with ((B2R 53 1024 f - 1 / IZR b') + 1 / IZR b')%R by ring.
     unfold rough_approx_inv_thresh in C1D.
-- 
cgit 


From c6e3d6b38c7cc62877d85f561b69f8651a5f4a3f Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 10 Jan 2022 19:07:15 +0100
Subject: some more proofs?

---
 kvx/FPDivision64.v | 67 ++++++++++++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 65 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 4d9f0537..521cbe0f 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -404,6 +404,62 @@ Proof.
   gappa.
 Qed.
 
+Theorem rough_approx_inv_longu_correct1 :
+  forall b,
+    (Int64.unsigned b = 1%Z) ->
+    exists f,
+    (rough_approx_inv_longu (Vlong b)) = Vfloat f /\
+        is_finite _ _ f = true /\
+        (B2R _ _ f) = 1%R /\
+        Bsign _ _ f = false.
+Proof.
+  intros b ONE.
+  cbn.
+  unfold Float.of_single, ExtFloat32.inv, Float32.div, ExtFloat32.one, Float32.of_longu, Float32.of_int.
+  set (re := (@eq_refl Datatypes.comparison Lt)).
+  rewrite ONE.
+  change (Int.signed (Int.repr 1)) with 1%Z.
+  
+  econstructor.
+  split. reflexivity.
+  split. reflexivity.
+  split. 2: reflexivity.
+  destruct (BofZ_exact 24 128 re re 1%Z) as (C1R & C1F & C1S). lia.
+
+  pose proof (Bdiv_correct 24 128 re re Float32.binop_nan mode_NE 
+                           (BofZ 24 128 re re 1) (BofZ 24 128 re re 1)) as C2.
+  rewrite C1R in C2.
+  rewrite C1F in C2.
+  rewrite C1S in C2.
+  rewrite Rlt_bool_true in C2; cycle 1.
+  { clear C2.
+    apply Rabs_relax with (b := bpow radix2 0%Z).
+    { apply bpow_lt. lia. }
+    cbn.
+    gappa.
+  }
+  assert (1 <> 0)%R as OBVIOUS by lra.
+  destruct (C2 OBVIOUS) as (C2R & C2F & C2S).
+  clear C2.
+
+  pose proof (Bconv_correct 24 128 53 1024 re re Float.of_single_nan mode_NE
+       (Bdiv 24 128 re re Float32.binop_nan mode_NE 
+          (BofZ 24 128 re re 1) (BofZ 24 128 re re 1)) C2F) as C3.
+  rewrite Rlt_bool_true in C3; cycle 1.
+  { clear C3.
+    rewrite C2R.
+    apply Rabs_relax with (b := bpow radix2 0%Z).
+    { apply bpow_lt. lia. }
+    cbn.
+    gappa.
+  }
+  destruct C3 as (C3R & C3F & C3S).
+  rewrite C3R.
+  rewrite C2R.
+  cbn.
+  gappa.
+Qed.
+
 Definition rough_approx_div_longu a b :=
   Val.maketotal (Val.longuoffloat_ne
     (Val.mulf (Val.maketotal (Val.floatoflongu a)) (rough_approx_inv_longu b))).
@@ -468,14 +524,21 @@ Proof.
   { split.
     { apply Bsign_false_nonneg. trivial .}
     unfold rough_approx_inv_thresh in C1D.
+    destruct (Z_le_gt_dec b' 1) as [LE | GT].
+    { unfold f_r.
+      assert (b' = 1%Z) as EQ by lia.
+      destruct (rough_approx_inv_longu_correct1 b EQ) as
+        (f2 & OE & OF & OR & OS).
+      replace f2 with f in * by congruence.
+      lra.
+    }
     replace f_r with ((f_r - 1 / IZR b') + 1/ IZR b')%R by (field; lra).
     set (delta := (f_r - 1 / IZR b')%R) in *.
-    destruct (Z_le_gt_dec b' 1) as [LE | GT].
-    { admit. }
     assert (2 <= IZR b')%R as NOT1.
     { apply IZR_le. lia. }
     gappa.
   }
+  
   pose proof (BofZ_correct 53 1024 re re a') as C2.
   rewrite Rlt_bool_true in C2; cycle 1.
   { clear C2.
-- 
cgit 


From d1754fa46e98ef709274c7528b0c6453a6050d9d Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 10 Jan 2022 20:30:35 +0100
Subject: progress in proofs

---
 kvx/FPDivision64.v | 71 ++++++++++++++++++++++++++++++++----------------------
 1 file changed, 42 insertions(+), 29 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 521cbe0f..a241e4f0 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -482,15 +482,37 @@ Proof.
   lia.
 Qed.
 
+Lemma Znearest_IZR_le :
+  forall rnd n x, (IZR n <= x)%R -> (n <= Znearest rnd x)%Z.
+Proof.
+  intros until x. intro ORDER.
+  pose proof (Znearest_ge_floor rnd x).
+  pose proof (Zfloor_le _ _ ORDER) as KK.
+  rewrite Zfloor_IZR in KK.
+  lia.
+Qed.
+
+Lemma Znearest_le_IZR :
+  forall rnd n x, (x <= IZR n)%R -> (Znearest rnd x <= n)%Z.
+Proof.
+  intros until x. intro ORDER.
+  pose proof (Znearest_le_ceil rnd x).
+  pose proof (Zceil_le _ _ ORDER) as KK.
+  rewrite Zceil_IZR in KK.
+  lia.
+Qed.
+  
+
 Theorem rough_approx_div_longu_correct :
   forall a b,
-    (0 < Int64.unsigned b)%Z ->
+    (2 <= Int64.unsigned b)%Z ->
     exists (q : int64),
       (rough_approx_div_longu (Vlong a) (Vlong b)) = Vlong q /\
         (Z.abs ((Int64.unsigned a) - (Int64.unsigned q) * (Int64.unsigned b)) <= rough_approx_div_thresh)%Z.
 Proof.
-  intros a b b_NONZ.
+  intros a b b_NOT01.
   unfold rough_approx_div_longu.
+  assert (0 < Int64.unsigned b)%Z as b_NONZ by lia.
   destruct (rough_approx_inv_longu_correct b b_NONZ) as (f & C1R & C1F & C1D & C1S).
   rewrite C1R.
   cbn.
@@ -515,27 +537,17 @@ Proof.
   pose proof (Int64.unsigned_range b) as b_RANGE.
   change Int64.modulus with 18446744073709551616%Z in b_RANGE.
   set (b' := Int64.unsigned b) in *.
-  assert (1 <= IZR b' <= 18446744073709551615)%R as IZR_b_RANGE.
+  assert (2 <= IZR b' <= 18446744073709551615)%R as IZR_b_RANGE.
   { split ; apply IZR_le; lia.
   }
 
   set (f_r := (B2R _ _ f)) in *.
-  assert (0 <= f_r <= 1)%R as f_RANGE.
+  assert (0 <= f_r <= 4194305/8388608)%R as f_RANGE.
   { split.
     { apply Bsign_false_nonneg. trivial .}
     unfold rough_approx_inv_thresh in C1D.
-    destruct (Z_le_gt_dec b' 1) as [LE | GT].
-    { unfold f_r.
-      assert (b' = 1%Z) as EQ by lia.
-      destruct (rough_approx_inv_longu_correct1 b EQ) as
-        (f2 & OE & OF & OR & OS).
-      replace f2 with f in * by congruence.
-      lra.
-    }
     replace f_r with ((f_r - 1 / IZR b') + 1/ IZR b')%R by (field; lra).
     set (delta := (f_r - 1 / IZR b')%R) in *.
-    assert (2 <= IZR b')%R as NOT1.
-    { apply IZR_le. lia. }
     gappa.
   }
   
@@ -554,8 +566,8 @@ Proof.
   rewrite Rlt_bool_true in C3; cycle 1.
   { clear C3.
     rewrite C2R.
-    replace (B2R 53 1024 f)%R with ((B2R 53 1024 f - 1 / IZR b') + 1 / IZR b')%R by ring.
-    unfold rough_approx_inv_thresh in C1D.
+    fold f_r.
+    
     apply Rabs_relax with (b := bpow radix2 65).
     { apply bpow_lt ; lia. }
     cbn.
@@ -573,23 +585,24 @@ Proof.
   rewrite C3R in C4.
   rewrite C2R in C4.
 
-  assert(0 <=
-      (round radix2 (FLT_exp (-1074) 53) ZnearestE
+  set (y := (round radix2 (FLT_exp (-1074) 53) ZnearestE
               (round radix2 (FLT_exp (3 - 1024 - 53) 53) 
-                 (round_mode mode_NE) (IZR a') * B2R 53 1024 f))
-         <= IZR Int64.max_unsigned)%R as q_RANGE.
+                 (round_mode mode_NE) (IZR a') * B2R 53 1024 f))) in *.
+  assert(0 <= y <= IZR Int64.max_unsigned)%R as q_RANGE.
   { clear C4.
+    unfold y.
     cbn.
     change (IZR Int64.max_unsigned) with 18446744073709551615%R.
-    fold a_d.
-    gappa.
-    set (y := ((IZR a') * B2R 53 1024 f)%R).
-    assert ((round radix2 (FLT_exp (-1074) 53) ZnearestE (IZR a') * B2R 53 1024 f) >= 0)%R.
-   
-    cbn.
-    replace (B2R 53 1024 f)%R with ((B2R 53 1024 f - 1 / IZR b') + 1 / IZR b')%R by ring.
-    unfold rough_approx_inv_thresh in C1D.
-    set (delta :=  (B2R 53 1024 f - 1 / IZR b')%R) in *.
+    fold f_r.
     gappa.
   }
+  cbn in C4.
+  rewrite Zle_imp_le_bool in C4; cycle 1.
+  { apply Znearest_IZR_le. intuition.
+  }
+  rewrite Zle_imp_le_bool in C4; cycle 1.
+  { apply Znearest_le_IZR. intuition.
+  }
+  
+    
 Admitted.
-- 
cgit 


From f09e226b326185e940517c6940e5f71abf7d4fac Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 10 Jan 2022 20:37:20 +0100
Subject: progress in proof

---
 kvx/FPDivision64.v | 12 +++++++++++-
 1 file changed, 11 insertions(+), 1 deletion(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index a241e4f0..906ad650 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -603,6 +603,16 @@ Proof.
   rewrite Zle_imp_le_bool in C4; cycle 1.
   { apply Znearest_le_IZR. intuition.
   }
+  cbn in C4.
+  change Int64.max_unsigned with 18446744073709551615%Z.
+  rewrite C4.
+  cbn.
+  exists (Int64.repr (ZnearestE y)).
+  split. reflexivity.
+  rewrite Int64.unsigned_repr; cycle 1.
+  { split.
+    - apply Znearest_IZR_le. intuition.
+    - apply Znearest_le_IZR. intuition.
+  }
   
-    
 Admitted.
-- 
cgit 


From 76917c124e7c7de216acd99555209f18c1ecd16b Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Tue, 11 Jan 2022 15:22:43 +0100
Subject: progress in proof

---
 kvx/FPDivision64.v | 86 +++++++++++++++++++++---------------------------------
 1 file changed, 34 insertions(+), 52 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 906ad650..76e21e54 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -461,10 +461,7 @@ Proof.
 Qed.
 
 Definition rough_approx_div_longu a b :=
-  Val.maketotal (Val.longuoffloat_ne
-    (Val.mulf (Val.maketotal (Val.floatoflongu a)) (rough_approx_inv_longu b))).
-
-Definition rough_approx_div_thresh := 1649267441663%Z.
+  Val.mulf (Val.maketotal (Val.floatoflongu a)) (rough_approx_inv_longu b).
 
 Lemma Bsign_false_nonneg:
   forall prec emax f,
@@ -502,15 +499,19 @@ Proof.
   lia.
 Qed.
   
+Definition rough_approx_div_thresh := 1%R.
 
 Theorem rough_approx_div_longu_correct :
   forall a b,
     (2 <= Int64.unsigned b)%Z ->
-    exists (q : int64),
-      (rough_approx_div_longu (Vlong a) (Vlong b)) = Vlong q /\
-        (Z.abs ((Int64.unsigned a) - (Int64.unsigned q) * (Int64.unsigned b)) <= rough_approx_div_thresh)%Z.
+    exists (q : float),
+      (rough_approx_div_longu (Vlong a) (Vlong b)) = Vfloat q /\
+        (Rabs (B2R _ _ q - (IZR (Int64.unsigned a)) / (IZR (Int64.unsigned b)))
+         <= rough_approx_div_thresh)%R /\
+        is_finite _ _ q = true /\
+        Bsign _ _ q = false.
 Proof.
-  intros a b b_NOT01.
+  intros a b b_NON01.
   unfold rough_approx_div_longu.
   assert (0 < Int64.unsigned b)%Z as b_NONZ by lia.
   destruct (rough_approx_inv_longu_correct b b_NONZ) as (f & C1R & C1F & C1D & C1S).
@@ -528,12 +529,6 @@ Proof.
   { split ; apply IZR_le; lia.
   }
 
-  set (a_d := (round radix2 (FLT_exp (-1074) 53) ZnearestE (IZR a'))).
-  assert (0 <= a_d <= 18446744073709551616)%R as a_d_RANGE.
-  { unfold a_d.
-    gappa.
-  }
-
   pose proof (Int64.unsigned_range b) as b_RANGE.
   change Int64.modulus with 18446744073709551616%Z in b_RANGE.
   set (b' := Int64.unsigned b) in *.
@@ -541,6 +536,12 @@ Proof.
   { split ; apply IZR_le; lia.
   }
 
+  set (a_d := (round radix2 (FLT_exp (-1074) 53) ZnearestE (IZR a'))).
+  assert (0 <= a_d <= 18446744073709551616)%R as a_d_RANGE.
+  { unfold a_d.
+    gappa.
+  }
+
   set (f_r := (B2R _ _ f)) in *.
   assert (0 <= f_r <= 4194305/8388608)%R as f_RANGE.
   { split.
@@ -559,7 +560,7 @@ Proof.
     cbn.
     gappa.
   }
-  destruct C2 as (C2R & C2F & _).
+  destruct C2 as (C2R & C2F & C2S).
 
   pose proof (Bmult_correct  53 1024 re re Float.binop_nan mode_NE
                              (BofZ 53 1024 re re a') f) as C3.
@@ -576,43 +577,24 @@ Proof.
   rewrite C1F in C3.
   rewrite C2F in C3.
   cbn in C3.
-  destruct C3 as (C3R & C3F & _).
+  destruct C3 as (C3R & C3F & C3S).
+
+  econstructor. split. reflexivity.
+  rewrite C3R.
+  rewrite C2R.
+  rewrite C3S.
+  rewrite C2S.
+  rewrite C1S.
+  rewrite C3F.
+  rewrite Zlt_bool_false by lia.
+  split. 2: auto; fail.
 
-  pose proof (ZofB_ne_range_correct 53 1024
-               (Bmult 53 1024 re re Float.binop_nan mode_NE
-                      (BofZ 53 1024 re re a') f) 0 Int64.max_unsigned) as C4.
-  rewrite C3F in C4.
-  rewrite C3R in C4.
-  rewrite C2R in C4.
-
-  set (y := (round radix2 (FLT_exp (-1074) 53) ZnearestE
-              (round radix2 (FLT_exp (3 - 1024 - 53) 53) 
-                 (round_mode mode_NE) (IZR a') * B2R 53 1024 f))) in *.
-  assert(0 <= y <= IZR Int64.max_unsigned)%R as q_RANGE.
-  { clear C4.
-    unfold y.
-    cbn.
-    change (IZR Int64.max_unsigned) with 18446744073709551615%R.
-    fold f_r.
-    gappa.
-  }
-  cbn in C4.
-  rewrite Zle_imp_le_bool in C4; cycle 1.
-  { apply Znearest_IZR_le. intuition.
-  }
-  rewrite Zle_imp_le_bool in C4; cycle 1.
-  { apply Znearest_le_IZR. intuition.
-  }
-  cbn in C4.
-  change Int64.max_unsigned with 18446744073709551615%Z.
-  rewrite C4.
   cbn.
-  exists (Int64.repr (ZnearestE y)).
-  split. reflexivity.
-  rewrite Int64.unsigned_repr; cycle 1.
-  { split.
-    - apply Znearest_IZR_le. intuition.
-    - apply Znearest_le_IZR. intuition.
-  }
-  
+  unfold rough_approx_div_thresh.
+  fold f_r.
+  set (rd := round radix2 (FLT_exp (-1074) 53) ZnearestE).
+  replace (rd (rd (IZR a') * f_r) - IZR a' / IZR b')%R with
+   ((rd (rd (IZR a') * f_r) - (rd (IZR a') * f_r))
+    + ((rd (IZR a') - IZR a') * f_r)
+    + (IZR a') * (f_r - 1 / IZR b'))%R by (field ; lra).
 Admitted.
-- 
cgit 


From aa8f655361670b593f805b1e57a1089b8f924938 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Tue, 11 Jan 2022 15:31:02 +0100
Subject: progress in proofs

---
 kvx/FPDivision64.v | 12 +++++++++++-
 1 file changed, 11 insertions(+), 1 deletion(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 76e21e54..d16417fb 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -582,7 +582,9 @@ Proof.
   econstructor. split. reflexivity.
   rewrite C3R.
   rewrite C2R.
-  rewrite C3S.
+  rewrite C3S; cycle 1.
+  { apply is_finite_not_is_nan.
+    assumption. }
   rewrite C2S.
   rewrite C1S.
   rewrite C3F.
@@ -597,4 +599,12 @@ Proof.
    ((rd (rd (IZR a') * f_r) - (rd (IZR a') * f_r))
     + ((rd (IZR a') - IZR a') * f_r)
     + (IZR a') * (f_r - 1 / IZR b'))%R by (field ; lra).
+
+  assert (Rabs (rd (IZR a') - IZR a') <= 1024)%R as DELTA1.
+  { unfold rd. gappa. }
+  set (delta1 := (rd (IZR a') - IZR a')%R) in *.
+  
+  assert (Rabs (rd (rd (IZR a') * f_r) - rd (IZR a') * f_r) <= 1024)%R as DELTA2.
+  { unfold rd. gappa. }
+  set (delta2 := (rd (rd (IZR a') * f_r) - rd (IZR a') * f_r)%R) in *.
 Admitted.
-- 
cgit 


From 6e01e627018c745c6fd490dd6ced63203d482bbd Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Tue, 11 Jan 2022 15:37:52 +0100
Subject: progress in proofs

---
 kvx/FPDivision64.v | 7 +++++--
 1 file changed, 5 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index d16417fb..f20efa4e 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -499,7 +499,7 @@ Proof.
   lia.
 Qed.
   
-Definition rough_approx_div_thresh := 1%R.
+Definition rough_approx_div_thresh := 1683627180032%R.
 
 Theorem rough_approx_div_longu_correct :
   forall a b,
@@ -607,4 +607,7 @@ Proof.
   assert (Rabs (rd (rd (IZR a') * f_r) - rd (IZR a') * f_r) <= 1024)%R as DELTA2.
   { unfold rd. gappa. }
   set (delta2 := (rd (rd (IZR a') * f_r) - rd (IZR a') * f_r)%R) in *.
-Admitted.
+  unfold rough_approx_inv_thresh in C1D.
+  set (delta3 := (f_r - 1 / IZR b')%R) in *.
+  gappa.
+Qed.
-- 
cgit 


From 185846ddf99f5c9f17b74e40fc9ad1deb91cb1ff Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Wed, 12 Jan 2022 11:35:05 +0100
Subject: finer proof

---
 kvx/FPDivision64.v | 105 +++++++++++++++++++++++++++++++++++++++++++++++------
 1 file changed, 94 insertions(+), 11 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index f20efa4e..425e11da 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -289,14 +289,17 @@ Definition rough_approx_inv_longu b :=
 
 Definition rough_approx_inv_thresh := (3/33554432)%R.
 (* 8.94069671630859e-8 *)
-    
+
+Local Notation "'rd'" := (round radix2 (FLT_exp (-1074) 53) ZnearestE).
+Local Notation "'rs'" := (round radix2 (FLT_exp (-149) 24) ZnearestE).
+
 Theorem rough_approx_inv_longu_correct :
   forall b,
     (0 < Int64.unsigned b)%Z ->
     exists (f : float),
       (rough_approx_inv_longu (Vlong b)) = Vfloat f /\
+        (B2R _ _ f) = (rd (rs (1 / rs (IZR (Int64.unsigned b))))) /\
         is_finite _ _ f = true /\
-        (Rabs((B2R _ _ f) - (1 / IZR (Int64.unsigned b))) <= rough_approx_inv_thresh)%R /\
         Bsign _ _ f = false.
 Proof.
   intros b NONZ.
@@ -350,6 +353,7 @@ Proof.
     cbn.
     gappa.
   }
+  rewrite C1R in C2.
   destruct C2 as (C2R & C2F & C2Sz).
   rewrite C1S in C2Sz.
   change (xorb _ _) with false in C2Sz.
@@ -382,7 +386,6 @@ Proof.
     cbn.
     gappa.
   }
-  
   destruct C3 as (C3R & C3F & C3S).
   set (invb_d :=  (Bconv 24 128 53 1024 re re Float.of_single_nan mode_NE invb_s)) in *.
   assert ((1/18446744073709551616 <= B2R 53 1024 invb_d <= 1)%R) as invb_d_RANGE.
@@ -393,15 +396,10 @@ Proof.
     gappa.
   }
   rewrite C2S in C3S.
+  rewrite C2R in C3R.
+  rewrite C0R in C3R.
 
-  split. assumption. split. 2: assumption.
-  unfold rough_approx_inv_thresh.
-  rewrite C3R.
-  rewrite C2R.
-  rewrite C1R.
-  rewrite C0R.
-  cbn.
-  gappa.
+  auto.
 Qed.
 
 Theorem rough_approx_inv_longu_correct1 :
@@ -611,3 +609,88 @@ Proof.
   set (delta3 := (f_r - 1 / IZR b')%R) in *.
   gappa.
 Qed.
+
+Definition rough_approx_euclid_div_longu a b :=
+  Val.maketotal (Val.longuoffloat_ne (rough_approx_div_longu a b)).
+
+Definition smallb_threshold := 4398046511104%Z.
+Definition rough_approx_div_longu_smallb_delta := 3917010173952%Z.
+
+Theorem rough_approx_div_longu_smallb :
+  forall a b,
+    (2 <= Int64.unsigned b <= smallb_threshold)%Z ->
+    exists (qr : int64),
+      (rough_approx_euclid_div_longu (Vlong a) (Vlong b)) = Vlong qr /\
+        (Z.abs (Int64.unsigned a - Int64.unsigned b* Int64.unsigned qr) <= rough_approx_div_longu_smallb_delta)%Z /\
+        (0 <= Int64.unsigned qr <= 11529215046068469760)%Z.
+Proof.
+  intros a b b_RANGE.
+  unfold rough_approx_euclid_div_longu.
+  destruct (rough_approx_div_longu_correct a b) as (q & C1R & C1D & C1F & C1S).
+  lia.
+  rewrite C1R.
+  cbn.
+  set (qr := B2R 53 1024 q) in *.
+  pose proof (Int64.unsigned_range a) as a_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in a_RANGE.
+  set (a' := Int64.unsigned a) in *.
+  assert (0 <= IZR a' <= 18446744073709551615)%R as IZR_a_RANGE.
+  { split ; apply IZR_le; lia.
+  }
+
+  set (b' := Int64.unsigned b) in *.
+  assert (2 <= IZR b' <= IZR smallb_threshold)%R as IZR_b_RANGE.
+  { split ; apply IZR_le; lia.
+  }
+
+  assert (0 <= qr <= 11529215046068469760)%R as q_RANGE.
+  { split.
+    { unfold qr.
+      apply Bsign_false_nonneg.
+      assumption.
+    }
+    unfold rough_approx_div_thresh in C1D.
+    replace qr with ((qr - IZR a' / IZR b') + (IZR a' / IZR b'))%R by (field ; lra).
+    set (delta := (qr - IZR a' / IZR b')%R) in *.
+    unfold smallb_threshold in *.
+    gappa.
+  }
+  unfold Float.to_longu_ne.
+  pose proof (ZofB_ne_range_correct _ _ q 0 Int64.max_unsigned) as C2.
+  rewrite C1F in C2.
+  cbn in C2.
+  fold qr in C2.
+  assert (0 <= ZnearestE qr <= 11529215046068469760)%Z as round_RANGE.
+  { split.
+    { apply Znearest_IZR_le.
+      lra.
+    }
+    apply Znearest_le_IZR.
+    lra.
+  }
+  rewrite Zle_bool_true in C2 by lia.
+  rewrite Zle_bool_true in C2 by lia.
+  change Int64.max_unsigned with 18446744073709551615%Z.
+  cbn in C2.
+  rewrite C2.
+  cbn.
+  econstructor. split. reflexivity.
+  rewrite Int64.unsigned_repr; cycle 1.
+  { change Int64.max_unsigned with 18446744073709551615%Z.
+    lia.
+  }
+  split.
+  2: assumption.
+  unfold rough_approx_div_thresh, rough_approx_div_longu_smallb_delta in *.
+  apply le_IZR.
+  rewrite <- Rabs_Zabs.
+  rewrite minus_IZR.
+  rewrite mult_IZR.
+  replace  (IZR a' - IZR b' * IZR (ZnearestE qr))%R with
+    (- IZR b' * (IZR (ZnearestE qr) - qr)
+     - IZR b' * (qr - IZR a' / IZR b'))%R by (field ; lra).
+  pose proof (Znearest_imp2  (fun x : Z => negb (Z.even x)) qr) as ROUND.
+  set (delta1 := (IZR (ZnearestE qr) - qr)%R) in *.
+  set (delta2 := (qr - IZR a' / IZR b')%R) in *.
+  unfold smallb_threshold in *.
+ 
-- 
cgit 


From 7ff41e564bcc2e01d33b26419d3039d2f393440a Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Wed, 12 Jan 2022 11:41:27 +0100
Subject: progress in proofs

---
 kvx/FPDivision64.v | 53 ++++++++---------------------------------------------
 1 file changed, 8 insertions(+), 45 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 425e11da..b1ba6938 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -407,54 +407,17 @@ Theorem rough_approx_inv_longu_correct1 :
     (Int64.unsigned b = 1%Z) ->
     exists f,
     (rough_approx_inv_longu (Vlong b)) = Vfloat f /\
-        is_finite _ _ f = true /\
         (B2R _ _ f) = 1%R /\
+        is_finite _ _ f = true /\
         Bsign _ _ f = false.
 Proof.
-  intros b ONE.
-  cbn.
-  unfold Float.of_single, ExtFloat32.inv, Float32.div, ExtFloat32.one, Float32.of_longu, Float32.of_int.
-  set (re := (@eq_refl Datatypes.comparison Lt)).
-  rewrite ONE.
-  change (Int.signed (Int.repr 1)) with 1%Z.
-  
-  econstructor.
-  split. reflexivity.
-  split. reflexivity.
-  split. 2: reflexivity.
-  destruct (BofZ_exact 24 128 re re 1%Z) as (C1R & C1F & C1S). lia.
-
-  pose proof (Bdiv_correct 24 128 re re Float32.binop_nan mode_NE 
-                           (BofZ 24 128 re re 1) (BofZ 24 128 re re 1)) as C2.
-  rewrite C1R in C2.
-  rewrite C1F in C2.
-  rewrite C1S in C2.
-  rewrite Rlt_bool_true in C2; cycle 1.
-  { clear C2.
-    apply Rabs_relax with (b := bpow radix2 0%Z).
-    { apply bpow_lt. lia. }
-    cbn.
-    gappa.
-  }
-  assert (1 <> 0)%R as OBVIOUS by lra.
-  destruct (C2 OBVIOUS) as (C2R & C2F & C2S).
-  clear C2.
-
-  pose proof (Bconv_correct 24 128 53 1024 re re Float.of_single_nan mode_NE
-       (Bdiv 24 128 re re Float32.binop_nan mode_NE 
-          (BofZ 24 128 re re 1) (BofZ 24 128 re re 1)) C2F) as C3.
-  rewrite Rlt_bool_true in C3; cycle 1.
-  { clear C3.
-    rewrite C2R.
-    apply Rabs_relax with (b := bpow radix2 0%Z).
-    { apply bpow_lt. lia. }
-    cbn.
-    gappa.
-  }
-  destruct C3 as (C3R & C3F & C3S).
-  rewrite C3R.
-  rewrite C2R.
-  cbn.
+  intros b EQ1.
+  assert (0 < Int64.unsigned b)%Z as b_RANGE by lia.
+  destruct (rough_approx_inv_longu_correct b b_RANGE) as (f & C1E & C1R & C1F & C1S).
+  rewrite EQ1 in C1R.
+  exists f.
+  repeat split; try assumption.
+  rewrite C1R.
   gappa.
 Qed.
 
-- 
cgit 


From 41701fa253bf8adb790e0f27559bb4e6e6276ea8 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Wed, 12 Jan 2022 13:10:49 +0100
Subject: more proofs

---
 kvx/FPDivision64.v | 82 ++++++++++++++++++++++++++++++++++++++++++++++++++----
 1 file changed, 77 insertions(+), 5 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index b1ba6938..81d587f7 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -421,9 +421,6 @@ Proof.
   gappa.
 Qed.
 
-Definition rough_approx_div_longu a b :=
-  Val.mulf (Val.maketotal (Val.floatoflongu a)) (rough_approx_inv_longu b).
-
 Lemma Bsign_false_nonneg:
   forall prec emax f,
     (Bsign prec emax f) = false -> (0 <= (B2R prec emax f))%R.
@@ -459,7 +456,83 @@ Proof.
   rewrite Zceil_IZR in KK.
   lia.
 Qed.
+
+
+Definition rough_approx_div_longu a b :=
+  Val.mulf (Val.maketotal (Val.floatoflongu a)) (rough_approx_inv_longu b).
+Definition smallb_threshold := 4398046511104%Z.
+Definition rough_approx_div_longu_smallb_delta := 3917010173952%Z.
+
+Theorem rough_approx_div_longu_correct:
+  forall a b,
+    (1 < Int64.unsigned b)%Z ->
+    exists (q : float),
+      (rough_approx_div_longu (Vlong a) (Vlong b)) = Vfloat q /\
+        (B2R _ _ q) = rd (rd (IZR (Int64.unsigned a)) * (rd (rs (1 / rs (IZR (Int64.unsigned b)))))) /\
+        is_finite _ _ q = true /\
+        Bsign _ _ q = false.
+Proof.
+  intros a b b_NON01.
+  assert (0 < Int64.unsigned b)%Z as b_NON0 by lia.
+  destruct (rough_approx_inv_longu_correct b b_NON0) as (f & C1E & C1R & C1F & C1S).
+  unfold rough_approx_div_longu.
+  rewrite C1E.
+  cbn.
+  set (b' := Int64.unsigned b) in *.
+  Local Transparent Float.mul.
+  unfold Float.mul, Float.of_longu.
+  econstructor.
+  split. reflexivity.
+  set (a' := Int64.unsigned a) in *.
+  set (re :=  (@eq_refl Datatypes.comparison Lt)).
+
+  pose proof (Int64.unsigned_range a) as a_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in a_RANGE.
+  assert (0 <= IZR a' <= 18446744073709551615)%R as IZR_a_RANGE.
+  { split; apply IZR_le; lia. }
+  pose proof (Int64.unsigned_range b) as b_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in b_RANGE.
+  assert (2 <= IZR b' <= 18446744073709551615)%R as IZR_b_RANGE.
+  { split; apply IZR_le; lia. }
   
+  pose proof (BofZ_correct 53 1024 re re a') as C2.
+  rewrite Rlt_bool_true in C2; cycle 1.
+  { clear C2.
+    apply Rabs_relax with (b := bpow radix2 64).
+    { apply bpow_lt. lia. }
+    cbn.
+    gappa.
+  }
+  destruct C2 as (C2R & C2F & C2S).
+  rewrite Zlt_bool_false in C2S by lia.
+
+  pose proof (Bmult_correct 53 1024 re re Float.binop_nan mode_NE (BofZ 53 1024 re re a') f) as C3.
+  rewrite C1S in C3.
+  rewrite C2S in C3.
+  rewrite C1F in C3.
+  rewrite C2F in C3.
+  rewrite C1R in C3.
+  rewrite C2R in C3.
+  rewrite Rlt_bool_true in C3; cycle 1.
+  { apply Rabs_relax with (b := bpow radix2 64).
+    { apply bpow_lt ; lia. }
+    cbn.
+    gappa.
+  }
+  cbn in C3.
+  destruct C3 as (C3R & C3F & C3Sz).
+  assert (is_nan 53 1024
+          (Bmult 53 1024 re re Float.binop_nan mode_NE 
+                 (BofZ 53 1024 re re a') f) = false) as NAN.
+  { apply is_finite_not_is_nan.
+    assumption. }
+  pose proof (C3Sz NAN) as C3S.
+  clear NAN C3Sz.
+
+  auto.
+Qed.
+  
+(*
 Definition rough_approx_div_thresh := 1683627180032%R.
 
 Theorem rough_approx_div_longu_correct :
@@ -572,12 +645,11 @@ Proof.
   set (delta3 := (f_r - 1 / IZR b')%R) in *.
   gappa.
 Qed.
+ *)
 
 Definition rough_approx_euclid_div_longu a b :=
   Val.maketotal (Val.longuoffloat_ne (rough_approx_div_longu a b)).
 
-Definition smallb_threshold := 4398046511104%Z.
-Definition rough_approx_div_longu_smallb_delta := 3917010173952%Z.
 
 Theorem rough_approx_div_longu_smallb :
   forall a b,
-- 
cgit 


From fc58331d1aa0c1379df8b17d52f71b07d80c3045 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Wed, 12 Jan 2022 15:37:43 +0100
Subject: give a name

---
 kvx/FPDivision64.v | 6 +++++-
 1 file changed, 5 insertions(+), 1 deletion(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 81d587f7..e8e7f8be 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -463,12 +463,16 @@ Definition rough_approx_div_longu a b :=
 Definition smallb_threshold := 4398046511104%Z.
 Definition rough_approx_div_longu_smallb_delta := 3917010173952%Z.
 
+Definition step1_real_quotient (a b : R) :=
+             rd ((rd a) * (rd (rs (1 / rs b)))).
+  
 Theorem rough_approx_div_longu_correct:
   forall a b,
     (1 < Int64.unsigned b)%Z ->
     exists (q : float),
       (rough_approx_div_longu (Vlong a) (Vlong b)) = Vfloat q /\
-        (B2R _ _ q) = rd (rd (IZR (Int64.unsigned a)) * (rd (rs (1 / rs (IZR (Int64.unsigned b)))))) /\
+        (B2R _ _ q) = step1_real_quotient (IZR (Int64.unsigned a))
+                                          (IZR (Int64.unsigned b)) /\
         is_finite _ _ q = true /\
         Bsign _ _ q = false.
 Proof.
-- 
cgit 


From 9cacc4caa5710f35a7ab73de6aa0ddf849e60d73 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Wed, 12 Jan 2022 16:06:35 +0100
Subject: proof of approx for a/b

---
 kvx/FPDivision64.v | 28 ++++++++++++++++++++++++++--
 1 file changed, 26 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index e8e7f8be..f4d727be 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -464,7 +464,7 @@ Definition smallb_threshold := 4398046511104%Z.
 Definition rough_approx_div_longu_smallb_delta := 3917010173952%Z.
 
 Definition step1_real_quotient (a b : R) :=
-             rd ((rd a) * (rd (rs (1 / rs b)))).
+             rd ((rd (a)) * (rd (rs (1 / rs (b))))).
   
 Theorem rough_approx_div_longu_correct:
   forall a b,
@@ -535,7 +535,31 @@ Proof.
 
   auto.
 Qed.
-  
+
+Definition smallb_thresh := 4398046511104%Z.
+
+Definition smallb_approx_range := 1649280000000%R.
+Lemma step1_smallb :
+  forall a b
+    (a_RANGE : (0 <= a <= 18446744073709551615)%R)
+    (b_RANGE : (1 <= b <= IZR smallb_thresh)%R),
+    (Rabs((step1_real_quotient a b) - a/b) <= smallb_approx_range)%R.
+Proof.
+  intros.
+  unfold smallb_thresh in b_RANGE.
+  unfold smallb_approx_range.
+  replace (step1_real_quotient a b - a/b)%R with
+    ((rd ((rd (a)) * (rd (rs (1 / rs (b)))))
+             - ((rd (a)) * (rd (rs (1 / rs (b)))))) +
+	   (rd(a)) * (rd (rs (1 / rs (b))) - 1/b) +
+       (rd(a) - a)/b)%R; cycle 1.
+  { unfold step1_real_quotient.
+    field.
+    lra.
+  }
+  gappa.
+Qed.
+
 (*
 Definition rough_approx_div_thresh := 1683627180032%R.
 
-- 
cgit 


From 82245a79c9bfd3400d309242377efba20882a8e3 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Wed, 12 Jan 2022 17:25:43 +0100
Subject: progress in proofs

---
 kvx/FPDivision64.v | 254 +++++++----------------------------------------------
 1 file changed, 30 insertions(+), 224 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index f4d727be..e7ba56e3 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -283,27 +283,27 @@ Proof.
   gappa.
 Qed.
 
-Definition rough_approx_inv_longu b :=
+Definition step1_real_inv_longu b :=
   let invb_s := ExtValues.invfs (Val.maketotal (Val.singleoflongu b)) in
   Val.floatofsingle invb_s.
 
-Definition rough_approx_inv_thresh := (3/33554432)%R.
+Definition step1_real_inv_thresh := (3/33554432)%R.
 (* 8.94069671630859e-8 *)
 
 Local Notation "'rd'" := (round radix2 (FLT_exp (-1074) 53) ZnearestE).
 Local Notation "'rs'" := (round radix2 (FLT_exp (-149) 24) ZnearestE).
 
-Theorem rough_approx_inv_longu_correct :
+Theorem step1_real_inv_longu_correct :
   forall b,
     (0 < Int64.unsigned b)%Z ->
     exists (f : float),
-      (rough_approx_inv_longu (Vlong b)) = Vfloat f /\
+      (step1_real_inv_longu (Vlong b)) = Vfloat f /\
         (B2R _ _ f) = (rd (rs (1 / rs (IZR (Int64.unsigned b))))) /\
         is_finite _ _ f = true /\
         Bsign _ _ f = false.
 Proof.
   intros b NONZ.
-  unfold rough_approx_inv_longu.
+  unfold step1_real_inv_longu.
   cbn.
   econstructor.
   split.
@@ -402,18 +402,18 @@ Proof.
   auto.
 Qed.
 
-Theorem rough_approx_inv_longu_correct1 :
+Theorem step1_real_inv_longu_correct1 :
   forall b,
     (Int64.unsigned b = 1%Z) ->
     exists f,
-    (rough_approx_inv_longu (Vlong b)) = Vfloat f /\
+    (step1_real_inv_longu (Vlong b)) = Vfloat f /\
         (B2R _ _ f) = 1%R /\
         is_finite _ _ f = true /\
         Bsign _ _ f = false.
 Proof.
   intros b EQ1.
   assert (0 < Int64.unsigned b)%Z as b_RANGE by lia.
-  destruct (rough_approx_inv_longu_correct b b_RANGE) as (f & C1E & C1R & C1F & C1S).
+  destruct (step1_real_inv_longu_correct b b_RANGE) as (f & C1E & C1R & C1F & C1S).
   rewrite EQ1 in C1R.
   exists f.
   repeat split; try assumption.
@@ -457,20 +457,20 @@ Proof.
   lia.
 Qed.
 
+Definition step1_real_div_longu a b :=
+  Val.mulf (Val.maketotal (Val.floatoflongu a)) (step1_real_inv_longu b).
 
-Definition rough_approx_div_longu a b :=
-  Val.mulf (Val.maketotal (Val.floatoflongu a)) (rough_approx_inv_longu b).
-Definition smallb_threshold := 4398046511104%Z.
-Definition rough_approx_div_longu_smallb_delta := 3917010173952%Z.
+Definition step1_div_longu a b :=
+  Val.maketotal (Val.longuoffloat (Val.mulf (Val.maketotal (Val.floatoflongu a)) (step1_real_inv_longu b))).
 
 Definition step1_real_quotient (a b : R) :=
              rd ((rd (a)) * (rd (rs (1 / rs (b))))).
   
-Theorem rough_approx_div_longu_correct:
+Theorem step1_real_div_longu_correct:
   forall a b,
     (1 < Int64.unsigned b)%Z ->
     exists (q : float),
-      (rough_approx_div_longu (Vlong a) (Vlong b)) = Vfloat q /\
+      (step1_real_div_longu (Vlong a) (Vlong b)) = Vfloat q /\
         (B2R _ _ q) = step1_real_quotient (IZR (Int64.unsigned a))
                                           (IZR (Int64.unsigned b)) /\
         is_finite _ _ q = true /\
@@ -478,8 +478,8 @@ Theorem rough_approx_div_longu_correct:
 Proof.
   intros a b b_NON01.
   assert (0 < Int64.unsigned b)%Z as b_NON0 by lia.
-  destruct (rough_approx_inv_longu_correct b b_NON0) as (f & C1E & C1R & C1F & C1S).
-  unfold rough_approx_div_longu.
+  destruct (step1_real_inv_longu_correct b b_NON0) as (f & C1E & C1R & C1F & C1S).
+  unfold step1_real_div_longu.
   rewrite C1E.
   cbn.
   set (b' := Int64.unsigned b) in *.
@@ -536,224 +536,30 @@ Proof.
   auto.
 Qed.
 
-Definition smallb_thresh := 4398046511104%Z.
+Definition smallb_thresh :=       4398046511104%Z.
 
-Definition smallb_approx_range := 1649280000000%R.
+Definition smallb_approx_range := 2200000000000%R.
 Lemma step1_smallb :
   forall a b
-    (a_RANGE : (0 <= a <= 18446744073709551615)%R)
+    (a_RANGE : (1 <= a <= 18446744073709551615)%R)
     (b_RANGE : (1 <= b <= IZR smallb_thresh)%R),
-    (Rabs((step1_real_quotient a b) - a/b) <= smallb_approx_range)%R.
+    (Rabs((step1_real_quotient a b) * b - a) <= smallb_approx_range)%R.
 Proof.
   intros.
   unfold smallb_thresh in b_RANGE.
   unfold smallb_approx_range.
-  replace (step1_real_quotient a b - a/b)%R with
-    ((rd ((rd (a)) * (rd (rs (1 / rs (b)))))
-             - ((rd (a)) * (rd (rs (1 / rs (b)))))) +
-	   (rd(a)) * (rd (rs (1 / rs (b))) - 1/b) +
-       (rd(a) - a)/b)%R; cycle 1.
-  { unfold step1_real_quotient.
+  unfold step1_real_quotient.
+  set (q := ((rd (a)) * (rd (rs (1 / rs (b)))))%R) in *.
+  replace ((rd q) *b - a)%R with
+     ((rd(q)-q)/q * rd(a) * (1 + (rd (rs (1 / rs (b))) - 1/b)/(1/b)) +
+  (rd (a)) * ((rd (rs (1 / rs (b))) - 1 / b) / (1/b)) +
+        (rd(a) - a))%R; cycle 1.
+  { unfold q.
     field.
-    lra.
-  }
-  gappa.
-Qed.
-
-(*
-Definition rough_approx_div_thresh := 1683627180032%R.
-
-Theorem rough_approx_div_longu_correct :
-  forall a b,
-    (2 <= Int64.unsigned b)%Z ->
-    exists (q : float),
-      (rough_approx_div_longu (Vlong a) (Vlong b)) = Vfloat q /\
-        (Rabs (B2R _ _ q - (IZR (Int64.unsigned a)) / (IZR (Int64.unsigned b)))
-         <= rough_approx_div_thresh)%R /\
-        is_finite _ _ q = true /\
-        Bsign _ _ q = false.
-Proof.
-  intros a b b_NON01.
-  unfold rough_approx_div_longu.
-  assert (0 < Int64.unsigned b)%Z as b_NONZ by lia.
-  destruct (rough_approx_inv_longu_correct b b_NONZ) as (f & C1R & C1F & C1D & C1S).
-  rewrite C1R.
-  cbn.
-
-  Local Transparent Float.to_longu Float.mul.
-  unfold Float.to_longu_ne, Float.mul, Float.of_longu.
-  set (re := (@eq_refl Datatypes.comparison Lt)).
-
-  pose proof (Int64.unsigned_range a) as a_RANGE.
-  change Int64.modulus with 18446744073709551616%Z in a_RANGE.
-  set (a' := Int64.unsigned a) in *.
-  assert (0 <= IZR a' <= 18446744073709551615)%R as IZR_a_RANGE.
-  { split ; apply IZR_le; lia.
-  }
-
-  pose proof (Int64.unsigned_range b) as b_RANGE.
-  change Int64.modulus with 18446744073709551616%Z in b_RANGE.
-  set (b' := Int64.unsigned b) in *.
-  assert (2 <= IZR b' <= 18446744073709551615)%R as IZR_b_RANGE.
-  { split ; apply IZR_le; lia.
-  }
-
-  set (a_d := (round radix2 (FLT_exp (-1074) 53) ZnearestE (IZR a'))).
-  assert (0 <= a_d <= 18446744073709551616)%R as a_d_RANGE.
-  { unfold a_d.
-    gappa.
-  }
-
-  set (f_r := (B2R _ _ f)) in *.
-  assert (0 <= f_r <= 4194305/8388608)%R as f_RANGE.
-  { split.
-    { apply Bsign_false_nonneg. trivial .}
-    unfold rough_approx_inv_thresh in C1D.
-    replace f_r with ((f_r - 1 / IZR b') + 1/ IZR b')%R by (field; lra).
-    set (delta := (f_r - 1 / IZR b')%R) in *.
-    gappa.
-  }
-  
-  pose proof (BofZ_correct 53 1024 re re a') as C2.
-  rewrite Rlt_bool_true in C2; cycle 1.
-  { clear C2.
-    apply Rabs_relax with (b := bpow radix2 64).
-    { apply bpow_lt. lia. }
-    cbn.
-    gappa.
-  }
-  destruct C2 as (C2R & C2F & C2S).
-
-  pose proof (Bmult_correct  53 1024 re re Float.binop_nan mode_NE
-                             (BofZ 53 1024 re re a') f) as C3.
-  rewrite Rlt_bool_true in C3; cycle 1.
-  { clear C3.
-    rewrite C2R.
-    fold f_r.
-    
-    apply Rabs_relax with (b := bpow radix2 65).
-    { apply bpow_lt ; lia. }
-    cbn.
+    split. lra.
+    split. gappa.
     gappa.
   }
-  rewrite C1F in C3.
-  rewrite C2F in C3.
-  cbn in C3.
-  destruct C3 as (C3R & C3F & C3S).
-
-  econstructor. split. reflexivity.
-  rewrite C3R.
-  rewrite C2R.
-  rewrite C3S; cycle 1.
-  { apply is_finite_not_is_nan.
-    assumption. }
-  rewrite C2S.
-  rewrite C1S.
-  rewrite C3F.
-  rewrite Zlt_bool_false by lia.
-  split. 2: auto; fail.
-
-  cbn.
-  unfold rough_approx_div_thresh.
-  fold f_r.
-  set (rd := round radix2 (FLT_exp (-1074) 53) ZnearestE).
-  replace (rd (rd (IZR a') * f_r) - IZR a' / IZR b')%R with
-   ((rd (rd (IZR a') * f_r) - (rd (IZR a') * f_r))
-    + ((rd (IZR a') - IZR a') * f_r)
-    + (IZR a') * (f_r - 1 / IZR b'))%R by (field ; lra).
-
-  assert (Rabs (rd (IZR a') - IZR a') <= 1024)%R as DELTA1.
-  { unfold rd. gappa. }
-  set (delta1 := (rd (IZR a') - IZR a')%R) in *.
-  
-  assert (Rabs (rd (rd (IZR a') * f_r) - rd (IZR a') * f_r) <= 1024)%R as DELTA2.
-  { unfold rd. gappa. }
-  set (delta2 := (rd (rd (IZR a') * f_r) - rd (IZR a') * f_r)%R) in *.
-  unfold rough_approx_inv_thresh in C1D.
-  set (delta3 := (f_r - 1 / IZR b')%R) in *.
+  unfold q.
   gappa.
 Qed.
- *)
-
-Definition rough_approx_euclid_div_longu a b :=
-  Val.maketotal (Val.longuoffloat_ne (rough_approx_div_longu a b)).
-
-
-Theorem rough_approx_div_longu_smallb :
-  forall a b,
-    (2 <= Int64.unsigned b <= smallb_threshold)%Z ->
-    exists (qr : int64),
-      (rough_approx_euclid_div_longu (Vlong a) (Vlong b)) = Vlong qr /\
-        (Z.abs (Int64.unsigned a - Int64.unsigned b* Int64.unsigned qr) <= rough_approx_div_longu_smallb_delta)%Z /\
-        (0 <= Int64.unsigned qr <= 11529215046068469760)%Z.
-Proof.
-  intros a b b_RANGE.
-  unfold rough_approx_euclid_div_longu.
-  destruct (rough_approx_div_longu_correct a b) as (q & C1R & C1D & C1F & C1S).
-  lia.
-  rewrite C1R.
-  cbn.
-  set (qr := B2R 53 1024 q) in *.
-  pose proof (Int64.unsigned_range a) as a_RANGE.
-  change Int64.modulus with 18446744073709551616%Z in a_RANGE.
-  set (a' := Int64.unsigned a) in *.
-  assert (0 <= IZR a' <= 18446744073709551615)%R as IZR_a_RANGE.
-  { split ; apply IZR_le; lia.
-  }
-
-  set (b' := Int64.unsigned b) in *.
-  assert (2 <= IZR b' <= IZR smallb_threshold)%R as IZR_b_RANGE.
-  { split ; apply IZR_le; lia.
-  }
-
-  assert (0 <= qr <= 11529215046068469760)%R as q_RANGE.
-  { split.
-    { unfold qr.
-      apply Bsign_false_nonneg.
-      assumption.
-    }
-    unfold rough_approx_div_thresh in C1D.
-    replace qr with ((qr - IZR a' / IZR b') + (IZR a' / IZR b'))%R by (field ; lra).
-    set (delta := (qr - IZR a' / IZR b')%R) in *.
-    unfold smallb_threshold in *.
-    gappa.
-  }
-  unfold Float.to_longu_ne.
-  pose proof (ZofB_ne_range_correct _ _ q 0 Int64.max_unsigned) as C2.
-  rewrite C1F in C2.
-  cbn in C2.
-  fold qr in C2.
-  assert (0 <= ZnearestE qr <= 11529215046068469760)%Z as round_RANGE.
-  { split.
-    { apply Znearest_IZR_le.
-      lra.
-    }
-    apply Znearest_le_IZR.
-    lra.
-  }
-  rewrite Zle_bool_true in C2 by lia.
-  rewrite Zle_bool_true in C2 by lia.
-  change Int64.max_unsigned with 18446744073709551615%Z.
-  cbn in C2.
-  rewrite C2.
-  cbn.
-  econstructor. split. reflexivity.
-  rewrite Int64.unsigned_repr; cycle 1.
-  { change Int64.max_unsigned with 18446744073709551615%Z.
-    lia.
-  }
-  split.
-  2: assumption.
-  unfold rough_approx_div_thresh, rough_approx_div_longu_smallb_delta in *.
-  apply le_IZR.
-  rewrite <- Rabs_Zabs.
-  rewrite minus_IZR.
-  rewrite mult_IZR.
-  replace  (IZR a' - IZR b' * IZR (ZnearestE qr))%R with
-    (- IZR b' * (IZR (ZnearestE qr) - qr)
-     - IZR b' * (qr - IZR a' / IZR b'))%R by (field ; lra).
-  pose proof (Znearest_imp2  (fun x : Z => negb (Z.even x)) qr) as ROUND.
-  set (delta1 := (IZR (ZnearestE qr) - qr)%R) in *.
-  set (delta2 := (qr - IZR a' / IZR b')%R) in *.
-  unfold smallb_threshold in *.
- 
-- 
cgit 


From 13876421ac06c904e9c8ea7636f5bef1fc4c8988 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Wed, 12 Jan 2022 19:15:06 +0100
Subject: progress in proofs

---
 kvx/FPDivision64.v | 66 +++++++++++++++++++++++++++++++++++++++++++++++++-----
 1 file changed, 61 insertions(+), 5 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index e7ba56e3..5da53535 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -461,7 +461,7 @@ Definition step1_real_div_longu a b :=
   Val.mulf (Val.maketotal (Val.floatoflongu a)) (step1_real_inv_longu b).
 
 Definition step1_div_longu a b :=
-  Val.maketotal (Val.longuoffloat (Val.mulf (Val.maketotal (Val.floatoflongu a)) (step1_real_inv_longu b))).
+  Val.maketotal (Val.longuoffloat_ne (step1_real_div_longu a b)).
 
 Definition step1_real_quotient (a b : R) :=
              rd ((rd (a)) * (rd (rs (1 / rs (b))))).
@@ -538,16 +538,16 @@ Qed.
 
 Definition smallb_thresh :=       4398046511104%Z.
 
-Definition smallb_approx_range := 2200000000000%R.
-Lemma step1_smallb :
+Definition smallb_approx_real_range := 2200000000000%R.
+Lemma step1_smallb_real :
   forall a b
     (a_RANGE : (1 <= a <= 18446744073709551615)%R)
     (b_RANGE : (1 <= b <= IZR smallb_thresh)%R),
-    (Rabs((step1_real_quotient a b) * b - a) <= smallb_approx_range)%R.
+    (Rabs((step1_real_quotient a b) * b - a) <= smallb_approx_real_range)%R.
 Proof.
   intros.
   unfold smallb_thresh in b_RANGE.
-  unfold smallb_approx_range.
+  unfold smallb_approx_real_range.
   unfold step1_real_quotient.
   set (q := ((rd (a)) * (rd (rs (1 / rs (b)))))%R) in *.
   replace ((rd q) *b - a)%R with
@@ -563,3 +563,59 @@ Proof.
   unfold q.
   gappa.
 Qed.
+
+Definition smallb_approx_range := 2200000000000%Z.
+Lemma step1_div_longu_correct :
+    forall a b,
+    (1 < Int64.unsigned b <= smallb_thresh)%Z ->
+    exists (q : int64),
+      (step1_div_longu (Vlong a) (Vlong b)) = Vlong q /\
+        (Z.abs (Int64.unsigned a - Int64.unsigned b*Int64.signed q) <= smallb_approx_range)%Z.
+Proof.
+  intros a b b_RANGE.
+  unfold step1_div_longu.
+  assert (1 < Int64.unsigned b)%Z as b_NOT01 by lia.   
+  destruct (step1_real_div_longu_correct a b b_NOT01) as (q & C1E & C1R & C1F & C1S).
+  rewrite C1E. cbn.
+  unfold Float.to_longu_ne.
+  pose proof (ZofB_ne_range_correct 53 1024 q 0 Int64.max_unsigned) as C2.
+  rewrite C1F in C2.
+  pose proof (Int64.unsigned_range a) as a_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in a_RANGE.
+  set (a' := Int64.unsigned a) in *.
+  assert (1 <= a')%Z by admit.
+  
+  set (b' := Int64.unsigned b) in *.
+  assert (1 <= IZR a' <= 18446744073709551615)%R as a_RANGE'.
+  { split; apply IZR_le; lia. }
+  assert (2 <= IZR b' <= IZR smallb_thresh)%R as b_RANGE'.
+  { split; apply IZR_le; lia. }
+  assert (1 <= IZR b' <= IZR smallb_thresh)%R as b_RANGE'' by lra.
+  pose proof (step1_smallb_real (IZR a') (IZR b') a_RANGE' b_RANGE'') as DELTA.
+  rewrite <- C1R in DELTA.
+
+  assert (0 <= B2R _ _ q)%R as q_NONNEG.
+  { apply Bsign_false_nonneg. assumption. }
+  cbn in C2.
+  rewrite Zle_bool_true in C2; cycle 1.
+  { apply Znearest_IZR_le. assumption. }
+  assert (B2R _ _ q <= 18446744073709551615)%R as q_SMALL.
+  { replace (B2R _ _ q) with
+      ((IZR a' / IZR b') + (B2R _ _ q * IZR b' - IZR a') / IZR b')%R; cycle 1.
+    { field. lra. }
+    unfold smallb_approx_real_range in DELTA.
+    unfold smallb_thresh in b_RANGE'.
+    set (y := (B2R 53 1024 q * IZR b' - IZR a')%R) in *.
+    gappa.
+  }
+  rewrite Zle_bool_true in C2; cycle 1.
+  { apply Znearest_le_IZR. assumption. }
+  cbn in C2.
+
+  change Int64.max_unsigned with 18446744073709551615%Z.
+  rewrite C2.
+  cbn.
+
+  econstructor. split. reflexivity.
+  
+ 
-- 
cgit 


From 35b823b50952c5d46eaeb36f3c7781bbb2e91674 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Wed, 12 Jan 2022 19:44:05 +0100
Subject: proof forward

---
 kvx/FPDivision64.v | 31 +++++++++++++++++++++++++------
 1 file changed, 25 insertions(+), 6 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 5da53535..76959118 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -564,13 +564,13 @@ Proof.
   gappa.
 Qed.
 
-Definition smallb_approx_range := 2200000000000%Z.
+Definition smallb_approx_range := 4400000000000%Z.
 Lemma step1_div_longu_correct :
     forall a b,
     (1 < Int64.unsigned b <= smallb_thresh)%Z ->
     exists (q : int64),
       (step1_div_longu (Vlong a) (Vlong b)) = Vlong q /\
-        (Z.abs (Int64.unsigned a - Int64.unsigned b*Int64.signed q) <= smallb_approx_range)%Z.
+        (Z.abs (Int64.unsigned a - Int64.unsigned b*Int64.unsigned q) <= smallb_approx_range)%Z.
 Proof.
   intros a b b_RANGE.
   unfold step1_div_longu.
@@ -599,7 +599,7 @@ Proof.
   cbn in C2.
   rewrite Zle_bool_true in C2; cycle 1.
   { apply Znearest_IZR_le. assumption. }
-  assert (B2R _ _ q <= 18446744073709551615)%R as q_SMALL.
+  assert (B2R _ _ q <= 9223376000000000000)%R as q_SMALL.
   { replace (B2R _ _ q) with
       ((IZR a' / IZR b') + (B2R _ _ q * IZR b' - IZR a') / IZR b')%R; cycle 1.
     { field. lra. }
@@ -609,7 +609,7 @@ Proof.
     gappa.
   }
   rewrite Zle_bool_true in C2; cycle 1.
-  { apply Znearest_le_IZR. assumption. }
+  { apply Znearest_le_IZR. lra. }
   cbn in C2.
 
   change Int64.max_unsigned with 18446744073709551615%Z.
@@ -617,5 +617,24 @@ Proof.
   cbn.
 
   econstructor. split. reflexivity.
-  
- 
+  rewrite Int64.unsigned_repr; cycle 1.
+  { split.
+    - apply Znearest_IZR_le. lra.
+    - apply Znearest_le_IZR.
+      change Int64.max_unsigned with 18446744073709551615%Z.
+      lra.
+  }
+  apply le_IZR.
+  rewrite abs_IZR.
+  unfold smallb_approx_real_range, smallb_approx_range, smallb_thresh in *.
+  rewrite minus_IZR.
+  rewrite mult_IZR.
+  set (q_r := B2R 53 1024 q) in *.
+  assert (Rabs (IZR (ZnearestE q_r) - q_r) <= / 2)%R as NEAR
+      by apply Znearest_imp2.
+  set (q_i := IZR (ZnearestE q_r)) in *.
+  replace  (IZR a' - IZR b' * q_i)%R with
+    (-(IZR b' * (q_i - q_r)) - (q_r * IZR b' - IZR a'))%R by ring.
+  set (delta1 := (q_i - q_r)%R) in *.
+  set (delta2 := (q_r * IZR b' - IZR a')%R) in *.
+  gappa.
-- 
cgit 


From c46cded7f85c692f12c1d5d912ba4a64150a0696 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Wed, 12 Jan 2022 20:15:39 +0100
Subject: proof progresses

---
 kvx/FPDivision64.v | 44 ++++++++++++++++++++++++++++++++++++++------
 1 file changed, 38 insertions(+), 6 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 76959118..c17e4f1b 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -564,6 +564,12 @@ Proof.
   gappa.
 Qed.
 
+Lemma step1_div_longu_a0 :
+  forall b,
+    (0 < Int64.unsigned b)%Z ->
+    (step1_div_longu (Vlong Int64.zero) (Vlong b)) = Vlong Int64.zero.
+Admitted.
+                                         
 Definition smallb_approx_range := 4400000000000%Z.
 Lemma step1_div_longu_correct :
     forall a b,
@@ -573,25 +579,50 @@ Lemma step1_div_longu_correct :
         (Z.abs (Int64.unsigned a - Int64.unsigned b*Int64.unsigned q) <= smallb_approx_range)%Z.
 Proof.
   intros a b b_RANGE.
+
+  pose proof (Int64.unsigned_range a) as a_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in a_RANGE.
+  set (a' := Int64.unsigned a) in *.
+  set (b' := Int64.unsigned b) in *.
+
+  destruct (Z_le_gt_dec a' 0).
+  { assert (a' = 0%Z) as ZERO by lia.
+    exists Int64.zero.
+    rewrite ZERO.
+    rewrite Int64.unsigned_zero.
+    replace (Z.abs (0 - b' * 0))%Z with 0%Z by lia.
+    replace a with Int64.zero; cycle 1.
+    {
+      unfold a' in ZERO.
+      unfold Int64.zero.
+      rewrite <- ZERO.
+      apply Int64.repr_unsigned.
+    }
+    split.
+    { apply step1_div_longu_a0.
+      lia.
+    }
+    unfold smallb_approx_range.
+    lia.
+  }
+
   unfold step1_div_longu.
-  assert (1 < Int64.unsigned b)%Z as b_NOT01 by lia.   
+  assert (1 < b')%Z as b_NOT01 by lia.   
   destruct (step1_real_div_longu_correct a b b_NOT01) as (q & C1E & C1R & C1F & C1S).
   rewrite C1E. cbn.
   unfold Float.to_longu_ne.
   pose proof (ZofB_ne_range_correct 53 1024 q 0 Int64.max_unsigned) as C2.
   rewrite C1F in C2.
-  pose proof (Int64.unsigned_range a) as a_RANGE.
-  change Int64.modulus with 18446744073709551616%Z in a_RANGE.
-  set (a' := Int64.unsigned a) in *.
-  assert (1 <= a')%Z by admit.
+
   
-  set (b' := Int64.unsigned b) in *.
   assert (1 <= IZR a' <= 18446744073709551615)%R as a_RANGE'.
   { split; apply IZR_le; lia. }
   assert (2 <= IZR b' <= IZR smallb_thresh)%R as b_RANGE'.
   { split; apply IZR_le; lia. }
   assert (1 <= IZR b' <= IZR smallb_thresh)%R as b_RANGE'' by lra.
   pose proof (step1_smallb_real (IZR a') (IZR b') a_RANGE' b_RANGE'') as DELTA.
+  fold a' in C1R.
+  fold b' in C1R.
   rewrite <- C1R in DELTA.
 
   assert (0 <= B2R _ _ q)%R as q_NONNEG.
@@ -638,3 +669,4 @@ Proof.
   set (delta1 := (q_i - q_r)%R) in *.
   set (delta2 := (q_r * IZR b' - IZR a')%R) in *.
   gappa.
+Qed.
-- 
cgit 


From 5354fea25dd1537a9ce36fbebcc190f5dda241c1 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Wed, 12 Jan 2022 20:51:23 +0100
Subject: more proofs on step1 small b

---
 kvx/FPDivision64.v | 59 +++++++++++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 58 insertions(+), 1 deletion(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index c17e4f1b..fc2050bb 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -568,7 +568,64 @@ Lemma step1_div_longu_a0 :
   forall b,
     (0 < Int64.unsigned b)%Z ->
     (step1_div_longu (Vlong Int64.zero) (Vlong b)) = Vlong Int64.zero.
-Admitted.
+Proof.
+  intros b b_NOT0.
+  unfold step1_div_longu.
+  unfold step1_real_div_longu.
+  destruct (step1_real_inv_longu_correct b b_NOT0) as
+    (f & C1E & C1R & C1F & C1S).
+  rewrite C1E.
+  cbn.
+  unfold Float.to_longu_ne, Float.of_longu, Float.mul.
+  rewrite Int64.unsigned_zero.
+  set (re :=  (@eq_refl Datatypes.comparison Lt)).
+  assert (- 2 ^ 53 <= 0 <= 2 ^ 53)%Z as SILLY by lia.
+  destruct (BofZ_exact 53 1024 re re 0 SILLY) as (C2R & C2F & C2S).
+  
+  pose proof (Bmult_correct 53 1024 re re Float.binop_nan mode_NE
+                            (BofZ 53 1024 re re 0) f) as C3.
+  rewrite C1F in C3.
+  rewrite C2F in C3.
+  rewrite C1S in C3.
+  rewrite C2S in C3.
+  rewrite Z.ltb_irrefl in C3.
+  rewrite Rlt_bool_true in C3; cycle 1.
+  { clear C3.
+    apply Rabs_relax with (b := bpow radix2 64).
+    { apply bpow_lt. lia. }
+    cbn.
+    rewrite Rmult_0_l.
+    gappa.
+  }
+  rewrite C2R in C3.
+  rewrite Rmult_0_l in C3.
+  destruct C3 as (C3R & C3F & C3Sz).
+  change (true && true) with true in C3F.
+  change (xorb false false) with false in C3Sz.
+  assert (is_nan 53 1024
+           (Bmult 53 1024 re re Float.binop_nan mode_NE 
+                  (BofZ 53 1024 re re 0) f) = false) as NAN.
+  { apply is_finite_not_is_nan.
+    assumption.
+  }
+  pose proof (C3Sz NAN) as C3S.
+  clear NAN C3Sz.
+  pose proof ((ZofB_ne_range_correct 53 1024
+             (Bmult 53 1024 re re Float.binop_nan mode_NE
+                    (BofZ 53 1024 re re 0) f) 0 Int64.max_unsigned)) as C4.
+  rewrite C3R in C4.
+  replace (round radix2 (FLT_exp (3 - 1024 - 53) 53) (round_mode mode_NE) 0)
+    with 0%R in C4 by (cbn ; gappa).
+  rewrite Znearest_IZR in C4.
+  cbn zeta in C4.
+  rewrite Z.leb_refl in C4.
+  change (0 <=? Int64.max_unsigned)%Z with true in C4.
+  rewrite andb_true_r in C4.
+  rewrite andb_true_r in C4.
+  rewrite C3F in C4.
+  rewrite C4.
+  reflexivity.
+Qed.
                                          
 Definition smallb_approx_range := 4400000000000%Z.
 Lemma step1_div_longu_correct :
-- 
cgit 


From 57780f0ec50350d7158c242ed77047ec6cbf5791 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Thu, 13 Jan 2022 20:52:22 +0100
Subject: relative bound of error

---
 kvx/FPDivision64.v | 233 ++++++++++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 229 insertions(+), 4 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index fc2050bb..f9557335 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -49,7 +49,7 @@ Qed.
 Definition approx_inv_thresh := (25/2251799813685248)%R.
 (* 1.11022302462516e-14 *)
     
-Theorem approx_inv_longu_correct :
+Theorem approx_inv_longu_correct_abs :
   forall b,
     (0 < Int64.unsigned b)%Z ->
     exists (f : float),
@@ -283,6 +283,234 @@ Proof.
   gappa.
 Qed.
 
+Local Notation "'rd'" := (round radix2 (FLT_exp (-1074) 53) ZnearestE).
+Local Notation "'rs'" := (round radix2 (FLT_exp (-149) 24) ZnearestE).
+
+Definition approx_inv_rel_thresh := (1049/72057594037927936)%R.
+Theorem approx_inv_longu_correct_rel :
+  forall b,
+    (0 < Int64.unsigned b)%Z ->
+    exists (f : float),
+      (approx_inv_longu (Vlong b)) = Vfloat f /\
+      is_finite _ _ f = true /\ (Rabs(IZR (Int64.unsigned b) * (B2R _ _ f) - 1) <= approx_inv_rel_thresh)%R.
+Proof.
+  intros b NONZ.
+  unfold approx_inv_longu.
+  cbn.
+  econstructor.
+  split.
+  reflexivity.
+  Local Transparent Float.neg Float.of_single Float32.of_longu Float32.div Float.of_longu Float32.of_int Float.of_int.
+  unfold Float.fma, Float.neg, Float.of_single, Float32.of_longu, ExtFloat32.inv, Float32.div, Float.of_longu, ExtFloat32.one, Float32.of_int, ExtFloat.one, Float.of_int.
+  set (re := (@eq_refl Datatypes.comparison Lt)).
+  change (Int.signed (Int.repr 1)) with 1%Z.
+  set (b' := Int64.unsigned b) in *.
+  pose proof (Int64.unsigned_range b) as RANGE.
+  change Int64.modulus with 18446744073709551616%Z in RANGE.                                              
+  assert(1 <= IZR b' <= 18446744073709551616)%R as RANGE'.
+  { split; apply IZR_le; lia.
+  }
+
+  assert (-16777216 <= 1 <= 16777216)%Z as SILLY by lia.
+  destruct (BofZ_exact 24 128 re re 1 SILLY) as (C0R & C0F & _).
+  clear SILLY.
+  set (one_s := (BofZ 24 128 re re 1)) in *.
+  
+  pose proof (BofZ_correct 24 128 re re b') as C1.
+  cbn in C1.
+  rewrite Rlt_bool_true in C1; cycle 1.
+  { clear C1.
+    eapply (Rabs_relax (IZR 18446744073709551616)).
+    lra.
+    set (b'' := IZR b') in *.
+    gappa.
+  }
+  destruct C1 as (C1R & C1F & _).
+  set (b_s :=  (BofZ 24 128 re re b')) in *.
+
+  assert(1 <= B2R 24 128 b_s <= 18446744073709551616)%R as b_s_RANGE.
+  { rewrite C1R.
+    gappa.
+  }
+  assert(B2R 24 128 b_s <> 0)%R as b_s_NONZ by lra.
+  
+  pose proof (Bdiv_correct 24 128 re re Float32.binop_nan mode_NE one_s b_s b_s_NONZ) as C2.
+  rewrite Rlt_bool_true in C2; cycle 1.
+  { clear C2.
+    apply Rabs_relax with (b := 1%R).
+    { cbn; lra. }
+    rewrite C0R.
+    set (r_b_s := B2R 24 128 b_s) in *.
+    cbn.
+    gappa.
+  }
+  
+  destruct C2 as (C2R & C2F & _).
+  set (invb_s := (Bdiv 24 128 re re Float32.binop_nan mode_NE one_s b_s)) in *.
+  rewrite C0F in C2F.
+
+  assert ((1/18446744073709551616 <= B2R 24 128 invb_s <= 1)%R) as invb_s_RANGE.
+  { rewrite C2R.
+    set (r_b_s := B2R 24 128 b_s) in *.
+    rewrite C0R.
+    cbn.
+    gappa.
+  }
+  
+  pose proof (Bconv_correct 24 128 53 1024 re re Float.of_single_nan mode_NE invb_s C2F) as C3.
+  rewrite Rlt_bool_true in C3; cycle 1.
+  { clear C3.
+    set (r_invb_s := (B2R 24 128 invb_s)) in *.
+    apply Rabs_relax with (b := 1%R).
+    { replace 1%R with (bpow radix2 0)%R by reflexivity.
+      apply bpow_lt.
+      lia.
+    }
+    cbn.
+    gappa.
+  }
+  
+  destruct C3 as (C3R & C3F & _).
+  set (invb_d :=  (Bconv 24 128 53 1024 re re Float.of_single_nan mode_NE invb_s)) in *.
+  assert ((1/18446744073709551616 <= B2R 53 1024 invb_d <= 1)%R) as invb_d_RANGE.
+  { 
+    rewrite C3R.
+    set (r_invb_s := B2R 24 128 invb_s) in *.
+    cbn.
+    gappa.
+  }
+
+  pose proof (is_finite_Bopp 53 1024 Float.neg_nan invb_d) as opp_finite.
+  rewrite C3F in opp_finite.
+
+  pose proof (BofZ_correct 53 1024 re re 1) as C4.
+  rewrite Rlt_bool_true in C4; cycle 1.
+  { clear C4.
+    cbn.
+    eapply (Rabs_relax (IZR 18446744073709551616)).
+    lra.
+    set (b'' := IZR b') in *.
+    gappa.
+  }
+  destruct C4 as (C4R & C4F & _).
+  
+  pose proof (BofZ_correct 53 1024 re re b') as C5.
+  cbn in C5.
+  rewrite Rlt_bool_true in C5; cycle 1.
+  { clear C5.
+    eapply (Rabs_relax (IZR 18446744073709551616)).
+    lra.
+    set (b'' := IZR b') in *.
+    gappa.
+  }
+  destruct C5 as (C5R & C5F & _).
+  set (b_d :=  (BofZ 53 1024 re re b')) in *.
+  
+  assert(1 <= B2R 53 1024 b_d <= 18446744073709551616)%R as b_d_RANGE.
+  { rewrite C5R.
+    gappa.
+  }
+
+  pose proof (Bfma_correct 53 1024 re re Float.fma_nan mode_NE
+          (Bopp 53 1024 Float.neg_nan invb_d) (BofZ 53 1024 re re b')
+          (BofZ 53 1024 re re 1) opp_finite C5F C4F) as C6.
+  rewrite Rlt_bool_true in C6; cycle 1.
+  { clear C6.
+    rewrite C4R.
+    rewrite B2R_Bopp.
+    cbn.
+    eapply (Rabs_relax (IZR 18446744073709551616)).
+    { lra. }
+    fold invb_d.
+    fold b_d.
+    set (r_invb_d := B2R 53 1024 invb_d) in *.
+    set (r_b_d := B2R 53 1024 b_d) in *.
+    gappa.
+  }
+  fold b_d in C6.
+  destruct C6 as (C6R & C6F & _).
+
+  pose proof (Bfma_correct 53 1024 re re Float.fma_nan mode_NE
+       (Bfma 53 1024 re re Float.fma_nan mode_NE
+          (Bopp 53 1024 Float.neg_nan invb_d) b_d (BofZ 53 1024 re re 1))
+       invb_d invb_d C6F C3F C3F) as C7.
+  rewrite Rlt_bool_true in C7; cycle 1.
+  { clear C7.
+    rewrite C6R.
+    rewrite B2R_Bopp.
+    eapply (Rabs_relax (bpow radix2 64)).
+    { apply bpow_lt. lia. }
+    rewrite C4R.
+    cbn.
+    set (r_invb_d := B2R 53 1024 invb_d) in *.
+    set (r_b_d := B2R 53 1024 b_d) in *.
+    gappa.
+  }
+  destruct C7 as (C7R & C7F & _).
+
+  split. assumption.
+  rewrite C7R.
+  rewrite C6R.
+  rewrite C5R.
+  rewrite C4R.
+  rewrite B2R_Bopp.
+  rewrite C3R.
+  rewrite C2R.
+  rewrite C1R.
+  rewrite C0R.
+  cbn.
+  set(b1 := IZR b') in *.
+
+  replace (rd 1) with 1%R by gappa.
+  replace (rd (rs (1 / rs b1))) with
+    ((((rd (rs (1 / rs b1)) - (/b1))/(/b1))+1)*(/ b1))%R ; cycle 1.
+  { field. lra. }
+  set (er0 := ((rd (rs (1 / rs b1)) - (/b1))/(/b1))%R).
+  replace (rd b1) with ((((rd b1) - b1)/b1 + 1) * b1)%R; cycle 1.
+  { field. lra. }
+  set (er1 := (((rd b1) - b1)/b1)%R).
+  replace (- ((er0 + 1) * / b1) * ((er1 + 1) * b1) + 1)%R
+    with (1 - (er0 + 1)*(er1 + 1))%R ; cycle 1.
+  { field. lra. }
+  set (z0 := (1 - (er0 + 1) * (er1 + 1))%R).
+  assert (Rabs er0 <= 257/2147483648)%R as er0_ABS.
+  { unfold er0.
+    gappa.
+  }
+  assert (Rabs er1 <= 1/9007199254740992)%R as er1_ABS.
+  { unfold er1.
+    gappa.
+  }
+  replace (rd z0) with ((rd(z0)-z0)+z0)%R by ring.
+  set (ea0 := (rd(z0)-z0)%R).
+  assert (Rabs ea0 <= 1/75557863725914323419136)%R as ea0_ABS.
+  { unfold ea0. unfold z0.
+    gappa.
+  }
+  set (z1 :=  ((ea0 + z0) * ((er0 + 1) * / b1) + (er0 + 1) * / b1)%R).
+  replace (rd z1) with ((((rd z1)-z1)/z1+1)*z1)%R; cycle 1.
+  { field.
+    unfold z1.
+    unfold z0.
+    gappa.
+  }
+  set (er2 := ((rd z1 - z1) / z1)%R).
+  assert (Rabs er2 <= 1/9007199254740992)%R as er2_ABS.
+  { unfold er2.
+    unfold z1, z0.
+    gappa.
+  }
+  unfold z1, z0.
+  replace  (b1 *
+      ((er2 + 1) *
+       ((ea0 + (1 - (er0 + 1) * (er1 + 1))) * ((er0 + 1) * / b1) +
+          (er0 + 1) * / b1)) - 1)%R
+      with (-er0*er0*er1*er2 - er0*er0*er1 + ea0*er0*er2 - er0*er0*er2 - 2*er0*er1*er2 + ea0*er0 - er0*er0 - 2*er0*er1 + ea0*er2 - er1*er2 + ea0 - er1 + er2)%R; cycle 1.
+  { field. lra. }
+  unfold approx_inv_rel_thresh.
+  gappa.
+Qed.
+
 Definition step1_real_inv_longu b :=
   let invb_s := ExtValues.invfs (Val.maketotal (Val.singleoflongu b)) in
   Val.floatofsingle invb_s.
@@ -290,9 +518,6 @@ Definition step1_real_inv_longu b :=
 Definition step1_real_inv_thresh := (3/33554432)%R.
 (* 8.94069671630859e-8 *)
 
-Local Notation "'rd'" := (round radix2 (FLT_exp (-1074) 53) ZnearestE).
-Local Notation "'rs'" := (round radix2 (FLT_exp (-149) 24) ZnearestE).
-
 Theorem step1_real_inv_longu_correct :
   forall b,
     (0 < Int64.unsigned b)%Z ->
-- 
cgit 


From 0742df3a567f98445e95669f92a291fda733d286 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 14 Jan 2022 11:14:36 +0100
Subject: some progress

---
 kvx/FPDivision64.v | 75 ++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 75 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index f9557335..767e7ae0 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -952,3 +952,78 @@ Proof.
   set (delta2 := (q_r * IZR b' - IZR a')%R) in *.
   gappa.
 Qed.
+
+Lemma range_up_le :
+  forall a x b,
+    (IZR a <= IZR x <= (IZR b) - 1)%R ->
+    (a <= x < b)%Z.
+Proof.
+  intros until b. intro RANGE.
+  split.
+  { apply le_IZR. lra. }
+  assert (x <= b-1)%Z.
+  { apply le_IZR. rewrite minus_IZR. lra. }
+  lia.
+Qed.
+
+Lemma find_quotient:
+  forall (a b : Z)
+         (b_POS : (b > 0)%Z)
+         (invb : R)
+         (eps : R)
+         (invb_EPS : (Rabs (invb * IZR b - 1) <= eps)%R)
+         (SMALL : (IZR (Z.abs a)*eps < IZR b / 2)%R),
+    (a / b)%Z =
+      let q := ZnearestE ((IZR a) * invb) in
+      if (b*q >? a)%Z
+      then (q-1)%Z
+      else q.
+Proof.
+  intros.
+  rewrite <- Rabs_Zabs in SMALL.
+  set (q := ZnearestE (IZR a * invb)%R).
+  cbn zeta.
+  set (b' := IZR b) in *.
+  set (a' := IZR a) in *.
+  assert (1 <= b')%R as b_POS'.
+  { apply IZR_le.
+    lia.
+  }
+  
+  pose proof (Znearest_imp2 (fun x : Z => negb (Z.even x)) (a' * invb)) as NEAR.
+  fold q in NEAR.
+  set (q' := IZR q) in *.
+  assert ((Rabs a') * Rabs (invb * b' - 1) <= (Rabs a') * eps)%R as S1.
+  { apply Rmult_le_compat_l.
+    apply Rabs_pos.
+    assumption. }
+  rewrite <- Rabs_mult in S1.
+  assert ((Rabs b') * Rabs (q' - a' * invb) <= (Rabs b') / 2)%R as S2.
+  { apply Rmult_le_compat_l.
+    apply Rabs_pos.
+    assumption. }
+  rewrite <- Rabs_mult in S2.
+  rewrite (Rabs_right b') in S2 by lra.
+  pose proof (Rabs_triang (a' * (invb * b' - 1))
+                          (b' * (q' - a' * invb))) as TRIANGLE.
+  replace (a' * (invb * b' - 1) + b' * (q' - a' * invb))%R with
+    (b' * q' - a')%R in TRIANGLE by ring.
+  assert (Rabs (b' * q' - a') < b')%R as DELTA by lra.
+  
+  pose proof (Zgt_cases (b * q) a)%Z as CASE.
+  destruct (_ >? _)%Z.
+  { admit. }
+  apply Zdiv_unique with (b := (a - q*b)%Z).
+  ring.
+  apply range_up_le.
+  rewrite minus_IZR.
+  rewrite mult_IZR.
+  fold a' b' q'.
+  
+  
+  Search (Rabs _ * Rabs _)%R.
+Definition step2_real_div_long a b :=
+  Val.mulf (Val.maketotal (Val.floatoflong a)) (step1_real_inv_longu b).
+
+Definition step2_div_long a b :=
+  Val.maketotal (Val.longuoffloat_ne (step2_real_div_long a)).
-- 
cgit 


From a5df0e05f97f5ce4ee35723089cbd4a83435d37f Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 14 Jan 2022 11:39:22 +0100
Subject: some more proof

---
 kvx/FPDivision64.v | 21 +++++++++++++++------
 1 file changed, 15 insertions(+), 6 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 767e7ae0..6d82ae67 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -953,6 +953,7 @@ Proof.
   gappa.
 Qed.
 
+(*
 Lemma range_up_le :
   forall a x b,
     (IZR a <= IZR x <= (IZR b) - 1)%R ->
@@ -965,6 +966,7 @@ Proof.
   { apply le_IZR. rewrite minus_IZR. lra. }
   lia.
 Qed.
+ *)
 
 Lemma find_quotient:
   forall (a b : Z)
@@ -1015,13 +1017,20 @@ Proof.
   { admit. }
   apply Zdiv_unique with (b := (a - q*b)%Z).
   ring.
-  apply range_up_le.
-  rewrite minus_IZR.
-  rewrite mult_IZR.
-  fold a' b' q'.
   
-  
-  Search (Rabs _ * Rabs _)%R.
+  split.
+  { rewrite Z.mul_comm. lia. }
+  rewrite <- Rabs_Ropp in DELTA.
+  unfold b', q', a' in DELTA.
+  rewrite <- mult_IZR in DELTA.
+  rewrite <- minus_IZR in DELTA.
+  rewrite <- opp_IZR in DELTA.
+  rewrite Rabs_Zabs in DELTA.
+  apply lt_IZR in DELTA.
+  rewrite Z.abs_eq in DELTA by lia.
+  lia.
+Admitted.
+
 Definition step2_real_div_long a b :=
   Val.mulf (Val.maketotal (Val.floatoflong a)) (step1_real_inv_longu b).
 
-- 
cgit 


From 0ca71551f9d47dcc04e7a97bb294a0379732bc04 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 14 Jan 2022 11:47:58 +0100
Subject: find_quotient

---
 kvx/FPDivision64.v | 24 +++++++++++++++++-------
 1 file changed, 17 insertions(+), 7 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 6d82ae67..8f0d2fa1 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1014,12 +1014,18 @@ Proof.
   
   pose proof (Zgt_cases (b * q) a)%Z as CASE.
   destruct (_ >? _)%Z.
-  { admit. }
-  apply Zdiv_unique with (b := (a - q*b)%Z).
-  ring.
+  { unfold b', q', a' in DELTA.
+    rewrite <- mult_IZR in DELTA.
+    rewrite <- minus_IZR in DELTA.
+    rewrite Rabs_Zabs in DELTA.
+    apply lt_IZR in DELTA.
+    rewrite Z.abs_eq in DELTA by lia.
+    
+    apply Zdiv_unique with (b := (a - (q-1)*b)%Z).
+    ring.
+    split; lia.
+  }
   
-  split.
-  { rewrite Z.mul_comm. lia. }
   rewrite <- Rabs_Ropp in DELTA.
   unfold b', q', a' in DELTA.
   rewrite <- mult_IZR in DELTA.
@@ -1028,8 +1034,12 @@ Proof.
   rewrite Rabs_Zabs in DELTA.
   apply lt_IZR in DELTA.
   rewrite Z.abs_eq in DELTA by lia.
-  lia.
-Admitted.
+
+  apply Zdiv_unique with (b := (a - q*b)%Z).
+  ring.
+  
+  split; lia.
+Qed.
 
 Definition step2_real_div_long a b :=
   Val.mulf (Val.maketotal (Val.floatoflong a)) (step1_real_inv_longu b).
-- 
cgit 


From d403269fa27446c1f7281e35a51ea0eb6483c987 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 14 Jan 2022 21:10:30 +0100
Subject: find_quotient

---
 kvx/FPDivision64.v | 64 ++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 64 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 8f0d2fa1..b5052917 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -968,6 +968,7 @@ Proof.
 Qed.
  *)
 
+(*
 Lemma find_quotient:
   forall (a b : Z)
          (b_POS : (b > 0)%Z)
@@ -1040,6 +1041,69 @@ Proof.
   
   split; lia.
 Qed.
+ *)
+
+Lemma find_quotient:
+  forall (a b : Z)
+         (b_POS : (b > 0)%Z)
+         (invb : R)
+         (qr : R)
+         (GAP : (Rabs (IZR a / IZR b - qr) < /2)%R),
+    (a / b)%Z =
+      let q := ZnearestE qr in
+      if (b*q >? a)%Z
+      then (q-1)%Z
+      else q.
+Proof.
+  intros.
+  set (q := ZnearestE qr).
+  cbn zeta.
+  set (b' := IZR b) in *.
+  set (a' := IZR a) in *.
+  assert (1 <= b')%R as b_POS'.
+  { apply IZR_le.
+    lia.
+  }
+  
+  pose proof (Znearest_imp2 (fun x : Z => negb (Z.even x)) qr) as ROUND.
+  fold q in ROUND.
+  set (q' := IZR q) in *.
+  
+  pose proof (Rabs_triang (a' / b' - qr)
+                          (qr - q'))%R as TRIANGLE.
+  replace ((a' / b' - qr) +  (qr - q'))%R with
+    (a' / b' - q')%R in TRIANGLE by ring.
+  rewrite <- Rabs_Ropp in ROUND.
+  replace (- (q' - qr))%R with (qr - q')%R in ROUND by ring.
+  assert (Z.abs (a - b*q) < b)%Z as DELTA.
+  { apply lt_IZR.
+    rewrite <- Rabs_Zabs.
+    rewrite minus_IZR.
+    rewrite mult_IZR.
+    fold a' q' b'.
+    apply Rmult_lt_reg_r with (r := (/b')%R).
+    { apply Rinv_0_lt_compat. lra. }
+    rewrite Rinv_r by lra.
+    replace (/ b')%R with (/ Rabs(b'))%R ; cycle 1.
+    { f_equal.
+      apply Rabs_pos_eq. lra. }
+    rewrite <- Rabs_Rinv by lra.
+    rewrite <- Rabs_mult.
+    replace ((a' - b' * q') * / b')%R with (a'/b' - q')%R by (field ; lra).
+    lra.
+  }
+   
+  pose proof (Zgt_cases (b * q) a)%Z as CASE.
+  destruct (_ >? _)%Z.
+  { apply Zdiv_unique with (b := (a - (q-1)*b)%Z).
+    ring.
+    split; lia.
+  }
+
+  apply Zdiv_unique with (b := (a - q*b)%Z).
+  ring.
+  split; lia.
+Qed.
 
 Definition step2_real_div_long a b :=
   Val.mulf (Val.maketotal (Val.floatoflong a)) (step1_real_inv_longu b).
-- 
cgit 


From 0284db0e9b9f73fa98f2c92defdf61ea90f91b1a Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Wed, 19 Jan 2022 15:51:12 +0100
Subject: so that it compiles

---
 kvx/FPDivision64.v | 60 ++++++++++++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 58 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index b5052917..64ac77a1 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1106,7 +1106,63 @@ Proof.
 Qed.
 
 Definition step2_real_div_long a b :=
-  Val.mulf (Val.maketotal (Val.floatoflong a)) (step1_real_inv_longu b).
+  Val.mulf (Val.maketotal (Val.floatoflong a)) (approx_inv_longu b).
+
+Definition smalla_thresh := 35184372088832%Z.
+
+Lemma step2_real_div_long_smalla_correct :
+    forall (a b : int64)
+           (a_SMALL : (Z.abs (Int64.signed a) <= smalla_thresh)%Z)
+           (b_NOT0 : (0 < Int64.unsigned b)%Z),
+    exists (q : float),
+      (step2_real_div_long (Vlong a) (Vlong b)) = Vfloat q /\
+        (Rabs ((B2R _ _ q) - (IZR (Int64.signed a)) / (IZR (Int64.unsigned b))) < (1/2))%R.
+Proof.
+  intros.
+  unfold step2_real_div_long.
+  destruct (approx_inv_longu_correct_rel b b_NOT0) as (f & C0E & C0F & C0R).
+  rewrite C0E.
+  econstructor.
+  split. reflexivity.
+  Local Transparent Float.of_long.
+  unfold Float.mul, Float.of_long.
+  set (re := (@eq_refl Datatypes.comparison Lt)) in *.
+  set (a' := Int64.signed a) in *.
+  set (b' := Int64.unsigned b) in *.
+  assert(Rabs (IZR a') <= IZR smalla_thresh)%R as a_RANGE'.
+  { rewrite Rabs_Zabs.
+    apply IZR_le.
+    assumption.
+  }
+  assert (- 2 ^ 53 <= a' <= 2 ^ 53)%Z as SILLY.
+  { unfold smalla_thresh in a_SMALL.
+    apply Z.abs_le.
+    lia.
+  } 
+  destruct (BofZ_exact 53 1024 re re (Int64.signed a) SILLY) as (C1R & C1F & C1S).
+  fold a' in C1R.
+  pose proof (Bmult_correct 53 1024 re re Float.binop_nan mode_NE (BofZ 53 1024 re re a') f) as C2.
+  rewrite Rlt_bool_true in C2 ; cycle 1.
+  { apply Rabs_relax with (b := bpow radix2 53).
+    { apply bpow_lt. lia. }
+    cbn.
+    rewrite C1R.
+    unfold approx_inv_rel_thresh in C0R.
+    admit.
+  }
+  cbn.
+Admitted.
+
+Definition step2_div_long' a b :=
+  Val.maketotal (Val.longuoffloat_ne (step2_real_div_long a b)).
 
 Definition step2_div_long a b :=
-  Val.maketotal (Val.longuoffloat_ne (step2_real_div_long a)).
+  let q := Val.maketotal (Val.longuoffloat_ne (step2_div_long' a b)) in
+  Val.select (Val.cmpl_bool Clt (Val.subl a (Val.mull q b)) (Vlong Int64.zero))
+             (Val.addl q (Vlong Int64.mone)) q Tlong.
+
+Lemma step2_div_long_smalla_correct :
+    forall a b,
+    (Z.abs (Int64.signed a) <= smalla_thresh)%Z ->
+      step2_div_long (Vlong a) (Vlong b) = Vlong (Int64.repr (Int64.signed a / Int64.unsigned b))%Z.
+Admitted.
-- 
cgit 


From a7db868181d74403449ce436028f0658f284088e Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 09:23:43 +0100
Subject: progress

---
 kvx/FPDivision64.v | 14 ++++++++++++--
 1 file changed, 12 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 64ac77a1..0dfa9882 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1127,8 +1127,13 @@ Proof.
   Local Transparent Float.of_long.
   unfold Float.mul, Float.of_long.
   set (re := (@eq_refl Datatypes.comparison Lt)) in *.
+  pose proof (Int64.unsigned_range b) as b_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in b_RANGE.
   set (a' := Int64.signed a) in *.
   set (b' := Int64.unsigned b) in *.
+  assert (1 <= IZR b' <= 18446744073709551615)%R as b_RANGE'.
+  { split; apply IZR_le; lia.
+  }
   assert(Rabs (IZR a') <= IZR smalla_thresh)%R as a_RANGE'.
   { rewrite Rabs_Zabs.
     apply IZR_le.
@@ -1143,12 +1148,17 @@ Proof.
   fold a' in C1R.
   pose proof (Bmult_correct 53 1024 re re Float.binop_nan mode_NE (BofZ 53 1024 re re a') f) as C2.
   rewrite Rlt_bool_true in C2 ; cycle 1.
-  { apply Rabs_relax with (b := bpow radix2 53).
+  { clear C2.
+    apply Rabs_relax with (b := bpow radix2 53).
     { apply bpow_lt. lia. }
     cbn.
     rewrite C1R.
     unfold approx_inv_rel_thresh in C0R.
-    admit.
+    replace (B2R 53 1024 f) with
+      ((1/IZR b') * ((IZR b' * B2R 53 1024 f - 1) + 1))%R ; cycle 1.
+    { field. lra. }
+    unfold smalla_thresh in *.
+    gappa.
   }
   cbn.
 Admitted.
-- 
cgit 


From fa79debdca4121f2435750b8090e83b541887979 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 09:35:23 +0100
Subject: progress

---
 kvx/FPDivision64.v | 11 +++++++++--
 1 file changed, 9 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 0dfa9882..8d40e88e 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1145,7 +1145,7 @@ Proof.
     lia.
   } 
   destruct (BofZ_exact 53 1024 re re (Int64.signed a) SILLY) as (C1R & C1F & C1S).
-  fold a' in C1R.
+  fold a' in C1R, C1F, C1S.
   pose proof (Bmult_correct 53 1024 re re Float.binop_nan mode_NE (BofZ 53 1024 re re a') f) as C2.
   rewrite Rlt_bool_true in C2 ; cycle 1.
   { clear C2.
@@ -1160,7 +1160,14 @@ Proof.
     unfold smalla_thresh in *.
     gappa.
   }
-  cbn.
+  rewrite C0F in C2.
+  rewrite C1R in C2.
+  rewrite C1F in C2.
+  rewrite C1S in C2.
+  cbn in C2.
+  destruct C2 as (C2R & C2F & _).
+  rewrite C2R.
+  admit.
 Admitted.
 
 Definition step2_div_long' a b :=
-- 
cgit 


From 3b36bd818ffc20f7a455da46a89641c02491a4e1 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 10:02:43 +0100
Subject: progress

---
 kvx/FPDivision64.v | 20 +++++++++++++++++---
 1 file changed, 17 insertions(+), 3 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 8d40e88e..6894460f 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1116,7 +1116,7 @@ Lemma step2_real_div_long_smalla_correct :
            (b_NOT0 : (0 < Int64.unsigned b)%Z),
     exists (q : float),
       (step2_real_div_long (Vlong a) (Vlong b)) = Vfloat q /\
-        (Rabs ((B2R _ _ q) - (IZR (Int64.signed a)) / (IZR (Int64.unsigned b))) < (1/2))%R.
+        (Rabs ((B2R _ _ q) - (IZR (Int64.signed a)) / (IZR (Int64.unsigned b))) <= (255/256))%R.
 Proof.
   intros.
   unfold step2_real_div_long.
@@ -1167,8 +1167,22 @@ Proof.
   cbn in C2.
   destruct C2 as (C2R & C2F & _).
   rewrite C2R.
-  admit.
-Admitted.
+  replace (IZR a' * (B2R 53 1024 f))%R with
+    ((IZR a'/IZR b') * ((IZR b' * B2R 53 1024 f - 1) + 1))%R ; cycle 1.
+  { field. lra. }
+  set (delta1 := (IZR b' * B2R 53 1024 f - 1)%R) in *.
+  set (q1 := (IZR a' / IZR b' * (delta1 + 1))%R).
+  replace (rd q1) with (((rd q1) - q1) + q1)%R by ring.
+  set (delta2 := ((rd q1) - q1)%R).
+  unfold q1.
+  replace (delta2 + IZR a' / IZR b' * (delta1 + 1) - IZR a' / IZR b')%R with
+    (delta2 + (IZR a' / IZR b') * delta1)%R by ring.
+  unfold delta2.
+  unfold q1.
+  unfold approx_inv_rel_thresh in *.
+  unfold smalla_thresh in *.
+  gappa.
+Qed.
 
 Definition step2_div_long' a b :=
   Val.maketotal (Val.longuoffloat_ne (step2_real_div_long a b)).
-- 
cgit 


From db85b47c5487ddfa051b01e444d2be555013e7f7 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 10:04:21 +0100
Subject: progress

---
 kvx/FPDivision64.v | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 6894460f..9ed12064 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1108,7 +1108,7 @@ Qed.
 Definition step2_real_div_long a b :=
   Val.mulf (Val.maketotal (Val.floatoflong a)) (approx_inv_longu b).
 
-Definition smalla_thresh := 35184372088832%Z.
+Definition smalla_thresh := 34184372088832%Z.
 
 Lemma step2_real_div_long_smalla_correct :
     forall (a b : int64)
@@ -1116,7 +1116,7 @@ Lemma step2_real_div_long_smalla_correct :
            (b_NOT0 : (0 < Int64.unsigned b)%Z),
     exists (q : float),
       (step2_real_div_long (Vlong a) (Vlong b)) = Vfloat q /\
-        (Rabs ((B2R _ _ q) - (IZR (Int64.signed a)) / (IZR (Int64.unsigned b))) <= (255/256))%R.
+        (Rabs ((B2R _ _ q) - (IZR (Int64.signed a)) / (IZR (Int64.unsigned b))) <= (32767/65536))%R.
 Proof.
   intros.
   unfold step2_real_div_long.
-- 
cgit 


From 6e45966f1cd5ea6feadb044f2fda2e1b1c81e57c Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 10:07:56 +0100
Subject: progress

---
 kvx/FPDivision64.v | 5 ++++-
 1 file changed, 4 insertions(+), 1 deletion(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 9ed12064..72ad43d1 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1116,7 +1116,8 @@ Lemma step2_real_div_long_smalla_correct :
            (b_NOT0 : (0 < Int64.unsigned b)%Z),
     exists (q : float),
       (step2_real_div_long (Vlong a) (Vlong b)) = Vfloat q /\
-        (Rabs ((B2R _ _ q) - (IZR (Int64.signed a)) / (IZR (Int64.unsigned b))) <= (32767/65536))%R.
+        (Rabs ((B2R _ _ q) - (IZR (Int64.signed a)) / (IZR (Int64.unsigned b))) <= (32767/65536))%R /\
+      is_finite _ _ q = true.
 Proof.
   intros.
   unfold step2_real_div_long.
@@ -1166,6 +1167,8 @@ Proof.
   rewrite C1S in C2.
   cbn in C2.
   destruct C2 as (C2R & C2F & _).
+  split.
+  2: exact C2F.
   rewrite C2R.
   replace (IZR a' * (B2R 53 1024 f))%R with
     ((IZR a'/IZR b') * ((IZR b' * B2R 53 1024 f - 1) + 1))%R ; cycle 1.
-- 
cgit 


From 64552329d9e6c1ae4071f2f144f7c767287f75c6 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 11:31:04 +0100
Subject: fix issue

---
 kvx/FPDivision64.v | 71 ++++++++++++++++++++++++++++++++++++++++++++++++++----
 1 file changed, 66 insertions(+), 5 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 72ad43d1..7f7aa71a 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1188,15 +1188,76 @@ Proof.
 Qed.
 
 Definition step2_div_long' a b :=
-  Val.maketotal (Val.longuoffloat_ne (step2_real_div_long a b)).
+  Val.maketotal (Val.longoffloat_ne (step2_real_div_long a b)).
 
 Definition step2_div_long a b :=
-  let q := Val.maketotal (Val.longuoffloat_ne (step2_div_long' a b)) in
+  let q := step2_div_long' a b in
   Val.select (Val.cmpl_bool Clt (Val.subl a (Val.mull q b)) (Vlong Int64.zero))
              (Val.addl q (Vlong Int64.mone)) q Tlong.
 
+Lemma Znearest_lub :
+  forall choice (n : Z) (x : R), (IZR n <= x)%R -> (n <= Znearest choice x)%Z.
+Proof.
+  intros until x. intro BND.
+  pose proof (Zfloor_lub n x BND).
+  pose proof (Znearest_ge_floor choice x).
+  lia.
+Qed.
+
+Lemma Znearest_glb :
+  forall choice (n : Z) (x : R), (x <= IZR n)%R -> (Znearest choice x <= n)%Z.
+Proof.
+  intros until x. intro BND.
+  pose proof (Zceil_glb n x BND).
+  pose proof (Znearest_le_ceil choice x).
+  lia.
+Qed.
+
 Lemma step2_div_long_smalla_correct :
-    forall a b,
-    (Z.abs (Int64.signed a) <= smalla_thresh)%Z ->
+    forall a b
+      (a_SMALL : (Z.abs (Int64.signed a) <= smalla_thresh)%Z)
+      (b_NOT0 : (0 < Int64.unsigned b)%Z),
       step2_div_long (Vlong a) (Vlong b) = Vlong (Int64.repr (Int64.signed a / Int64.unsigned b))%Z.
-Admitted.
+Proof.
+  intros.
+  pose proof (Int64.unsigned_range b) as b_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in b_RANGE.
+  set (a' := (Int64.signed a)) in *.
+  set (b' := (Int64.unsigned b)) in *.
+  assert (Rabs (IZR a') <= IZR smalla_thresh)%R as a_RANGE'.
+  { rewrite Rabs_Zabs.
+    apply IZR_le.
+    assumption.
+  }
+  assert (1 <= IZR b' <= 18446744073709551615)%R as b_RANGE'.
+  { split; apply IZR_le; lia.
+  } 
+  destruct (step2_real_div_long_smalla_correct a b a_SMALL b_NOT0) as (q & C1R & C1E & C1F).
+  fold a' b' in C1E.
+  assert ((Int64.min_signed <=? ZnearestE (B2R 53 1024 q))=true)%Z as q_LOW.
+  { apply Zle_imp_le_bool.
+    change Int64.min_signed with (-9223372036854775808)%Z.
+  set (q' :=  B2R 53 1024 q) in *.
+    apply Znearest_lub.
+    replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring.
+    unfold smalla_thresh in a_RANGE'.
+    gappa.
+  }
+  assert ((ZnearestE (B2R 53 1024 q) <=? Int64.max_signed)=true)%Z as q_HIGH.
+  { apply Zle_imp_le_bool.
+    change Int64.max_signed with (9223372036854775807)%Z.
+    apply Znearest_glb.
+    set (q' :=  B2R 53 1024 q) in *.
+    replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring.
+    unfold smalla_thresh in a_RANGE'.
+    gappa.
+  }
+  unfold step2_div_long, step2_div_long'.
+  rewrite C1R.
+  cbn.
+  unfold Float.to_long_ne.
+  rewrite (ZofB_ne_range_correct _ _ q Int64.min_signed Int64.max_signed).
+  rewrite C1F.
+  rewrite q_LOW.
+  rewrite q_HIGH.
+  cbn.
-- 
cgit 


From f2ea4988ec81d931ad035595855088c4cb667477 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 11:40:16 +0100
Subject: some progress

---
 kvx/FPDivision64.v | 26 +++++++++++++++++++++++++-
 1 file changed, 25 insertions(+), 1 deletion(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 7f7aa71a..63b48140 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1213,6 +1213,23 @@ Proof.
   lia.
 Qed.
 
+Lemma function_ite :
+  forall {A B : Type} (fn : A->B) (b : bool) (x y: A),
+    fn (if b then x else y) = (if b then fn x else fn y).
+Proof.
+  intros.
+  destruct b; reflexivity.
+Qed.
+
+Lemma normalize_ite :
+  forall ty (b : bool) x y,
+    Val.normalize (if b then x else y) ty =
+      (if b then Val.normalize x ty else Val.normalize y ty).
+Proof.
+  intros.
+  destruct b; reflexivity.
+Qed.
+  
 Lemma step2_div_long_smalla_correct :
     forall a b
       (a_SMALL : (Z.abs (Int64.signed a) <= smalla_thresh)%Z)
@@ -1237,8 +1254,8 @@ Proof.
   assert ((Int64.min_signed <=? ZnearestE (B2R 53 1024 q))=true)%Z as q_LOW.
   { apply Zle_imp_le_bool.
     change Int64.min_signed with (-9223372036854775808)%Z.
-  set (q' :=  B2R 53 1024 q) in *.
     apply Znearest_lub.
+    set (q' :=  B2R 53 1024 q) in *.
     replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring.
     unfold smalla_thresh in a_RANGE'.
     gappa.
@@ -1261,3 +1278,10 @@ Proof.
   rewrite q_LOW.
   rewrite q_HIGH.
   cbn.
+  rewrite normalize_ite.
+  cbn.
+  rewrite <- (function_ite Vlong).
+  f_equal.
+  set (q' :=  B2R 53 1024 q) in *.
+  
+  
-- 
cgit 


From c61b77be54b4696e5c5d5ed2a60f00fbc3dedb56 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 14:35:52 +0100
Subject: progress

---
 kvx/FPDivision64.v | 70 ++++++++++++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 68 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 63b48140..766ff80f 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1192,7 +1192,7 @@ Definition step2_div_long' a b :=
 
 Definition step2_div_long a b :=
   let q := step2_div_long' a b in
-  Val.select (Val.cmpl_bool Clt (Val.subl a (Val.mull q b)) (Vlong Int64.zero))
+  Val.select (Val.cmpl_bool Cgt (Val.subl (Val.mull q b) a) (Vlong Int64.zero))
              (Val.addl q (Vlong Int64.mone)) q Tlong.
 
 Lemma Znearest_lub :
@@ -1229,7 +1229,44 @@ Proof.
   intros.
   destruct b; reflexivity.
 Qed.
-  
+
+
+Lemma int64_mul_signed_unsigned:
+    forall x y : int64,
+      Int64.mul x y = Int64.repr (Int64.signed x * Int64.unsigned y).
+Proof.
+  intros.
+  unfold Int64.mul.
+  apply Int64.eqm_samerepr.
+  apply Int64.eqm_mult.
+  - apply Int64.eqm_sym.
+    apply Int64.eqm_signed_unsigned.
+  - apply Int64.eqm_refl.
+Qed.
+
+Lemma int64_eqm_signed_repr:
+  forall z : Z, Int64.eqm z (Int64.signed (Int64.repr z)).
+Proof.
+  intros.
+  apply Int64.eqm_trans with (y := Int64.unsigned (Int64.repr z)).
+  - apply Int64.eqm_unsigned_repr.
+  - apply Int64.eqm_sym.
+    apply Int64.eqm_signed_unsigned.
+Qed.
+
+Lemma signed_repr_sub:
+  forall x y,
+    Int64.repr (Int64.signed (Int64.repr x) - y) =
+    Int64.repr (x - y).
+Proof.
+  intros.
+  apply Int64.eqm_samerepr.
+  apply Int64.eqm_sub.
+  - apply Int64.eqm_sym.
+    apply int64_eqm_signed_repr.
+  - apply Int64.eqm_refl.
+Qed.
+                                                                        
 Lemma step2_div_long_smalla_correct :
     forall a b
       (a_SMALL : (Z.abs (Int64.signed a) <= smalla_thresh)%Z)
@@ -1282,6 +1319,35 @@ Proof.
   cbn.
   rewrite <- (function_ite Vlong).
   f_equal.
+  unfold Int64.lt.
   set (q' :=  B2R 53 1024 q) in *.
+  fold a'.
+  assert (Int64.signed (Int64.repr (ZnearestE q')) = ZnearestE q') as q_SIGNED.
+  { apply Int64.signed_repr.
+    split; lia.
+  }
+  rewrite Int64.add_signed.
+  rewrite q_SIGNED.
+  rewrite Int64.signed_mone.
+  rewrite Int64.signed_zero.
+  rewrite <- (function_ite Int64.repr).
+  f_equal.
+  replace  (ZnearestE q' + -1)%Z with (ZnearestE q' - 1)%Z by ring.
   
+  set (q'' :=  (ZnearestE q')) in *.
+  Check Int64.sub_signed.
+  fold a'.
+  rewrite int64_mul_signed_unsigned.
+  rewrite q_SIGNED.
+  fold b'.
+
+  rewrite Int64.sub_signed.
+  fold a'.
+  rewrite signed_repr_sub.
+
+   
   
+  Check Int64.signed_repr.
+  
+  rewrite find_quotient with (qr := q').
+  cbn.
-- 
cgit 


From 74dec8e174e1a49facaf68d45b3c920c5547123d Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 14:48:59 +0100
Subject: progress

---
 kvx/FPDivision64.v | 24 +++++++++++++++++-------
 1 file changed, 17 insertions(+), 7 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 766ff80f..23cf73e3 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1045,8 +1045,7 @@ Qed.
 
 Lemma find_quotient:
   forall (a b : Z)
-         (b_POS : (b > 0)%Z)
-         (invb : R)
+         (b_POS : (0 < b)%Z)
          (qr : R)
          (GAP : (Rabs (IZR a / IZR b - qr) < /2)%R),
     (a / b)%Z =
@@ -1345,9 +1344,20 @@ Proof.
   fold a'.
   rewrite signed_repr_sub.
 
-   
-  
-  Check Int64.signed_repr.
+  rewrite Int64.signed_repr; cycle 1.
+  { admit. }
+
+  assert ((Rabs (IZR a' / IZR b' - q') < / 2)%R) as HALF.
+  { replace (IZR a' / IZR b' - q')%R with
+      (-(q' - IZR a' / IZR b'))%R by ring.
+    rewrite Rabs_Ropp.
+    lra.
+  }
+  rewrite (find_quotient a' b' b_NOT0 q' HALF).
+  fold q''.
+  unfold zlt.
   
-  rewrite find_quotient with (qr := q').
-  cbn.
+  destruct (Z_lt_dec 0).
+  - replace ( _ >? _ )%Z with true by lia. reflexivity.
+  - replace ( _ >? _ )%Z with false by lia. reflexivity.
+Admitted.
-- 
cgit 


From 9bb316c0f32cd88cd95895111401ff7995938fec Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 15:47:56 +0100
Subject: progress

---
 kvx/FPDivision64.v | 42 +++++++++++++++++++++++++++++++-----------
 1 file changed, 31 insertions(+), 11 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 23cf73e3..0d11c3cb 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1269,7 +1269,8 @@ Qed.
 Lemma step2_div_long_smalla_correct :
     forall a b
       (a_SMALL : (Z.abs (Int64.signed a) <= smalla_thresh)%Z)
-      (b_NOT0 : (0 < Int64.unsigned b)%Z),
+      (b_NOT0 : (0 < Int64.unsigned b)%Z)
+      (b_NOT_VERY_BIG : (Int64.unsigned b <= Int64.max_signed)%Z),
       step2_div_long (Vlong a) (Vlong b) = Vlong (Int64.repr (Int64.signed a / Int64.unsigned b))%Z.
 Proof.
   intros.
@@ -1344,20 +1345,39 @@ Proof.
   fold a'.
   rewrite signed_repr_sub.
 
-  rewrite Int64.signed_repr; cycle 1.
-  { admit. }
-
   assert ((Rabs (IZR a' / IZR b' - q') < / 2)%R) as HALF.
   { replace (IZR a' / IZR b' - q')%R with
       (-(q' - IZR a' / IZR b'))%R by ring.
     rewrite Rabs_Ropp.
     lra.
   }
-  rewrite (find_quotient a' b' b_NOT0 q' HALF).
-  fold q''.
-  unfold zlt.
-  
-  destruct (Z_lt_dec 0).
-  - replace ( _ >? _ )%Z with true by lia. reflexivity.
-  - replace ( _ >? _ )%Z with false by lia. reflexivity.
+  pose proof (find_quotient a' b' b_NOT0 q' HALF) as QUOTIENT.
+  fold q'' in QUOTIENT.
+  cbn zeta in QUOTIENT.
+
+  assert (b' <> 0)%Z as NONZ by lia.
+  pose proof (Zmod_eq_full a' b' NONZ) as MOD.
+  assert (b' > 0)%Z as b_GT0 by lia.
+  pose proof (Z_mod_lt a' b' b_GT0) as MOD_LT.
+  destruct (Z_lt_dec a' (b' * q'')) as [LT | GE].
+  { replace (b' * q'' >? a')%Z with true in QUOTIENT by lia.
+    replace q'' with (1 + (a' / b'))%Z by lia.
+    replace ((1 + a' / b') * b' - a')%Z
+      with (-(a' - a' / b' * b')+b')%Z by ring.
+    rewrite <- MOD.
+    rewrite Int64.signed_repr; cycle 1.
+    { admit.
+    }
+    rewrite zlt_true by lia.
+    ring.
+  }
+  replace (b' * q'' >? a')%Z with false in QUOTIENT by lia.
+  rewrite <- QUOTIENT.
+  replace (a' / b' * b' - a')%Z with (-(a' - a' / b' * b'))%Z by ring.
+  rewrite <- MOD.
+  rewrite Int64.signed_repr ; cycle 1.
+  { admit.
+  }
+  rewrite zlt_false by lia.
+  reflexivity.
 Admitted.
-- 
cgit 


From d8f1751b88d0013166412677c965079700e36557 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 15:51:41 +0100
Subject: done for small a

---
 kvx/FPDivision64.v | 10 +++++++---
 1 file changed, 7 insertions(+), 3 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 0d11c3cb..a53169b8 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1366,7 +1366,9 @@ Proof.
       with (-(a' - a' / b' * b')+b')%Z by ring.
     rewrite <- MOD.
     rewrite Int64.signed_repr; cycle 1.
-    { admit.
+    { change Int64.min_signed with (-9223372036854775808)%Z in *.
+      change Int64.max_signed with (9223372036854775807)%Z in *.
+      lia.
     }
     rewrite zlt_true by lia.
     ring.
@@ -1376,8 +1378,10 @@ Proof.
   replace (a' / b' * b' - a')%Z with (-(a' - a' / b' * b'))%Z by ring.
   rewrite <- MOD.
   rewrite Int64.signed_repr ; cycle 1.
-  { admit.
+  { change Int64.min_signed with (-9223372036854775808)%Z in *.
+    change Int64.max_signed with (9223372036854775807)%Z in *.
+    lia.
   }
   rewrite zlt_false by lia.
   reflexivity.
-Admitted.
+Qed.
-- 
cgit 


From c5a426f410f95d6e410da5b2afd5a9225e902c3e Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 19:21:56 +0100
Subject: twostep_div_longu_smallb_correct

---
 kvx/FPDivision64.v | 82 ++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 82 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index a53169b8..758061b5 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1385,3 +1385,85 @@ Proof.
   rewrite zlt_false by lia.
   reflexivity.
 Qed.
+
+Definition twostep_div_longu a b :=
+  let q1 := step1_div_longu a b in
+  let q2 := step2_div_long (Val.subl a (Val.mull b q1)) b in
+  Val.addl q1 q2.
+
+Lemma unsigned_repr_sub :
+  forall a b,
+    Int64.repr (a - b) = Int64.repr (a - Int64.unsigned (Int64.repr b)).
+Proof.
+  intros.
+  apply Int64.eqm_samerepr.
+  apply Int64.eqm_sub.
+  - apply Int64.eqm_refl.
+  - apply Int64.eqm_unsigned_repr.
+Qed.
+
+Lemma unsigned_repr_add :
+  forall a b,
+    Int64.repr (a + b) = Int64.repr (a + Int64.unsigned (Int64.repr b)).
+Proof.
+  intros.
+  apply Int64.eqm_samerepr.
+  apply Int64.eqm_add.
+  - apply Int64.eqm_refl.
+  - apply Int64.eqm_unsigned_repr.
+Qed.
+    
+Lemma twostep_div_longu_smallb_correct :
+    forall a b
+      (b_RANGE : (1 < Int64.unsigned b <= smallb_thresh)%Z),
+          (twostep_div_longu (Vlong a) (Vlong b)) =
+            (Val.maketotal (Val.divlu (Vlong a) (Vlong b))).
+Proof.
+  intros.
+  unfold twostep_div_longu.
+  destruct (step1_div_longu_correct a b b_RANGE) as (q1 & C1R & C1E).
+  rewrite C1R.
+  set (q1' := Int64.unsigned q1) in *.
+  set (b' := Int64.unsigned b) in *.
+  set (a' := Int64.unsigned a) in *.
+  assert ( Z.abs (Int64.signed (Int64.sub a (Int64.mul b q1))) <= smalla_thresh)%Z as r1_SMALL.
+  { unfold smalla_thresh, smallb_approx_range in *.
+    unfold Int64.sub, Int64.mul.
+    fold q1' b' a'.
+    rewrite <- unsigned_repr_sub.
+    rewrite Int64.signed_repr ; cycle 1.
+    { change Int64.min_signed with (-9223372036854775808)%Z.
+      change Int64.max_signed with (9223372036854775807)%Z.
+      lia.
+    }
+    lia.
+  }
+  assert (0 < b')%Z as b_NOT0 by lia.
+  assert (b' <= Int64.max_signed)%Z as b_NOTBIG.
+  { change Int64.max_signed with (9223372036854775807)%Z.
+    unfold smallb_thresh in b_RANGE.
+    lia.
+  }
+  cbn.
+  rewrite (step2_div_long_smalla_correct (Int64.sub a (Int64.mul b q1)) b r1_SMALL b_NOT0 b_NOTBIG).
+  unfold Int64.add, Int64.sub, Int64.mul, Int64.divu.
+  fold q1' b' a'.
+  rewrite <- unsigned_repr_sub.
+  rewrite <- unsigned_repr_add.
+  rewrite Int64.signed_repr ; cycle 1.
+  {
+    change Int64.min_signed with (-9223372036854775808)%Z.
+    change Int64.max_signed with (9223372036854775807)%Z.
+    unfold smallb_approx_range in *.
+    lia.
+  }
+  rewrite Z.add_comm.
+  rewrite <- Z.div_add by lia.
+  replace (a' - b' * q1' + q1' * b')%Z with a' by ring.
+  rewrite Int64.eq_false ; cycle 1.
+  { intro Z. unfold b' in b_NOT0. rewrite Z in b_NOT0.
+    rewrite Int64.unsigned_zero in b_NOT0.
+    lia.
+  }
+  reflexivity.
+Qed.
-- 
cgit 


From 30c2e58daf62d53530505bea2401e646df79bd31 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 21:06:06 +0100
Subject: progress

---
 kvx/FPDivision64.v | 190 +++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 190 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 758061b5..580f4dd9 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1467,3 +1467,193 @@ Proof.
   }
   reflexivity.
 Qed.
+
+
+Lemma step2_real_div_long_bigb_correct :
+    forall (a b : int64)
+           (b_BIG : ((Int64.unsigned b) > smallb_thresh)%Z),
+    exists (q : float),
+      (step2_real_div_long (Vlong a) (Vlong b)) = Vfloat q /\
+        (Rabs ((B2R _ _ q) - (IZR (Int64.signed a)) / (IZR (Int64.unsigned b))) <= (32767/65536))%R /\
+      is_finite _ _ q = true.
+Proof.
+  intros.
+  unfold step2_real_div_long.
+  assert (0 < Int64.unsigned b)%Z as b_NOT0 by (unfold smallb_thresh in *; lia).
+  destruct (approx_inv_longu_correct_rel b b_NOT0) as (f & C0E & C0F & C0R).
+  rewrite C0E.
+  econstructor.
+  split. reflexivity.
+  Local Transparent Float.of_long.
+  unfold Float.mul, Float.of_long.
+  set (re := (@eq_refl Datatypes.comparison Lt)) in *.
+  pose proof (Int64.unsigned_range b) as b_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in b_RANGE.
+  pose proof (Int64.signed_range a) as a_RANGE.
+  set (a' := Int64.signed a) in *.
+  set (b' := Int64.unsigned b) in *.
+  assert (IZR (1 + smallb_thresh) <= IZR b' <= 18446744073709551615)%R as b_RANGE'.
+  { split; apply IZR_le; lia.
+  }
+  assert(IZR Int64.min_signed <= IZR a' <= IZR Int64.max_signed)%R as a_RANGE'.
+  { split; apply IZR_le; lia.
+  }
+  change Int64.min_signed with (-9223372036854775808)%Z in a_RANGE'.
+  change Int64.max_signed with (9223372036854775807)%Z in a_RANGE'.
+  pose proof (BofZ_correct 53 1024 re re a') as C1.
+  rewrite Rlt_bool_true in C1 ; cycle 1.
+  { clear C1.
+    apply Rabs_relax with (b := bpow radix2 64).
+    { apply bpow_lt; lia. }
+    cbn.
+    gappa.
+  }
+  cbn in C1.
+  destruct C1 as (C1R & C1F & C1S).
+
+  unfold smallb_thresh in b_RANGE'; cbn in b_RANGE'.
+
+  pose proof (Bmult_correct 53 1024 re re Float.binop_nan mode_NE (BofZ 53 1024 re re a') f) as C2.
+  rewrite Rlt_bool_true in C2 ; cycle 1.
+  { clear C2.
+    apply Rabs_relax with (b := bpow radix2 53).
+    { apply bpow_lt. lia. }
+    cbn.
+    rewrite C1R.
+    unfold approx_inv_rel_thresh in C0R.
+    replace (B2R 53 1024 f) with
+      ((1/IZR b') * ((IZR b' * B2R 53 1024 f - 1) + 1))%R ; cycle 1.
+    { field.  lra. }
+    gappa.
+  }
+  rewrite C0F in C2.
+  rewrite C1R in C2.
+  rewrite C1F in C2.
+  rewrite C1S in C2.
+  cbn in C2.
+  destruct C2 as (C2R & C2F & _).
+  split.
+  2: exact C2F.
+  rewrite C2R.
+  set (f' := (B2R 53 1024 f)) in *.  
+  replace (rd(rd (IZR a') * f') - IZR a' / IZR b')%R with
+    ((rd(rd (IZR a') * f') - IZR a' * f') + IZR a' / IZR b' * (IZR b' * f' - 1))%R ; cycle 1.
+  { field. lra. }
+  unfold approx_inv_rel_thresh in *.
+  gappa.
+Qed.
+
+Lemma step2_real_div_long_bigb_correct :
+    forall (a b : int64)
+           (b_BIG : ((Int64.unsigned b) > smallb_thresh)%Z),
+    exists (q : float),
+      (step2_real_div_long (Vlong a) (Vlong b)) = Vfloat q /\
+        (Rabs ((B2R _ _ q) - (IZR (Int64.signed a)) / (IZR (Int64.unsigned b))) <= (32767/65536))%R /\
+      is_finite _ _ q = true.
+Proof.
+  intros.
+  pose proof (Int64.unsigned_range b) as b_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in b_RANGE.
+  set (a' := (Int64.signed a)) in *.
+  set (b' := (Int64.unsigned b)) in *.
+  assert (IZR (1 + smallb_thresh) <= IZR b' <= 18446744073709551615)%R as b_RANGE'.
+  { split; apply IZR_le; lia.
+  } 
+  destruct (step2_real_div_long_smalla_correct a b a_SMALL b_NOT0) as (q & C1R & C1E & C1F).
+  fold a' b' in C1E.
+  assert ((Int64.min_signed <=? ZnearestE (B2R 53 1024 q))=true)%Z as q_LOW.
+  { apply Zle_imp_le_bool.
+    change Int64.min_signed with (-9223372036854775808)%Z.
+    apply Znearest_lub.
+    set (q' :=  B2R 53 1024 q) in *.
+    replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring.
+    unfold smalla_thresh in a_RANGE'.
+    gappa.
+  }
+  assert ((ZnearestE (B2R 53 1024 q) <=? Int64.max_signed)=true)%Z as q_HIGH.
+  { apply Zle_imp_le_bool.
+    change Int64.max_signed with (9223372036854775807)%Z.
+    apply Znearest_glb.
+    set (q' :=  B2R 53 1024 q) in *.
+    replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring.
+    unfold smalla_thresh in a_RANGE'.
+    gappa.
+  }
+  unfold step2_div_long, step2_div_long'.
+  rewrite C1R.
+  cbn.
+  unfold Float.to_long_ne.
+  rewrite (ZofB_ne_range_correct _ _ q Int64.min_signed Int64.max_signed).
+  rewrite C1F.
+  rewrite q_LOW.
+  rewrite q_HIGH.
+  cbn.
+  rewrite normalize_ite.
+  cbn.
+  rewrite <- (function_ite Vlong).
+  f_equal.
+  unfold Int64.lt.
+  set (q' :=  B2R 53 1024 q) in *.
+  fold a'.
+  assert (Int64.signed (Int64.repr (ZnearestE q')) = ZnearestE q') as q_SIGNED.
+  { apply Int64.signed_repr.
+    split; lia.
+  }
+  rewrite Int64.add_signed.
+  rewrite q_SIGNED.
+  rewrite Int64.signed_mone.
+  rewrite Int64.signed_zero.
+  rewrite <- (function_ite Int64.repr).
+  f_equal.
+  replace  (ZnearestE q' + -1)%Z with (ZnearestE q' - 1)%Z by ring.
+  
+  set (q'' :=  (ZnearestE q')) in *.
+  Check Int64.sub_signed.
+  fold a'.
+  rewrite int64_mul_signed_unsigned.
+  rewrite q_SIGNED.
+  fold b'.
+
+  rewrite Int64.sub_signed.
+  fold a'.
+  rewrite signed_repr_sub.
+
+  assert ((Rabs (IZR a' / IZR b' - q') < / 2)%R) as HALF.
+  { replace (IZR a' / IZR b' - q')%R with
+      (-(q' - IZR a' / IZR b'))%R by ring.
+    rewrite Rabs_Ropp.
+    lra.
+  }
+  pose proof (find_quotient a' b' b_NOT0 q' HALF) as QUOTIENT.
+  fold q'' in QUOTIENT.
+  cbn zeta in QUOTIENT.
+
+  assert (b' <> 0)%Z as NONZ by lia.
+  pose proof (Zmod_eq_full a' b' NONZ) as MOD.
+  assert (b' > 0)%Z as b_GT0 by lia.
+  pose proof (Z_mod_lt a' b' b_GT0) as MOD_LT.
+  destruct (Z_lt_dec a' (b' * q'')) as [LT | GE].
+  { replace (b' * q'' >? a')%Z with true in QUOTIENT by lia.
+    replace q'' with (1 + (a' / b'))%Z by lia.
+    replace ((1 + a' / b') * b' - a')%Z
+      with (-(a' - a' / b' * b')+b')%Z by ring.
+    rewrite <- MOD.
+    rewrite Int64.signed_repr; cycle 1.
+    { change Int64.min_signed with (-9223372036854775808)%Z in *.
+      change Int64.max_signed with (9223372036854775807)%Z in *.
+      lia.
+    }
+    rewrite zlt_true by lia.
+    ring.
+  }
+  replace (b' * q'' >? a')%Z with false in QUOTIENT by lia.
+  rewrite <- QUOTIENT.
+  replace (a' / b' * b' - a')%Z with (-(a' - a' / b' * b'))%Z by ring.
+  rewrite <- MOD.
+  rewrite Int64.signed_repr ; cycle 1.
+  { change Int64.min_signed with (-9223372036854775808)%Z in *.
+    change Int64.max_signed with (9223372036854775807)%Z in *.
+    lia.
+  }
+  rewrite zlt_false by lia.
+  reflexivity.
-- 
cgit 


From 6290759de7aec8ec914c16eac6882b353ad980f1 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 21:07:29 +0100
Subject: rm spurious function

---
 kvx/FPDivision64.v | 115 -----------------------------------------------------
 1 file changed, 115 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 580f4dd9..1e5d8478 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1542,118 +1542,3 @@ Proof.
   unfold approx_inv_rel_thresh in *.
   gappa.
 Qed.
-
-Lemma step2_real_div_long_bigb_correct :
-    forall (a b : int64)
-           (b_BIG : ((Int64.unsigned b) > smallb_thresh)%Z),
-    exists (q : float),
-      (step2_real_div_long (Vlong a) (Vlong b)) = Vfloat q /\
-        (Rabs ((B2R _ _ q) - (IZR (Int64.signed a)) / (IZR (Int64.unsigned b))) <= (32767/65536))%R /\
-      is_finite _ _ q = true.
-Proof.
-  intros.
-  pose proof (Int64.unsigned_range b) as b_RANGE.
-  change Int64.modulus with 18446744073709551616%Z in b_RANGE.
-  set (a' := (Int64.signed a)) in *.
-  set (b' := (Int64.unsigned b)) in *.
-  assert (IZR (1 + smallb_thresh) <= IZR b' <= 18446744073709551615)%R as b_RANGE'.
-  { split; apply IZR_le; lia.
-  } 
-  destruct (step2_real_div_long_smalla_correct a b a_SMALL b_NOT0) as (q & C1R & C1E & C1F).
-  fold a' b' in C1E.
-  assert ((Int64.min_signed <=? ZnearestE (B2R 53 1024 q))=true)%Z as q_LOW.
-  { apply Zle_imp_le_bool.
-    change Int64.min_signed with (-9223372036854775808)%Z.
-    apply Znearest_lub.
-    set (q' :=  B2R 53 1024 q) in *.
-    replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring.
-    unfold smalla_thresh in a_RANGE'.
-    gappa.
-  }
-  assert ((ZnearestE (B2R 53 1024 q) <=? Int64.max_signed)=true)%Z as q_HIGH.
-  { apply Zle_imp_le_bool.
-    change Int64.max_signed with (9223372036854775807)%Z.
-    apply Znearest_glb.
-    set (q' :=  B2R 53 1024 q) in *.
-    replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring.
-    unfold smalla_thresh in a_RANGE'.
-    gappa.
-  }
-  unfold step2_div_long, step2_div_long'.
-  rewrite C1R.
-  cbn.
-  unfold Float.to_long_ne.
-  rewrite (ZofB_ne_range_correct _ _ q Int64.min_signed Int64.max_signed).
-  rewrite C1F.
-  rewrite q_LOW.
-  rewrite q_HIGH.
-  cbn.
-  rewrite normalize_ite.
-  cbn.
-  rewrite <- (function_ite Vlong).
-  f_equal.
-  unfold Int64.lt.
-  set (q' :=  B2R 53 1024 q) in *.
-  fold a'.
-  assert (Int64.signed (Int64.repr (ZnearestE q')) = ZnearestE q') as q_SIGNED.
-  { apply Int64.signed_repr.
-    split; lia.
-  }
-  rewrite Int64.add_signed.
-  rewrite q_SIGNED.
-  rewrite Int64.signed_mone.
-  rewrite Int64.signed_zero.
-  rewrite <- (function_ite Int64.repr).
-  f_equal.
-  replace  (ZnearestE q' + -1)%Z with (ZnearestE q' - 1)%Z by ring.
-  
-  set (q'' :=  (ZnearestE q')) in *.
-  Check Int64.sub_signed.
-  fold a'.
-  rewrite int64_mul_signed_unsigned.
-  rewrite q_SIGNED.
-  fold b'.
-
-  rewrite Int64.sub_signed.
-  fold a'.
-  rewrite signed_repr_sub.
-
-  assert ((Rabs (IZR a' / IZR b' - q') < / 2)%R) as HALF.
-  { replace (IZR a' / IZR b' - q')%R with
-      (-(q' - IZR a' / IZR b'))%R by ring.
-    rewrite Rabs_Ropp.
-    lra.
-  }
-  pose proof (find_quotient a' b' b_NOT0 q' HALF) as QUOTIENT.
-  fold q'' in QUOTIENT.
-  cbn zeta in QUOTIENT.
-
-  assert (b' <> 0)%Z as NONZ by lia.
-  pose proof (Zmod_eq_full a' b' NONZ) as MOD.
-  assert (b' > 0)%Z as b_GT0 by lia.
-  pose proof (Z_mod_lt a' b' b_GT0) as MOD_LT.
-  destruct (Z_lt_dec a' (b' * q'')) as [LT | GE].
-  { replace (b' * q'' >? a')%Z with true in QUOTIENT by lia.
-    replace q'' with (1 + (a' / b'))%Z by lia.
-    replace ((1 + a' / b') * b' - a')%Z
-      with (-(a' - a' / b' * b')+b')%Z by ring.
-    rewrite <- MOD.
-    rewrite Int64.signed_repr; cycle 1.
-    { change Int64.min_signed with (-9223372036854775808)%Z in *.
-      change Int64.max_signed with (9223372036854775807)%Z in *.
-      lia.
-    }
-    rewrite zlt_true by lia.
-    ring.
-  }
-  replace (b' * q'' >? a')%Z with false in QUOTIENT by lia.
-  rewrite <- QUOTIENT.
-  replace (a' / b' * b' - a')%Z with (-(a' - a' / b' * b'))%Z by ring.
-  rewrite <- MOD.
-  rewrite Int64.signed_repr ; cycle 1.
-  { change Int64.min_signed with (-9223372036854775808)%Z in *.
-    change Int64.max_signed with (9223372036854775807)%Z in *.
-    lia.
-  }
-  rewrite zlt_false by lia.
-  reflexivity.
-- 
cgit 


From 56c5f324c1792df0a1c660f37bb9d5028c496a3f Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 21:42:52 +0100
Subject: progress

---
 kvx/FPDivision64.v | 121 +++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 121 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 1e5d8478..74303a7a 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1542,3 +1542,124 @@ Proof.
   unfold approx_inv_rel_thresh in *.
   gappa.
 Qed.
+                                                                        
+Lemma step2_div_long_bigb_correct :
+    forall a b
+           (b_BIG : ((Int64.unsigned b) > smallb_thresh)%Z)
+           (b_NOT_TOO_BIG : ((Int64.unsigned b) <= Int64.max_signed)%Z),
+      step2_div_long (Vlong a) (Vlong b) = Vlong (Int64.repr (Int64.signed a / Int64.unsigned b))%Z.
+Proof.
+  intros.
+  pose proof (Int64.unsigned_range b) as b_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in b_RANGE.
+  pose proof (Int64.signed_range a) as a_RANGE.
+  set (a' := (Int64.signed a)) in *.
+  set (b' := (Int64.unsigned b)) in *.
+  assert (IZR (1 + smallb_thresh) <= IZR b' <= 18446744073709551615)%R as b_RANGE'.
+  { split; apply IZR_le; lia.
+  }
+  assert(IZR Int64.min_signed <= IZR a' <= IZR Int64.max_signed)%R as a_RANGE'.
+  { split; apply IZR_le; lia.
+  }
+  unfold smallb_thresh in *; cbn in b_RANGE'.
+  change Int64.min_signed with (-9223372036854775808)%Z in *.
+  change Int64.max_signed with (9223372036854775807)%Z in *.
+  assert (0 < b')%Z as b_NOT0 by lia.
+
+  destruct (step2_real_div_long_bigb_correct a b b_BIG) as (q & C1R & C1E & C1F).
+  fold a' b' in C1E.
+  assert ((Int64.min_signed <=? ZnearestE (B2R 53 1024 q))=true)%Z as q_LOW.
+  { apply Zle_imp_le_bool.
+    change Int64.min_signed with (-9223372036854775808)%Z.
+    apply Znearest_lub.
+    set (q' :=  B2R 53 1024 q) in *.
+    replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring.
+    gappa.
+  }
+  assert ((ZnearestE (B2R 53 1024 q) <=? Int64.max_signed)=true)%Z as q_HIGH.
+  { apply Zle_imp_le_bool.
+    change Int64.max_signed with (9223372036854775807)%Z.
+    apply Znearest_glb.
+    set (q' :=  B2R 53 1024 q) in *.
+    replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring.
+    gappa.
+  }
+  unfold step2_div_long, step2_div_long'.
+  rewrite C1R.
+  cbn.
+  unfold Float.to_long_ne.
+  rewrite (ZofB_ne_range_correct _ _ q Int64.min_signed Int64.max_signed).
+  rewrite C1F.
+  rewrite q_LOW.
+  rewrite q_HIGH.
+  cbn.
+  rewrite normalize_ite.
+  cbn.
+  rewrite <- (function_ite Vlong).
+  f_equal.
+  unfold Int64.lt.
+  set (q' :=  B2R 53 1024 q) in *.
+  fold a'.
+  assert (Int64.signed (Int64.repr (ZnearestE q')) = ZnearestE q') as q_SIGNED.
+  { apply Int64.signed_repr.
+    split; lia.
+  }
+  rewrite Int64.add_signed.
+  rewrite q_SIGNED.
+  rewrite Int64.signed_mone.
+  rewrite Int64.signed_zero.
+  rewrite <- (function_ite Int64.repr).
+  f_equal.
+  replace  (ZnearestE q' + -1)%Z with (ZnearestE q' - 1)%Z by ring.
+  
+  set (q'' :=  (ZnearestE q')) in *.
+  Check Int64.sub_signed.
+  fold a'.
+  rewrite int64_mul_signed_unsigned.
+  rewrite q_SIGNED.
+  fold b'.
+
+  rewrite Int64.sub_signed.
+  fold a'.
+  rewrite signed_repr_sub.
+
+  assert ((Rabs (IZR a' / IZR b' - q') < / 2)%R) as HALF.
+  { replace (IZR a' / IZR b' - q')%R with
+      (-(q' - IZR a' / IZR b'))%R by ring.
+    rewrite Rabs_Ropp.
+    lra.
+  }
+  pose proof (find_quotient a' b' b_NOT0 q' HALF) as QUOTIENT.
+  fold q'' in QUOTIENT.
+  cbn zeta in QUOTIENT.
+
+  assert (b' <> 0)%Z as NONZ by lia.
+  pose proof (Zmod_eq_full a' b' NONZ) as MOD.
+  assert (b' > 0)%Z as b_GT0 by lia.
+  pose proof (Z_mod_lt a' b' b_GT0) as MOD_LT.
+  destruct (Z_lt_dec a' (b' * q'')) as [LT | GE].
+  { replace (b' * q'' >? a')%Z with true in QUOTIENT by lia.
+    replace q'' with (1 + (a' / b'))%Z by lia.
+    replace ((1 + a' / b') * b' - a')%Z
+      with (-(a' - a' / b' * b')+b')%Z by ring.
+    rewrite <- MOD.
+    rewrite Int64.signed_repr; cycle 1.
+    { change Int64.min_signed with (-9223372036854775808)%Z in *.
+      change Int64.max_signed with (9223372036854775807)%Z in *.
+      lia.
+    }
+    rewrite zlt_true by lia.
+    ring.
+  }
+  replace (b' * q'' >? a')%Z with false in QUOTIENT by lia.
+  rewrite <- QUOTIENT.
+  replace (a' / b' * b' - a')%Z with (-(a' - a' / b' * b'))%Z by ring.
+  rewrite <- MOD.
+  rewrite Int64.signed_repr ; cycle 1.
+  { change Int64.min_signed with (-9223372036854775808)%Z in *.
+    change Int64.max_signed with (9223372036854775807)%Z in *.
+    lia.
+  }
+  rewrite zlt_false by lia.
+  reflexivity.
+Qed.
-- 
cgit 


From 0ec09c99af2071f52271dd5e14846f5da3bcb718 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 22:37:25 +0100
Subject: some progress

---
 kvx/FPDivision64.v | 176 +++++++++++++++++++++++++++++++++++++++++++++++------
 1 file changed, 157 insertions(+), 19 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 74303a7a..e4f883e7 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -29,6 +29,25 @@ Require Import IEEE754_extra.
 
 From Gappa Require Import Gappa_tactic.
 
+
+Lemma Znearest_lub :
+  forall choice (n : Z) (x : R), (IZR n <= x)%R -> (n <= Znearest choice x)%Z.
+Proof.
+  intros until x. intro BND.
+  pose proof (Zfloor_lub n x BND).
+  pose proof (Znearest_ge_floor choice x).
+  lia.
+Qed.
+
+Lemma Znearest_glb :
+  forall choice (n : Z) (x : R), (x <= IZR n)%R -> (Znearest choice x <= n)%Z.
+Proof.
+  intros until x. intro BND.
+  pose proof (Zceil_glb n x BND).
+  pose proof (Znearest_le_ceil choice x).
+  lia.
+Qed.
+
 Definition approx_inv_longu b :=
   let invb_s := ExtValues.invfs (Val.maketotal (Val.singleoflongu b)) in
   let invb_d := Val.floatofsingle invb_s in
@@ -851,7 +870,99 @@ Proof.
   rewrite C4.
   reflexivity.
 Qed.
-                                         
+
+Lemma step1_div_longu_correct_anyb :
+    forall a b
+     (b_NOT01 : (1 < Int64.unsigned b)%Z),
+    exists (q : int64),
+      (step1_div_longu (Vlong a) (Vlong b)) = Vlong q.
+Proof.
+  intros.
+
+  pose proof (Int64.unsigned_range a) as a_RANGE.
+  pose proof (Int64.unsigned_range b) as b_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in *.
+  set (a' := Int64.unsigned a) in *.
+  set (b' := Int64.unsigned b) in *.
+  assert (0 <= IZR a' <= 18446744073709551615)%R as a_RANGE'.
+  { split; apply IZR_le; lia. }
+  assert (2 <= IZR b' <= 18446744073709551615)%R as b_RANGE'.
+  { split; apply IZR_le; lia. }
+
+  assert (0 < b')%Z as b_NOT0 by lia.
+                                  
+  destruct (Z_le_gt_dec a' 0).
+  { assert (a' = 0%Z) as ZERO by lia.
+    replace a with Int64.zero; cycle 1.
+    {
+      unfold a' in ZERO.
+      unfold Int64.zero.
+      rewrite <- ZERO.
+      apply Int64.repr_unsigned.
+    }
+    exists Int64.zero.
+    apply step1_div_longu_a0.
+    exact b_NOT0.
+  }
+
+  unfold step1_div_longu.
+  unfold step1_real_div_longu.
+  destruct (step1_real_inv_longu_correct b b_NOT0) as (f & C1E & C1R & C1F & C1S).
+  rewrite C1E.
+  cbn.
+  unfold Float.of_longu, Float.mul, Float.to_longu_ne.
+  set (re := (@eq_refl Datatypes.comparison Lt)).
+  fold a'.
+  fold b' in C1R.
+  pose proof (BofZ_correct 53 1024 re re a') as C2.
+  rewrite Rlt_bool_true in C2; cycle 1.
+  { clear C2.
+    apply Rabs_relax with (b := bpow radix2 64).
+    { apply bpow_lt. lia. }
+    cbn.
+    gappa.
+  }
+  cbn in C2.
+  destruct C2 as (C2R & C2F & C2S).
+  pose proof  (Bmult_correct 53 1024 re re Float.binop_nan mode_NE
+                             (BofZ 53 1024 re re a') f) as C3.
+  rewrite C2R in C3.
+  rewrite C2F in C3.
+  rewrite C2S in C3.
+  rewrite C1R in C3.
+  rewrite C1F in C3.
+  rewrite C1S in C3.
+  rewrite Rlt_bool_true in C3; cycle 1.
+  { clear C3.
+    apply Rabs_relax with (b := bpow radix2 64).
+    { apply bpow_lt. lia. }
+    cbn.
+    gappa.
+  }
+  cbn in C3.
+  destruct C3 as (C3R & C3F & _).
+  pose proof (ZofB_ne_range_correct 53 1024
+               (Bmult 53 1024 re re Float.binop_nan mode_NE
+                      (BofZ 53 1024 re re a') f) 0 Int64.max_unsigned) as C4.
+  rewrite C3R in C4.
+  rewrite C3F in C4.
+  cbn zeta in C4.
+  rewrite Zle_bool_true in C4 ; cycle 1.
+  { clear C4.
+    apply Znearest_lub.
+    gappa.
+  }
+  rewrite Zle_bool_true in C4 ; cycle 1.
+  { clear C4.
+    apply Znearest_glb.
+    cbn.
+    gappa.
+  }
+  rewrite C4.
+  cbn.
+  eauto.
+Qed.
+
 Definition smallb_approx_range := 4400000000000%Z.
 Lemma step1_div_longu_correct :
     forall a b,
@@ -1194,24 +1305,6 @@ Definition step2_div_long a b :=
   Val.select (Val.cmpl_bool Cgt (Val.subl (Val.mull q b) a) (Vlong Int64.zero))
              (Val.addl q (Vlong Int64.mone)) q Tlong.
 
-Lemma Znearest_lub :
-  forall choice (n : Z) (x : R), (IZR n <= x)%R -> (n <= Znearest choice x)%Z.
-Proof.
-  intros until x. intro BND.
-  pose proof (Zfloor_lub n x BND).
-  pose proof (Znearest_ge_floor choice x).
-  lia.
-Qed.
-
-Lemma Znearest_glb :
-  forall choice (n : Z) (x : R), (x <= IZR n)%R -> (Znearest choice x <= n)%Z.
-Proof.
-  intros until x. intro BND.
-  pose proof (Zceil_glb n x BND).
-  pose proof (Znearest_le_ceil choice x).
-  lia.
-Qed.
-
 Lemma function_ite :
   forall {A B : Type} (fn : A->B) (b : bool) (x y: A),
     fn (if b then x else y) = (if b then fn x else fn y).
@@ -1663,3 +1756,48 @@ Proof.
   rewrite zlt_false by lia.
   reflexivity.
 Qed.
+  
+Lemma twostep_div_longu_bigb_correct :
+    forall a b
+           (b_BIG : ((Int64.unsigned b) > smallb_thresh)%Z)
+           (b_NOT_TOO_BIG : ((Int64.unsigned b) <= Int64.max_signed)%Z),
+          (twostep_div_longu (Vlong a) (Vlong b)) =
+            (Val.maketotal (Val.divlu (Vlong a) (Vlong b))).
+Proof.
+  intros.
+  unfold twostep_div_longu.
+  set (b' := Int64.unsigned b) in *.
+  set (a' := Int64.unsigned a) in *.
+Admitted.
+
+(*
+  assert (0 < b')%Z as b_NOT0 by lia.
+  assert (b' <= Int64.max_signed)%Z as b_NOTBIG.
+  { change Int64.max_signed with (9223372036854775807)%Z.
+    unfold smallb_thresh in b_RANGE.
+    lia.
+  }
+  cbn.
+  rewrite (step2_div_long_bigb_correct (Int64.sub a (Int64.mul b q1)) b r1_SMALL b_NOT0 b_NOTBIG).
+  unfold Int64.add, Int64.sub, Int64.mul, Int64.divu.
+  fold q1' b' a'.
+  rewrite <- unsigned_repr_sub.
+  rewrite <- unsigned_repr_add.
+  rewrite Int64.signed_repr ; cycle 1.
+  {
+    change Int64.min_signed with (-9223372036854775808)%Z.
+    change Int64.max_signed with (9223372036854775807)%Z.
+    unfold smallb_approx_range in *.
+    lia.
+  }
+  rewrite Z.add_comm.
+  rewrite <- Z.div_add by lia.
+  replace (a' - b' * q1' + q1' * b')%Z with a' by ring.
+  rewrite Int64.eq_false ; cycle 1.
+  { intro Z. unfold b' in b_NOT0. rewrite Z in b_NOT0.
+    rewrite Int64.unsigned_zero in b_NOT0.
+    lia.
+  }
+  reflexivity.
+Qed.
+*)
-- 
cgit 


From 2bbe86d4efe2ae08444359008798c12adf0c9b60 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Mon, 24 Jan 2022 22:39:55 +0100
Subject: some progress

---
 kvx/FPDivision64.v | 2 --
 1 file changed, 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index e4f883e7..3685f601 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1428,7 +1428,6 @@ Proof.
   replace  (ZnearestE q' + -1)%Z with (ZnearestE q' - 1)%Z by ring.
   
   set (q'' :=  (ZnearestE q')) in *.
-  Check Int64.sub_signed.
   fold a'.
   rewrite int64_mul_signed_unsigned.
   rewrite q_SIGNED.
@@ -1706,7 +1705,6 @@ Proof.
   replace  (ZnearestE q' + -1)%Z with (ZnearestE q' - 1)%Z by ring.
   
   set (q'' :=  (ZnearestE q')) in *.
-  Check Int64.sub_signed.
   fold a'.
   rewrite int64_mul_signed_unsigned.
   rewrite q_SIGNED.
-- 
cgit 


From 0fb548cbf78c19430e60827f7c253fc42aa25e05 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Tue, 25 Jan 2022 10:54:19 +0100
Subject: some progress?

---
 kvx/FPDivision64.v | 311 +++++++++++++++++++++++++++++++----------------------
 1 file changed, 185 insertions(+), 126 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 3685f601..69e49f29 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1064,96 +1064,6 @@ Proof.
   gappa.
 Qed.
 
-(*
-Lemma range_up_le :
-  forall a x b,
-    (IZR a <= IZR x <= (IZR b) - 1)%R ->
-    (a <= x < b)%Z.
-Proof.
-  intros until b. intro RANGE.
-  split.
-  { apply le_IZR. lra. }
-  assert (x <= b-1)%Z.
-  { apply le_IZR. rewrite minus_IZR. lra. }
-  lia.
-Qed.
- *)
-
-(*
-Lemma find_quotient:
-  forall (a b : Z)
-         (b_POS : (b > 0)%Z)
-         (invb : R)
-         (eps : R)
-         (invb_EPS : (Rabs (invb * IZR b - 1) <= eps)%R)
-         (SMALL : (IZR (Z.abs a)*eps < IZR b / 2)%R),
-    (a / b)%Z =
-      let q := ZnearestE ((IZR a) * invb) in
-      if (b*q >? a)%Z
-      then (q-1)%Z
-      else q.
-Proof.
-  intros.
-  rewrite <- Rabs_Zabs in SMALL.
-  set (q := ZnearestE (IZR a * invb)%R).
-  cbn zeta.
-  set (b' := IZR b) in *.
-  set (a' := IZR a) in *.
-  assert (1 <= b')%R as b_POS'.
-  { apply IZR_le.
-    lia.
-  }
-  
-  pose proof (Znearest_imp2 (fun x : Z => negb (Z.even x)) (a' * invb)) as NEAR.
-  fold q in NEAR.
-  set (q' := IZR q) in *.
-  assert ((Rabs a') * Rabs (invb * b' - 1) <= (Rabs a') * eps)%R as S1.
-  { apply Rmult_le_compat_l.
-    apply Rabs_pos.
-    assumption. }
-  rewrite <- Rabs_mult in S1.
-  assert ((Rabs b') * Rabs (q' - a' * invb) <= (Rabs b') / 2)%R as S2.
-  { apply Rmult_le_compat_l.
-    apply Rabs_pos.
-    assumption. }
-  rewrite <- Rabs_mult in S2.
-  rewrite (Rabs_right b') in S2 by lra.
-  pose proof (Rabs_triang (a' * (invb * b' - 1))
-                          (b' * (q' - a' * invb))) as TRIANGLE.
-  replace (a' * (invb * b' - 1) + b' * (q' - a' * invb))%R with
-    (b' * q' - a')%R in TRIANGLE by ring.
-  assert (Rabs (b' * q' - a') < b')%R as DELTA by lra.
-  
-  pose proof (Zgt_cases (b * q) a)%Z as CASE.
-  destruct (_ >? _)%Z.
-  { unfold b', q', a' in DELTA.
-    rewrite <- mult_IZR in DELTA.
-    rewrite <- minus_IZR in DELTA.
-    rewrite Rabs_Zabs in DELTA.
-    apply lt_IZR in DELTA.
-    rewrite Z.abs_eq in DELTA by lia.
-    
-    apply Zdiv_unique with (b := (a - (q-1)*b)%Z).
-    ring.
-    split; lia.
-  }
-  
-  rewrite <- Rabs_Ropp in DELTA.
-  unfold b', q', a' in DELTA.
-  rewrite <- mult_IZR in DELTA.
-  rewrite <- minus_IZR in DELTA.
-  rewrite <- opp_IZR in DELTA.
-  rewrite Rabs_Zabs in DELTA.
-  apply lt_IZR in DELTA.
-  rewrite Z.abs_eq in DELTA by lia.
-
-  apply Zdiv_unique with (b := (a - q*b)%Z).
-  ring.
-  
-  split; lia.
-Qed.
- *)
-
 Lemma find_quotient:
   forall (a b : Z)
          (b_POS : (0 < b)%Z)
@@ -1754,48 +1664,197 @@ Proof.
   rewrite zlt_false by lia.
   reflexivity.
 Qed.
-  
-Lemma twostep_div_longu_bigb_correct :
-    forall a b
-           (b_BIG : ((Int64.unsigned b) > smallb_thresh)%Z)
-           (b_NOT_TOO_BIG : ((Int64.unsigned b) <= Int64.max_signed)%Z),
-          (twostep_div_longu (Vlong a) (Vlong b)) =
-            (Val.maketotal (Val.divlu (Vlong a) (Vlong b))).
+
+Definition step2_real_div_longu a b :=
+  Val.mulf (Val.maketotal (Val.floatoflongu a)) (approx_inv_longu b).
+
+Definition step2_div_longu' a b :=
+  Val.maketotal (Val.longuoffloat_ne (step2_real_div_longu a b)).
+
+Definition step2_div_longu a b :=
+  let q := step2_div_longu' a b in
+  Val.select (Val.cmpl_bool Cgt (Val.subl (Val.mull q b) a) (Vlong Int64.zero))
+             (Val.addl q (Vlong Int64.mone)) q Tlong.
+
+Lemma step2_real_div_longu_bigb_correct :
+    forall (a b : int64)
+           (b_BIG : ((Int64.unsigned b) > smallb_thresh)%Z),
+    exists (q : float),
+      (step2_real_div_longu (Vlong a) (Vlong b)) = Vfloat q /\
+        (Rabs ((B2R _ _ q) - (IZR (Int64.unsigned a)) / (IZR (Int64.unsigned b))) <= (32767/65536))%R /\
+      is_finite _ _ q = true.
 Proof.
   intros.
-  unfold twostep_div_longu.
-  set (b' := Int64.unsigned b) in *.
+  unfold step2_real_div_longu.
+  assert (0 < Int64.unsigned b)%Z as b_NOT0 by (unfold smallb_thresh in *; lia).
+  destruct (approx_inv_longu_correct_rel b b_NOT0) as (f & C0E & C0F & C0R).
+  rewrite C0E.
+  econstructor.
+  split. reflexivity.
+  Local Transparent Float.of_longu.
+  unfold Float.mul, Float.of_longu.
+  set (re := (@eq_refl Datatypes.comparison Lt)) in *.
+  pose proof (Int64.unsigned_range b) as b_RANGE.
+  pose proof (Int64.unsigned_range a) as a_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in *.
   set (a' := Int64.unsigned a) in *.
-Admitted.
+  set (b' := Int64.unsigned b) in *.
+  assert (IZR (1 + smallb_thresh) <= IZR b' <= 18446744073709551615)%R as b_RANGE'.
+  { split; apply IZR_le; lia.
+  }
+  assert(0 <= IZR a' <= 18446744073709551615)%R as a_RANGE'.
+  { split; apply IZR_le; lia.
+  }
+  pose proof (BofZ_correct 53 1024 re re a') as C1.
+  rewrite Rlt_bool_true in C1 ; cycle 1.
+  { clear C1.
+    apply Rabs_relax with (b := bpow radix2 64).
+    { apply bpow_lt; lia. }
+    cbn.
+    gappa.
+  }
+  cbn in C1.
+  destruct C1 as (C1R & C1F & C1S).
 
-(*
+  unfold smallb_thresh in b_RANGE'; cbn in b_RANGE'.
+
+  pose proof (Bmult_correct 53 1024 re re Float.binop_nan mode_NE (BofZ 53 1024 re re a') f) as C2.
+  rewrite Rlt_bool_true in C2 ; cycle 1.
+  { clear C2.
+    apply Rabs_relax with (b := bpow radix2 53).
+    { apply bpow_lt. lia. }
+    cbn.
+    rewrite C1R.
+    unfold approx_inv_rel_thresh in C0R.
+    replace (B2R 53 1024 f) with
+      ((1/IZR b') * ((IZR b' * B2R 53 1024 f - 1) + 1))%R ; cycle 1.
+    { field.  lra. }
+    gappa.
+  }
+  rewrite C0F in C2.
+  rewrite C1R in C2.
+  rewrite C1F in C2.
+  rewrite C1S in C2.
+  cbn in C2.
+  destruct C2 as (C2R & C2F & _).
+  split.
+  2: exact C2F.
+  rewrite C2R.
+  set (f' := (B2R 53 1024 f)) in *.  
+  replace (rd(rd (IZR a') * f') - IZR a' / IZR b')%R with
+    ((rd(rd (IZR a') * f') - IZR a' * f') + IZR a' / IZR b' * (IZR b' * f' - 1))%R ; cycle 1.
+  { field. lra. }
+  unfold approx_inv_rel_thresh in *.
+  gappa.
+Qed.
+                                                                        
+Lemma step2_div_longu_bigb_correct :
+  forall a b
+           (b_BIG : ((Int64.unsigned b) > smallb_thresh)%Z)
+           (b_NOT_TOO_BIG : ((Int64.unsigned b) <= Int64.max_signed)%Z),
+      step2_div_longu (Vlong a) (Vlong b) = Vlong (Int64.repr (Int64.unsigned a / Int64.unsigned b))%Z.
+Proof.
+  intros.
+  pose proof (Int64.unsigned_range b) as b_RANGE.
+  pose proof (Int64.unsigned_range a) as a_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in *.
+  set (a' := (Int64.unsigned a)) in *.
+  set (b' := (Int64.unsigned b)) in *.
+  assert (IZR (1 + smallb_thresh) <= IZR b' <= 18446744073709551615)%R as b_RANGE'.
+  { split; apply IZR_le; lia.
+  }
+  assert(0 <= IZR a' <= 18446744073709551615)%R as a_RANGE'.
+  { split; apply IZR_le; lia.
+  }
+  unfold smallb_thresh in *; cbn in b_RANGE'.
   assert (0 < b')%Z as b_NOT0 by lia.
-  assert (b' <= Int64.max_signed)%Z as b_NOTBIG.
-  { change Int64.max_signed with (9223372036854775807)%Z.
-    unfold smallb_thresh in b_RANGE.
-    lia.
+
+  destruct (step2_real_div_longu_bigb_correct a b b_BIG) as (q & C1R & C1E & C1F).
+  fold a' b' in C1E.
+
+  assert ((0 <=? ZnearestE (B2R 53 1024 q))=true)%Z as q_LOW.
+  { apply Zle_imp_le_bool.
+    set (q' :=  B2R 53 1024 q) in *.
+    assert (-32767 / 65536 <= q')%R as LOWROUND.
+    { replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring.
+      gappa.
+    }
+    destruct (Rcase_abs q').
+    { replace (ZnearestE q') with 0%Z. lia.
+      symmetry.
+      apply Znearest_imp.
+      apply Rabs_lt.
+      split; lra.
+    }
+    apply Znearest_lub.
+    lra.
   }
+  assert ((ZnearestE (B2R 53 1024 q) <=? Int64.max_unsigned)=true)%Z as q_HIGH.
+  { apply Zle_imp_le_bool.
+    change Int64.max_unsigned with (18446744073709551615)%Z.
+    apply Znearest_glb.
+    set (q' :=  B2R 53 1024 q) in *.
+    replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring.
+    gappa.
+  }
+  
+  unfold step2_div_longu, step2_div_longu'.
+  rewrite C1R.
   cbn.
-  rewrite (step2_div_long_bigb_correct (Int64.sub a (Int64.mul b q1)) b r1_SMALL b_NOT0 b_NOTBIG).
-  unfold Int64.add, Int64.sub, Int64.mul, Int64.divu.
-  fold q1' b' a'.
-  rewrite <- unsigned_repr_sub.
-  rewrite <- unsigned_repr_add.
-  rewrite Int64.signed_repr ; cycle 1.
-  {
-    change Int64.min_signed with (-9223372036854775808)%Z.
-    change Int64.max_signed with (9223372036854775807)%Z.
-    unfold smallb_approx_range in *.
-    lia.
+  unfold Float.to_longu_ne.
+  rewrite (ZofB_ne_range_correct _ _ q _ _).
+  rewrite C1F.
+  rewrite q_LOW.
+  rewrite q_HIGH.
+  cbn.
+  rewrite normalize_ite.
+  cbn.
+  rewrite <- (function_ite Vlong).
+  f_equal.
+  unfold Int64.lt.
+  set (q' :=  B2R 53 1024 q) in *.
+  fold a'.
+  rewrite Int64.signed_zero.
+  set (q'' :=  (ZnearestE q')) in *.
+  assert ((Rabs (IZR a' / IZR b' - q') < / 2)%R) as HALF.
+  { replace (IZR a' / IZR b' - q')%R with
+      (-(q' - IZR a' / IZR b'))%R by ring.
+    rewrite Rabs_Ropp.
+    lra.
   }
-  rewrite Z.add_comm.
-  rewrite <- Z.div_add by lia.
-  replace (a' - b' * q1' + q1' * b')%Z with a' by ring.
-  rewrite Int64.eq_false ; cycle 1.
-  { intro Z. unfold b' in b_NOT0. rewrite Z in b_NOT0.
-    rewrite Int64.unsigned_zero in b_NOT0.
-    lia.
+  pose proof (find_quotient a' b' b_NOT0 q' HALF) as QUOTIENT.
+  fold q'' in QUOTIENT.
+  cbn zeta in QUOTIENT.
+
+  assert (b' <> 0)%Z as NONZ by lia.
+  pose proof (Zmod_eq_full a' b' NONZ) as MOD.
+  assert (b' > 0)%Z as b_GT0 by lia.
+  pose proof (Z_mod_lt a' b' b_GT0) as MOD_LT.
+  assert ((Int64.signed (Int64.sub (Int64.mul (Int64.repr q'') b) a))
+          = q''*b'-a')%Z as SIMPL.
+  { admit. }
+  rewrite SIMPL.
+  destruct (Z_lt_dec a' (b' * q'')) as [LT | GE].
+  { replace (b' * q'' >? a')%Z with true in QUOTIENT by lia.
+    rewrite zlt_true by lia.
+    replace q'' with (1 + (a' / b'))%Z by lia.
+    unfold Int64.add.
+    apply Int64.eqm_samerepr.
+    apply Int64.eqm_trans with (y := ((1 + a' / b') + (-1))%Z).
+    { apply Int64.eqm_add.
+      apply Int64.eqm_sym.
+      apply Int64.eqm_unsigned_repr.
+      rewrite Int64.unsigned_mone.
+      replace (-1)%Z with (0 - 1)%Z by ring.
+      apply Int64.eqm_add.
+      exists 1%Z.
+      lia.
+      apply Int64.eqm_refl.
+    }
+    replace (1 + a' / b' + -1)%Z with (a'/b')%Z by ring.
+    apply Int64.eqm_refl.
   }
-  reflexivity.
-Qed.
-*)
+  rewrite zlt_false by lia.
+  replace (b' * q'' >? a')%Z with false in QUOTIENT by lia.
+  congruence.
+Admitted.
-- 
cgit 


From 32a69b2acc078ffc5078ca5f12747170b16d9d78 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Tue, 25 Jan 2022 11:28:55 +0100
Subject: big progress on longu

---
 kvx/FPDivision64.v | 70 +++++++++++++++++++++++++++++++++++++++++++++++-------
 1 file changed, 61 insertions(+), 9 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 69e49f29..d96ced31 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1747,7 +1747,40 @@ Proof.
   unfold approx_inv_rel_thresh in *.
   gappa.
 Qed.
-                                                                        
+
+Lemma repr_unsigned_mul:
+  forall a b,
+    (Int64.repr (Int64.unsigned (Int64.repr a) * b)) = Int64.repr (a * b).
+Proof.
+  intros.
+  apply Int64.eqm_samerepr.
+  apply Int64.eqm_mult.
+  - apply Int64.eqm_sym. apply Int64.eqm_unsigned_repr.
+  - apply Int64.eqm_refl.
+Qed.
+
+Lemma repr_unsigned_sub:
+  forall a b,
+    (Int64.repr (Int64.unsigned (Int64.repr a) - b)) = Int64.repr (a - b).
+Proof.
+  intros.
+  apply Int64.eqm_samerepr.
+  apply Int64.eqm_sub.
+  - apply Int64.eqm_sym. apply Int64.eqm_unsigned_repr.
+  - apply Int64.eqm_refl.
+Qed.
+
+Lemma repr_unsigned_add:
+  forall a b,
+    (Int64.repr (Int64.unsigned (Int64.repr a) + b)) = Int64.repr (a + b).
+Proof.
+  intros.
+  apply Int64.eqm_samerepr.
+  apply Int64.eqm_add.
+  - apply Int64.eqm_sym. apply Int64.eqm_unsigned_repr.
+  - apply Int64.eqm_refl.
+Qed.
+  
 Lemma step2_div_longu_bigb_correct :
   forall a b
            (b_BIG : ((Int64.unsigned b) > smallb_thresh)%Z)
@@ -1830,15 +1863,23 @@ Proof.
   pose proof (Zmod_eq_full a' b' NONZ) as MOD.
   assert (b' > 0)%Z as b_GT0 by lia.
   pose proof (Z_mod_lt a' b' b_GT0) as MOD_LT.
-  assert ((Int64.signed (Int64.sub (Int64.mul (Int64.repr q'') b) a))
-          = q''*b'-a')%Z as SIMPL.
-  { admit. }
-  rewrite SIMPL.
   destruct (Z_lt_dec a' (b' * q'')) as [LT | GE].
   { replace (b' * q'' >? a')%Z with true in QUOTIENT by lia.
+    unfold Int64.sub, Int64.mul.
+    fold a' b'.
+    replace q'' with (1 + a'/b')%Z by lia.
+    rewrite repr_unsigned_mul.
+    rewrite repr_unsigned_sub.
+    
+    replace ((1 + a' / b') * b' - a')%Z with (b' -  (a' - a' / b' * b'))%Z by ring.
+    rewrite <- MOD.
+    rewrite Int64.signed_repr ; cycle 1.
+    { change Int64.max_signed with 9223372036854775807%Z in *.
+      change Int64.min_signed with (-9223372036854775808)%Z in *.
+      lia.
+    }
     rewrite zlt_true by lia.
     replace q'' with (1 + (a' / b'))%Z by lia.
-    unfold Int64.add.
     apply Int64.eqm_samerepr.
     apply Int64.eqm_trans with (y := ((1 + a' / b') + (-1))%Z).
     { apply Int64.eqm_add.
@@ -1854,7 +1895,18 @@ Proof.
     replace (1 + a' / b' + -1)%Z with (a'/b')%Z by ring.
     apply Int64.eqm_refl.
   }
-  rewrite zlt_false by lia.
   replace (b' * q'' >? a')%Z with false in QUOTIENT by lia.
-  congruence.
-Admitted.
+  rewrite <- QUOTIENT.
+  unfold Int64.sub, Int64.mul.
+  fold a' b'.
+  rewrite repr_unsigned_mul.
+  rewrite repr_unsigned_sub.
+  replace (a' / b' * b' - a')%Z with (- (a' mod b'))%Z by lia.
+  rewrite Int64.signed_repr ; cycle 1.
+  { change Int64.max_signed with 9223372036854775807%Z in *.
+    change Int64.min_signed with (-9223372036854775808)%Z in *.
+    lia.
+  }
+  rewrite zlt_false by lia.
+  reflexivity.
+Qed.
-- 
cgit 


From a32b5bf92865e354f81330353857ae2d9ec80d57 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Tue, 25 Jan 2022 11:49:19 +0100
Subject: anystep_div_longu_correct

---
 kvx/FPDivision64.v | 38 ++++++++++++++++++++++++++++++++++++++
 1 file changed, 38 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index d96ced31..e57f0d66 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -26,6 +26,7 @@ Require Import Values Globalenvs.
 Require Compopts.
 Require Import Psatz.
 Require Import IEEE754_extra.
+Require Import Memory.
 
 From Gappa Require Import Gappa_tactic.
 
@@ -1910,3 +1911,40 @@ Proof.
   rewrite zlt_false by lia.
   reflexivity.
 Qed.
+
+Definition anystep_div_longu a b m :=
+  Val.select (Val.cmplu_bool (Mem.valid_pointer m) Cgt b (Vlong (Int64.repr smallb_thresh)))
+             (step2_div_longu a b)
+             (twostep_div_longu a b) Tlong.
+
+Lemma anystep_div_longu_correct :
+  forall a b m
+           (b_RANGE : (1 < Int64.unsigned b <= Int64.max_signed)%Z),
+      anystep_div_longu (Vlong a) (Vlong b) m = Vlong (Int64.repr (Int64.unsigned a / Int64.unsigned b))%Z.
+Proof.
+  intros.
+  unfold anystep_div_longu.
+  set (a' := Int64.unsigned a).
+  set (b' := Int64.unsigned b).
+  assert (0 < b')%Z as b_NOT0 by lia.
+  cbn.
+  unfold Int64.ltu.
+  fold b'.
+  rewrite Int64.unsigned_repr; cycle 1.
+  { unfold smallb_thresh.
+    change Int64.max_unsigned with 18446744073709551615%Z.
+    lia.
+  }
+  destruct zlt.
+  { rewrite step2_div_longu_bigb_correct by lia.
+    reflexivity.
+  }
+  rewrite twostep_div_longu_smallb_correct by lia.
+  cbn.
+  rewrite Int64.eq_false ; cycle 1.
+  { intro Z. unfold b' in b_NOT0. rewrite Z in b_NOT0.
+    rewrite Int64.unsigned_zero in b_NOT0.
+    lia.
+  }
+  reflexivity.
+Qed.
-- 
cgit 


From 82d72ac1253b2c45dbc8f305349611ea2eae0675 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Tue, 25 Jan 2022 12:19:05 +0100
Subject: attempt at definition

---
 kvx/FPDivision64.v | 13 +++++++++++++
 1 file changed, 13 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index e57f0d66..1739e5cf 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1948,3 +1948,16 @@ Proof.
   }
   reflexivity.
 Qed.
+
+Definition full_div_longu a b m :=
+  let is_big := Val.cmpl_bool Clt b (Vlong Int64.zero) in
+  let is_one := Val.cmplu_bool (Mem.valid_pointer m) Cle b (Vlong Int64.one) in
+  let is_special := Val.or (Val.of_optbool is_big) (Val.of_optbool is_one) in
+  let if_big := Val.select (Val.cmplu_bool (Mem.valid_pointer m) Clt a b)
+                           (Vlong Int64.zero) (Vlong Int64.one) Tlong in
+  let if_special := Val.select is_big if_big a Tlong in
+  Val.select (Val.cmp_bool Cne is_special (Vlong Int64.zero))
+             if_special
+             (anystep_div_longu a b m) Tlong.
+
+             
-- 
cgit 


From 2d8a28a1db545c2fc6f25fc55057ea230decb5c9 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Tue, 25 Jan 2022 13:06:18 +0100
Subject: some progress

---
 kvx/FPDivision64.v | 68 ++++++++++++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 66 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 1739e5cf..bf9be23d 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1949,6 +1949,7 @@ Proof.
   reflexivity.
 Qed.
 
+
 Definition full_div_longu a b m :=
   let is_big := Val.cmpl_bool Clt b (Vlong Int64.zero) in
   let is_one := Val.cmplu_bool (Mem.valid_pointer m) Cle b (Vlong Int64.one) in
@@ -1956,8 +1957,71 @@ Definition full_div_longu a b m :=
   let if_big := Val.select (Val.cmplu_bool (Mem.valid_pointer m) Clt a b)
                            (Vlong Int64.zero) (Vlong Int64.one) Tlong in
   let if_special := Val.select is_big if_big a Tlong in
-  Val.select (Val.cmp_bool Cne is_special (Vlong Int64.zero))
+  Val.select (Val.cmp_bool Cne is_special (Vint Int.zero))
              if_special
              (anystep_div_longu a b m) Tlong.
 
-             
+Lemma one_bigb_div :
+  forall a b
+     (b_RANGE : (9223372036854775808 <= b < 18446744073709551616)%Z)
+     (ORDER : (b <= a < 18446744073709551616)%Z),
+  (a / b = 1)%Z.
+Proof.
+  intros.
+  assert (((a - b) / b) = 0)%Z as ZERO.
+  { apply Zdiv_small. lia.
+  }
+  replace a with (1 * b + (a - b))%Z by ring.
+  rewrite Z.div_add_l by lia.
+  rewrite ZERO.
+  ring.
+Qed.
+
+Lemma full_div_longu_correct :
+  forall a b m
+           (b_NOT0 : (0 < Int64.unsigned b)%Z),
+      full_div_longu (Vlong a) (Vlong b) m = Vlong (Int64.repr (Int64.unsigned a / Int64.unsigned b))%Z.
+Proof.
+  intros.
+  unfold full_div_longu.
+  cbn.
+  unfold Int64.lt, Int64.ltu.
+  pose proof (Int64.unsigned_range a).
+  pose proof (Int64.unsigned_range b).
+  set (a' := Int64.unsigned a) in *.
+  set (b' := Int64.unsigned b) in *.
+  rewrite Int64.signed_zero.
+  rewrite Int64.unsigned_one.
+  destruct zlt.
+  { replace  (Val.cmp_bool Cne
+       (Val.or Vtrue
+          (if negb (if zlt 1 b' then true else false) then Vtrue else Vfalse))
+       (Vint Int.zero)) with (Some true) ; cycle 1.
+    { destruct zlt; reflexivity.
+    }
+    cbn.
+    destruct zlt; cbn.
+    { rewrite Zdiv_small by lia.
+      reflexivity.
+    }
+    rewrite one_bigb_div.
+    reflexivity.
+    {
+      change Int64.modulus with 18446744073709551616%Z in *.
+      split. 2: lia.
+      unfold b'.
+      rewrite Int64.unsigned_signed.
+      unfold Int64.lt.
+      rewrite Int64.signed_zero.
+      rewrite zlt_true by lia.
+      pose proof (Int64.signed_range b).
+      change Int64.min_signed with (-9223372036854775808)%Z in *.
+      change Int64.max_signed with (9223372036854775807)%Z in *.
+      change Int64.modulus with 18446744073709551616%Z in *.
+      lia.
+    }
+    change Int64.modulus with 18446744073709551616%Z in *.
+    lia.
+  }
+  
+!
-- 
cgit 


From 657a8d362bfdfdcd362749f68ec2b661d166df0b Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Tue, 25 Jan 2022 13:14:51 +0100
Subject: full_div_longu_correct

---
 kvx/FPDivision64.v | 30 ++++++++++++++++++++++++++++--
 1 file changed, 28 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index bf9be23d..4e91b853 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1982,6 +1982,8 @@ Lemma full_div_longu_correct :
            (b_NOT0 : (0 < Int64.unsigned b)%Z),
       full_div_longu (Vlong a) (Vlong b) m = Vlong (Int64.repr (Int64.unsigned a / Int64.unsigned b))%Z.
 Proof.
+
+  Local Opaque anystep_div_longu.
   intros.
   unfold full_div_longu.
   cbn.
@@ -2023,5 +2025,29 @@ Proof.
     change Int64.modulus with 18446744073709551616%Z in *.
     lia.
   }
-  
-!
+  destruct zlt; cbn.
+  { change (negb (Int.eq (Int.or Int.zero Int.zero) Int.zero)) with false.
+    cbn.
+    rewrite anystep_div_longu_correct.
+    reflexivity.
+    
+    change Int64.modulus with 18446744073709551616%Z in *.
+    split. lia.
+    rewrite Int64.unsigned_signed.
+    unfold Int64.lt.
+    rewrite Int64.signed_zero.
+    rewrite zlt_false by lia.
+    pose proof (Int64.signed_range b).
+    change Int64.min_signed with (-9223372036854775808)%Z in *.
+    change Int64.max_signed with (9223372036854775807)%Z in *.
+    change Int64.modulus with 18446744073709551616%Z in *.
+    lia.
+  }
+  change (negb (Int.eq (Int.or Int.zero Int.one) Int.zero)) with true.
+  cbn.
+  replace b' with 1%Z by lia.
+  rewrite Z.div_1_r.
+  unfold a'.
+  rewrite Int64.repr_unsigned.
+  reflexivity.
+Qed.
-- 
cgit 


From 607186f87c74817af529cd39d40390f0b86e59d4 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 4 Feb 2022 20:17:14 +0100
Subject: some progress

---
 kvx/FPDivision64.v | 108 +++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 108 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 4e91b853..ddd96f8e 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1065,6 +1065,114 @@ Proof.
   gappa.
 Qed.
 
+Lemma le_IZR3 :
+  forall n m p : Z, (IZR n <= IZR m <= IZR p)%R -> (n <= m <= p)%Z.
+Proof.
+  intros ; split ; apply le_IZR ; lra.
+Qed.
+
+(* 9223372036854775808%Z. *)
+Definition bigb_thresh := 1000000000000000000%Z.
+Lemma step1_div_longu_correct_anyb2 :
+    forall a b,
+    (1 < Int64.unsigned b <= bigb_thresh)%Z ->
+    exists (q : int64),
+      (step1_div_longu (Vlong a) (Vlong b)) = Vlong q /\
+        (Int64.min_signed <= (Int64.unsigned a - Int64.unsigned b*Int64.unsigned q) <= Int64.max_signed)%Z.
+Proof.
+  intros a b b_RANGE.
+
+  pose proof (Int64.unsigned_range a) as a_RANGE.
+  change Int64.modulus with 18446744073709551616%Z in a_RANGE.
+  set (a' := Int64.unsigned a) in *.
+  set (b' := Int64.unsigned b) in *.
+
+  destruct (Z_le_gt_dec a' 0).
+  { assert (a' = 0%Z) as ZERO by lia.
+    exists Int64.zero.
+    rewrite ZERO.
+    rewrite Int64.unsigned_zero.
+    replace (Z.abs (0 - b' * 0))%Z with 0%Z by lia.
+    replace a with Int64.zero; cycle 1.
+    {
+      unfold a' in ZERO.
+      unfold Int64.zero.
+      rewrite <- ZERO.
+      apply Int64.repr_unsigned.
+    }
+    split.
+    { apply step1_div_longu_a0.
+      lia.
+    }
+    change Int64.min_signed with (-9223372036854775808)%Z.
+    change Int64.max_signed with ( 9223372036854775807)%Z.
+    lia.
+  }
+
+  unfold step1_div_longu.
+  assert (1 < b')%Z as b_NOT01 by lia.   
+  destruct (step1_real_div_longu_correct a b b_NOT01) as (q & C1E & C1R & C1F & C1S).
+  rewrite C1E. cbn.
+  unfold Float.to_longu_ne.
+  pose proof (ZofB_ne_range_correct 53 1024 q 0 Int64.max_unsigned) as C2.
+  rewrite C1F in C2.
+
+  
+  assert (1 <= IZR a' <= 18446744073709551615)%R as a_RANGE'.
+  { split; apply IZR_le; lia. }
+  assert (2 <= IZR b' <= IZR bigb_thresh)%R as b_RANGE'.
+  { split; apply IZR_le; lia. }
+  assert (1 <= IZR b' <= IZR bigb_thresh)%R as b_RANGE'' by lra.
+  cbn zeta in C2.
+  rewrite C2.
+  cbn.
+  rewrite C1R.
+  unfold step1_real_quotient.
+  fold a' b'.
+  unfold bigb_thresh in *.
+  
+  rewrite Zle_bool_true ; cycle 1.
+  { apply Znearest_IZR_le.
+    gappa.
+  }
+  rewrite Zle_bool_true ; cycle 1.
+  { apply Znearest_le_IZR.
+    gappa.
+  }
+  cbn.
+  econstructor; split. reflexivity.
+  set (q_r := (rd (rd (IZR a') * rd (rs (1 / rs (IZR b')))))%R).
+  assert (Rabs (IZR (ZnearestE q_r) - q_r) <= /2)%R as NEAR by apply Znearest_imp2.
+  set (delta1 := (IZR (ZnearestE q_r) - q_r)%R) in NEAR.
+  apply le_IZR3.
+  rewrite minus_IZR.
+  rewrite mult_IZR.
+  rewrite Int64.unsigned_repr ; cycle 1.
+  { split.
+    - apply Znearest_IZR_le. unfold q_r.
+      gappa.
+    - apply Znearest_le_IZR. unfold q_r.
+      change Int64.max_unsigned with 18446744073709551615%Z.
+      gappa.
+  }
+  replace (IZR (ZnearestE q_r)) with ((IZR (ZnearestE q_r) - q_r) + q_r)%R by ring.
+  fold delta1.
+  unfold q_r.
+  set (a1 := IZR a') in *. 
+  set (b1 := IZR b') in *.
+  replace (rd (rd a1 * rd (rs (1 / rs b1))))%R with
+    ((((rd (rd a1 * rd (rs (1 / rs b1)))-(a1 * (1 / b1))) / (a1 * (1 / b1)))+1) * (a1 / b1))%R;
+    cycle 1.
+  { field. lra. }
+  set (delta2 := ((rd (rd a1 * rd (rs (1 / rs b1)))-(a1 * (1 / b1))) / (a1 * (1 / b1)))%R) in *.
+  assert (Rabs (delta2) <= 1/4194304)%R.
+  {  unfold delta2. gappa. }
+  replace (a1 - b1 * (delta1 + (delta2 + 1) * (a1 / b1)))%R with
+    (-b1*delta1 - a1*delta2)%R; cycle 1.
+  { field. lra. }
+  gappa.
+Qed.
+
 Lemma find_quotient:
   forall (a b : Z)
          (b_POS : (0 < b)%Z)
-- 
cgit 


From 9e37305cc4b817869de98b5cce49d3599d2c0aba Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 4 Feb 2022 20:19:17 +0100
Subject: progress

---
 kvx/FPDivision64.v | 3 +--
 1 file changed, 1 insertion(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index ddd96f8e..fc8e6173 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1071,8 +1071,7 @@ Proof.
   intros ; split ; apply le_IZR ; lra.
 Qed.
 
-(* 9223372036854775808%Z. *)
-Definition bigb_thresh := 1000000000000000000%Z.
+Definition bigb_thresh := 9223372036854775808%Z.
 Lemma step1_div_longu_correct_anyb2 :
     forall a b,
     (1 < Int64.unsigned b <= bigb_thresh)%Z ->
-- 
cgit 


From 6eac4cb3302f57d43e6b8d4c17688388db9619e5 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 4 Feb 2022 20:35:11 +0100
Subject: progress

---
 kvx/FPDivision64.v | 15 ++++++++++-----
 1 file changed, 10 insertions(+), 5 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index fc8e6173..79bfdc3b 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1071,15 +1071,16 @@ Proof.
   intros ; split ; apply le_IZR ; lra.
 Qed.
 
-Definition bigb_thresh := 9223372036854775808%Z.
+Definition bigb_thresh := 18446740000000000000%Z.
 Lemma step1_div_longu_correct_anyb2 :
     forall a b,
-    (1 < Int64.unsigned b <= bigb_thresh)%Z ->
+      (1 < Int64.unsigned b <= bigb_thresh)%Z ->
+      (Int64.unsigned b <= Int64.unsigned a)%Z ->
     exists (q : int64),
       (step1_div_longu (Vlong a) (Vlong b)) = Vlong q /\
         (Int64.min_signed <= (Int64.unsigned a - Int64.unsigned b*Int64.unsigned q) <= Int64.max_signed)%Z.
 Proof.
-  intros a b b_RANGE.
+  intros a b b_RANGE LESS.
 
   pose proof (Int64.unsigned_range a) as a_RANGE.
   change Int64.modulus with 18446744073709551616%Z in a_RANGE.
@@ -1164,11 +1165,15 @@ Proof.
     cycle 1.
   { field. lra. }
   set (delta2 := ((rd (rd a1 * rd (rs (1 / rs b1)))-(a1 * (1 / b1))) / (a1 * (1 / b1)))%R) in *.
-  assert (Rabs (delta2) <= 1/4194304)%R.
-  {  unfold delta2. gappa. }
+  (* assert (Rabs (delta2) <= 1/4194304)%R.
+  {  unfold delta2. gappa. } *)
   replace (a1 - b1 * (delta1 + (delta2 + 1) * (a1 / b1)))%R with
     (-b1*delta1 - a1*delta2)%R; cycle 1.
   { field. lra. }
+  assert (b1 <= a1)%R as LESS'.
+  { unfold a1, b1.
+    apply IZR_le. lia. }
+  unfold delta2.
   gappa.
 Qed.
 
-- 
cgit 


From 7b4583e6c008469a4300b2fe2d405c8de3753238 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Wed, 9 Feb 2022 15:58:23 +0100
Subject: twostep_div_longu_mostb_correct

---
 kvx/FPDivision64.v | 86 +++++++++++++++++++++++++++++++++++++++++++++++-------
 1 file changed, 75 insertions(+), 11 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 79bfdc3b..53961294 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1071,16 +1071,15 @@ Proof.
   intros ; split ; apply le_IZR ; lra.
 Qed.
 
-Definition bigb_thresh := 18446740000000000000%Z.
-Lemma step1_div_longu_correct_anyb2 :
+Definition mostb_thresh := 18446740000000000000%Z.
+Lemma step1_div_longu_correct_mostb :
     forall a b,
-      (1 < Int64.unsigned b <= bigb_thresh)%Z ->
-      (Int64.unsigned b <= Int64.unsigned a)%Z ->
+      (1 < Int64.unsigned b <= mostb_thresh)%Z ->
     exists (q : int64),
       (step1_div_longu (Vlong a) (Vlong b)) = Vlong q /\
         (Int64.min_signed <= (Int64.unsigned a - Int64.unsigned b*Int64.unsigned q) <= Int64.max_signed)%Z.
 Proof.
-  intros a b b_RANGE LESS.
+  intros a b b_RANGE.
 
   pose proof (Int64.unsigned_range a) as a_RANGE.
   change Int64.modulus with 18446744073709551616%Z in a_RANGE.
@@ -1120,16 +1119,16 @@ Proof.
   
   assert (1 <= IZR a' <= 18446744073709551615)%R as a_RANGE'.
   { split; apply IZR_le; lia. }
-  assert (2 <= IZR b' <= IZR bigb_thresh)%R as b_RANGE'.
+  assert (2 <= IZR b' <= IZR mostb_thresh)%R as b_RANGE'.
   { split; apply IZR_le; lia. }
-  assert (1 <= IZR b' <= IZR bigb_thresh)%R as b_RANGE'' by lra.
+  assert (1 <= IZR b' <= IZR mostb_thresh)%R as b_RANGE'' by lra.
   cbn zeta in C2.
   rewrite C2.
   cbn.
   rewrite C1R.
   unfold step1_real_quotient.
   fold a' b'.
-  unfold bigb_thresh in *.
+  unfold mostb_thresh in *.
   
   rewrite Zle_bool_true ; cycle 1.
   { apply Znearest_IZR_le.
@@ -1170,9 +1169,6 @@ Proof.
   replace (a1 - b1 * (delta1 + (delta2 + 1) * (a1 / b1)))%R with
     (-b1*delta1 - a1*delta2)%R; cycle 1.
   { field. lra. }
-  assert (b1 <= a1)%R as LESS'.
-  { unfold a1, b1.
-    apply IZR_le. lia. }
   unfold delta2.
   gappa.
 Qed.
@@ -2163,3 +2159,71 @@ Proof.
   rewrite Int64.repr_unsigned.
   reflexivity.
 Qed.
+
+Lemma repr_unsigned_sub2:
+  forall a b,
+    (Int64.repr (a - Int64.unsigned (Int64.repr b))) = Int64.repr (a - b).
+Proof.
+  intros.
+  apply Int64.eqm_samerepr.
+  apply Int64.eqm_sub.
+  - apply Int64.eqm_refl.
+  - apply Int64.eqm_sym. apply Int64.eqm_unsigned_repr.
+Qed.
+
+Lemma repr_unsigned_add2:
+  forall a b,
+    (Int64.repr (a + Int64.unsigned (Int64.repr b))) = Int64.repr (a + b).
+Proof.
+  intros.
+  apply Int64.eqm_samerepr.
+  apply Int64.eqm_add.
+  - apply Int64.eqm_refl.
+  - apply Int64.eqm_sym. apply Int64.eqm_unsigned_repr.
+Qed.
+    
+Lemma twostep_div_longu_mostb_correct :
+    forall a b
+      (b_RANGE : (1 < Int64.unsigned b <= Int64.max_signed)%Z),
+          (twostep_div_longu (Vlong a) (Vlong b)) =
+            (Val.maketotal (Val.divlu (Vlong a) (Vlong b))).
+Proof.
+  intros.
+  destruct (Z_le_gt_dec (Int64.unsigned b) smallb_thresh).
+  { apply twostep_div_longu_smallb_correct. lia. }
+  set (a' := Int64.unsigned a).
+  set (b' := Int64.unsigned b).
+  assert (0 < b')%Z as b_NOT0 by lia.
+  cbn.
+  rewrite Int64.eq_false ; cycle 1.
+  { intro Z. unfold b' in b_NOT0. rewrite Z in b_NOT0.
+    rewrite Int64.unsigned_zero in b_NOT0.
+    lia.
+  }
+  cbn.
+
+  unfold twostep_div_longu.
+  assert (1 < Int64.unsigned b <= mostb_thresh)%Z as MOST_B.
+  { unfold mostb_thresh.
+    change Int64.max_signed with 9223372036854775807%Z in b_RANGE.
+    lia.
+  }
+  destruct (step1_div_longu_correct_mostb a b MOST_B) as
+    (q & step1_REW & step1_RANGE).
+  rewrite step1_REW.
+  cbn.
+  rewrite step2_div_long_bigb_correct; cycle 1.
+  1, 2: lia.
+  f_equal.
+
+  unfold Int64.sub, Int64.mul.
+  rewrite repr_unsigned_sub2.
+  rewrite Int64.signed_repr by lia.
+  unfold Int64.add, Int64.divu.
+  fold a' b'.
+  set (q' := Int64.unsigned q) in *.
+  rewrite repr_unsigned_add2.
+  rewrite <- Z.div_add_l by lia.
+  f_equal. f_equal.
+  ring.
+Qed.
-- 
cgit 


From 57c868e304df143918fa3a99d6272437a14299cc Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Wed, 9 Feb 2022 16:10:59 +0100
Subject: full2_div_longu_correct

---
 kvx/FPDivision64.v | 93 ++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 93 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 53961294..38eb6c1b 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -2227,3 +2227,96 @@ Proof.
   f_equal. f_equal.
   ring.
 Qed.
+
+Definition full2_div_longu a b m :=
+  let is_big := Val.cmpl_bool Clt b (Vlong Int64.zero) in
+  let is_one := Val.cmplu_bool (Mem.valid_pointer m) Cle b (Vlong Int64.one) in
+  let is_special := Val.or (Val.of_optbool is_big) (Val.of_optbool is_one) in
+  let if_big := Val.select (Val.cmplu_bool (Mem.valid_pointer m) Clt a b)
+                           (Vlong Int64.zero) (Vlong Int64.one) Tlong in
+  let if_special := Val.select is_big if_big a Tlong in
+  Val.select (Val.cmp_bool Cne is_special (Vint Int.zero))
+             if_special
+             (twostep_div_longu a b) Tlong.
+
+Lemma full2_div_longu_correct :
+  forall a b m
+           (b_NOT0 : (0 < Int64.unsigned b)%Z),
+      full2_div_longu (Vlong a) (Vlong b) m = Vlong (Int64.repr (Int64.unsigned a / Int64.unsigned b))%Z.
+Proof.
+
+  Local Opaque twostep_div_longu.
+  intros.
+  unfold full2_div_longu.
+  cbn.
+  unfold Int64.lt, Int64.ltu.
+  pose proof (Int64.unsigned_range a).
+  pose proof (Int64.unsigned_range b).
+  set (a' := Int64.unsigned a) in *.
+  set (b' := Int64.unsigned b) in *.
+  rewrite Int64.signed_zero.
+  rewrite Int64.unsigned_one.
+  destruct zlt.
+  { replace  (Val.cmp_bool Cne
+       (Val.or Vtrue
+          (if negb (if zlt 1 b' then true else false) then Vtrue else Vfalse))
+       (Vint Int.zero)) with (Some true) ; cycle 1.
+    { destruct zlt; reflexivity.
+    }
+    cbn.
+    destruct zlt; cbn.
+    { rewrite Zdiv_small by lia.
+      reflexivity.
+    }
+    rewrite one_bigb_div.
+    reflexivity.
+    {
+      change Int64.modulus with 18446744073709551616%Z in *.
+      split. 2: lia.
+      unfold b'.
+      rewrite Int64.unsigned_signed.
+      unfold Int64.lt.
+      rewrite Int64.signed_zero.
+      rewrite zlt_true by lia.
+      pose proof (Int64.signed_range b).
+      change Int64.min_signed with (-9223372036854775808)%Z in *.
+      change Int64.max_signed with (9223372036854775807)%Z in *.
+      change Int64.modulus with 18446744073709551616%Z in *.
+      lia.
+    }
+    change Int64.modulus with 18446744073709551616%Z in *.
+    lia.
+  }
+  destruct zlt; cbn.
+  { change (negb (Int.eq (Int.or Int.zero Int.zero) Int.zero)) with false.
+    cbn.
+    rewrite twostep_div_longu_mostb_correct.
+    {
+      cbn.
+      unfold Int64.eq.
+      fold b'.
+      rewrite Int64.unsigned_zero.
+      rewrite zeq_false by lia.
+      reflexivity.
+    }
+    
+    change Int64.modulus with 18446744073709551616%Z in *.
+    split. lia.
+    rewrite Int64.unsigned_signed.
+    unfold Int64.lt.
+    rewrite Int64.signed_zero.
+    rewrite zlt_false by lia.
+    pose proof (Int64.signed_range b).
+    change Int64.min_signed with (-9223372036854775808)%Z in *.
+    change Int64.max_signed with (9223372036854775807)%Z in *.
+    change Int64.modulus with 18446744073709551616%Z in *.
+    lia.
+  }
+  change (negb (Int.eq (Int.or Int.zero Int.one) Int.zero)) with true.
+  cbn.
+  replace b' with 1%Z by lia.
+  rewrite Z.div_1_r.
+  unfold a'.
+  rewrite Int64.repr_unsigned.
+  reflexivity.
+Qed.
-- 
cgit 


From dc50627d829a4600b3248c4b79ce20f87f3e6d21 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Wed, 9 Feb 2022 17:23:30 +0100
Subject: step1 expression is ok

---
 kvx/FPDivision64.v | 36 ++++++++++++++++++++++++++++++++++++
 1 file changed, 36 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 38eb6c1b..2c6a49c5 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -2320,3 +2320,39 @@ Proof.
   rewrite Int64.repr_unsigned.
   reflexivity.
 Qed.
+
+Open Scope cminorsel_scope.
+Definition e_invfs a := Eop Oinvfs (a ::: Enil).
+Definition e_float_of_longu a := Eop Ofloatoflongu (a ::: Enil).
+Definition e_single_of_longu a := Eop Osingleoflongu (a ::: Enil).
+Definition e_float_of_single a := Eop Ofloatofsingle (a ::: Enil).
+Definition e_longu_of_float_ne a := Eop Olonguoffloat_ne (a ::: Enil).
+Definition e_mulf a b := Eop Omulf (a ::: b ::: Enil).
+
+Definition a_var1 := Eletvar (4%nat).
+Definition a_d_var1 := Eletvar (3%nat).
+Definition b_var1 := Eletvar (2%nat).
+Definition b_d_var1 := Eletvar (1%nat).
+Definition binv_d_var1 := Eletvar (0%nat).
+
+Definition e_setup1 a b rest :=
+  Elet a (Elet (e_float_of_longu (Eletvar 0%nat))
+    (Elet (lift (lift b)) (Elet (e_float_of_longu (Eletvar 0%nat))
+      (Elet (e_float_of_single (e_invfs (e_single_of_longu (Eletvar 1%nat))))
+      rest)))).                  
+Definition e_step1 := e_longu_of_float_ne (e_mulf a_d_var1 binv_d_var1).
+
+Lemma e_step1_correct :
+  forall (ge : genv)  (sp: val) cmenv memenv (le : letenv)
+         (expr_a : expr) (a : int64) (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a))
+         (expr_b : expr) (b : int64) (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)),
+         (eval_expr ge sp cmenv memenv le (e_setup1 expr_a expr_b (e_step1))
+                    (step1_div_longu (Vlong a) (Vlong b))).           
+Proof.
+  intros.
+  unfold e_setup1, step1_div_longu.
+  repeat econstructor.
+  { eassumption. }
+  { cbn. apply eval_lift. apply eval_lift. eassumption. }
+Qed.
+  
-- 
cgit 


From 68149e3a492be432561e61356f570ba394ca1e3a Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Thu, 10 Feb 2022 10:18:46 +0100
Subject: progress

---
 kvx/FPDivision64.v | 38 ++++++++++++++++++++++++++++++++++++--
 1 file changed, 36 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 2c6a49c5..619fcfa6 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -2324,11 +2324,15 @@ Qed.
 Open Scope cminorsel_scope.
 Definition e_invfs a := Eop Oinvfs (a ::: Enil).
 Definition e_float_of_longu a := Eop Ofloatoflongu (a ::: Enil).
+Definition e_float_of_long a := Eop Ofloatoflong (a ::: Enil).
 Definition e_single_of_longu a := Eop Osingleoflongu (a ::: Enil).
 Definition e_float_of_single a := Eop Ofloatofsingle (a ::: Enil).
+Definition e_long_of_float_ne a := Eop Olongoffloat_ne (a ::: Enil).
 Definition e_longu_of_float_ne a := Eop Olonguoffloat_ne (a ::: Enil).
 Definition e_mulf a b := Eop Omulf (a ::: b ::: Enil).
-
+Definition e_float_const c := Eop (Ofloatconst c) Enil.
+Definition e_fmaddf a b c := Eop Ofmaddf (a ::: b ::: c ::: Enil).
+Definition e_fmsubf a b c := Eop Ofmsubf (a ::: b ::: c ::: Enil).
 Definition a_var1 := Eletvar (4%nat).
 Definition a_d_var1 := Eletvar (3%nat).
 Definition b_var1 := Eletvar (2%nat).
@@ -2355,4 +2359,34 @@ Proof.
   { eassumption. }
   { cbn. apply eval_lift. apply eval_lift. eassumption. }
 Qed.
-  
+
+Definition e_setup2 a b rest := (e_setup1 a b (Elet e_step1 rest)).
+
+Definition a_var2 := Eletvar (5%nat).
+Definition a_d_var2 := Eletvar (4%nat).
+Definition b_var2 := Eletvar (3%nat).
+Definition b_d_var2 := Eletvar (2%nat).
+Definition binv_d_var2 := Eletvar (1%nat).
+Definition step1_var2 := Eletvar (0%nat).
+
+Definition e_step2 :=
+ e_long_of_float_ne
+       (e_mulf (e_float_of_long step1_var2)
+          (e_fmaddf
+             binv_d_var2
+             (e_fmsubf (e_float_const ExtFloat.one)
+                binv_d_var2
+                b_d_var2 )
+             binv_d_var2)).
+
+Lemma e_step2_correct :
+  forall (ge : genv)  (sp: val) cmenv memenv (le : letenv)
+         (expr_a : expr) (a : int64) (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a))
+         (expr_b : expr) (b : int64) (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)),
+         (eval_expr ge sp cmenv memenv le (e_setup2 expr_a expr_b (e_step2))
+           (step2_div_long' (step1_div_longu (Vlong a) (Vlong b)) (Vlong b))).
+Proof.
+intros.  
+unfold e_setup2, e_setup1, e_step2, step2_div_long', step2_real_div_long, approx_inv_longu.
+repeat (econstructor + apply eval_lift + eassumption).
+Qed.
-- 
cgit 


From ea53967476937f15ec4ceea1b31ece4d75b85f98 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Thu, 10 Feb 2022 10:51:46 +0100
Subject: progress

---
 kvx/FPDivision64.v | 18 ++++++++++++++++++
 1 file changed, 18 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 619fcfa6..572df195 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -2333,6 +2333,10 @@ Definition e_mulf a b := Eop Omulf (a ::: b ::: Enil).
 Definition e_float_const c := Eop (Ofloatconst c) Enil.
 Definition e_fmaddf a b c := Eop Ofmaddf (a ::: b ::: c ::: Enil).
 Definition e_fmsubf a b c := Eop Ofmsubf (a ::: b ::: c ::: Enil).
+Definition e_addlimm a b := Eop (Oaddlimm b) (a ::: Enil).
+Definition e_msub a b c := Eop Omsub (a ::: b ::: c ::: Enil).
+Definition e_ite ty c vc v1 v2 := Eop (Osel c ty) (vc ::: v1 ::: v2 ::: Enil). 
+  
 Definition a_var1 := Eletvar (4%nat).
 Definition a_d_var1 := Eletvar (3%nat).
 Definition b_var1 := Eletvar (2%nat).
@@ -2390,3 +2394,17 @@ intros.
 unfold e_setup2, e_setup1, e_step2, step2_div_long', step2_real_div_long, approx_inv_longu.
 repeat (econstructor + apply eval_lift + eassumption).
 Qed.
+
+Definition e_setup3 a b rest := (e_setup2 a b (Elet e_step2 rest)).
+
+Definition a_var3 := Eletvar (6%nat).
+Definition a_d_var3 := Eletvar (5%nat).
+Definition b_var3 := Eletvar (4%nat).
+Definition b_d_var3 := Eletvar (3%nat).
+Definition binv_d_var3 := Eletvar (2%nat).
+Definition step1_var3 := Eletvar (1%nat).
+Definition step2_var3 := Eletvar (0%nat).
+
+Definition e_step3 :=
+  e_ite Tlong (Ccompl0 Clt) (e_msub a_var3 step2_var3 b_var3)
+             (e_addlimm step2_var3 Int64.mone) step2_var3.
-- 
cgit 


From 3cb1d2813f8ff6ac9cbad44974fee00b8a0236c9 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Thu, 10 Feb 2022 11:01:38 +0100
Subject: Cgt / Clt

---
 kvx/FPDivision64.v | 27 ++++++++++++++++++++-------
 1 file changed, 20 insertions(+), 7 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 572df195..34dc108b 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1321,7 +1321,7 @@ Definition step2_div_long' a b :=
 
 Definition step2_div_long a b :=
   let q := step2_div_long' a b in
-  Val.select (Val.cmpl_bool Cgt (Val.subl (Val.mull q b) a) (Vlong Int64.zero))
+  Val.select (Val.cmpl_bool Clt (Val.subl a (Val.mull q b)) (Vlong Int64.zero))
              (Val.addl q (Vlong Int64.mone)) q Tlong.
 
 Lemma function_ite :
@@ -1377,6 +1377,19 @@ Proof.
     apply int64_eqm_signed_repr.
   - apply Int64.eqm_refl.
 Qed.
+
+Lemma signed_repr_sub2:
+  forall x y,
+    Int64.repr (x - Int64.signed (Int64.repr y)) =
+    Int64.repr (x - y).
+Proof.
+  intros.
+  apply Int64.eqm_samerepr.
+  apply Int64.eqm_sub.
+  - apply Int64.eqm_refl.
+  - apply Int64.eqm_sym.
+    apply int64_eqm_signed_repr.
+Qed.
                                                                         
 Lemma step2_div_long_smalla_correct :
     forall a b
@@ -1454,7 +1467,7 @@ Proof.
 
   rewrite Int64.sub_signed.
   fold a'.
-  rewrite signed_repr_sub.
+  rewrite signed_repr_sub2.
 
   assert ((Rabs (IZR a' / IZR b' - q') < / 2)%R) as HALF.
   { replace (IZR a' / IZR b' - q')%R with
@@ -1473,8 +1486,8 @@ Proof.
   destruct (Z_lt_dec a' (b' * q'')) as [LT | GE].
   { replace (b' * q'' >? a')%Z with true in QUOTIENT by lia.
     replace q'' with (1 + (a' / b'))%Z by lia.
-    replace ((1 + a' / b') * b' - a')%Z
-      with (-(a' - a' / b' * b')+b')%Z by ring.
+    replace (a' - (1 + a' / b') * b')%Z
+      with ((a' - a' / b' * b')-b')%Z by ring.
     rewrite <- MOD.
     rewrite Int64.signed_repr; cycle 1.
     { change Int64.min_signed with (-9223372036854775808)%Z in *.
@@ -1731,7 +1744,7 @@ Proof.
 
   rewrite Int64.sub_signed.
   fold a'.
-  rewrite signed_repr_sub.
+  rewrite signed_repr_sub2.
 
   assert ((Rabs (IZR a' / IZR b' - q') < / 2)%R) as HALF.
   { replace (IZR a' / IZR b' - q')%R with
@@ -1750,8 +1763,8 @@ Proof.
   destruct (Z_lt_dec a' (b' * q'')) as [LT | GE].
   { replace (b' * q'' >? a')%Z with true in QUOTIENT by lia.
     replace q'' with (1 + (a' / b'))%Z by lia.
-    replace ((1 + a' / b') * b' - a')%Z
-      with (-(a' - a' / b' * b')+b')%Z by ring.
+    replace (a' - (1 + a' / b') * b')%Z
+      with ((a' - a' / b' * b')-b')%Z by ring.
     rewrite <- MOD.
     rewrite Int64.signed_repr; cycle 1.
     { change Int64.min_signed with (-9223372036854775808)%Z in *.
-- 
cgit 


From aa7c85e476d1f63d834cd73e3a2da81b73afd32b Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Thu, 10 Feb 2022 11:49:59 +0100
Subject: progress

---
 kvx/FPDivision64.v | 18 +++++++++++++++---
 1 file changed, 15 insertions(+), 3 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 34dc108b..f991ff93 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -2347,8 +2347,8 @@ Definition e_float_const c := Eop (Ofloatconst c) Enil.
 Definition e_fmaddf a b c := Eop Ofmaddf (a ::: b ::: c ::: Enil).
 Definition e_fmsubf a b c := Eop Ofmsubf (a ::: b ::: c ::: Enil).
 Definition e_addlimm a b := Eop (Oaddlimm b) (a ::: Enil).
-Definition e_msub a b c := Eop Omsub (a ::: b ::: c ::: Enil).
-Definition e_ite ty c vc v1 v2 := Eop (Osel c ty) (vc ::: v1 ::: v2 ::: Enil). 
+Definition e_msubl a b c := Eop Omsubl (a ::: b ::: c ::: Enil).
+Definition e_ite ty c vc v1 v2 := Eop (Osel c ty) (v1 ::: v2 ::: vc ::: Enil). 
   
 Definition a_var1 := Eletvar (4%nat).
 Definition a_d_var1 := Eletvar (3%nat).
@@ -2419,5 +2419,17 @@ Definition step1_var3 := Eletvar (1%nat).
 Definition step2_var3 := Eletvar (0%nat).
 
 Definition e_step3 :=
-  e_ite Tlong (Ccompl0 Clt) (e_msub a_var3 step2_var3 b_var3)
+  e_ite Tlong (Ccompl0 Clt) (e_msubl step1_var3 step2_var3 b_var3)
              (e_addlimm step2_var3 Int64.mone) step2_var3.
+
+Lemma e_step3_correct :
+  forall (ge : genv)  (sp: val) cmenv memenv (le : letenv)
+         (expr_a : expr) (a : int64) (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a))
+         (expr_b : expr) (b : int64) (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)),
+         (eval_expr ge sp cmenv memenv le (e_setup3 expr_a expr_b (e_step3))
+           (step2_div_long (step1_div_longu (Vlong a) (Vlong b)) (Vlong b))).
+Proof.
+intros.  
+unfold e_setup2, e_setup1, e_step2, step2_div_long, step2_div_long', step2_real_div_long, approx_inv_longu, step1_div_longu.
+repeat (econstructor + apply eval_lift + eassumption).
+Qed.
-- 
cgit 


From 3e445a4f31b5b309b64cdf307830716f0001003f Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Thu, 10 Feb 2022 17:16:26 +0100
Subject: progress

---
 kvx/FPDivision64.v | 59 +++++++++++++++++++++++++++++++++---------------------
 1 file changed, 36 insertions(+), 23 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index f991ff93..457b280b 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -2386,48 +2386,61 @@ Definition b_d_var2 := Eletvar (2%nat).
 Definition binv_d_var2 := Eletvar (1%nat).
 Definition step1_var2 := Eletvar (0%nat).
 
-Definition e_step2 :=
+Definition e_step2 := e_msubl a_var2 b_var2 step1_var2.
+
+Definition e_setup3 a b rest := (e_setup2 a b (Elet e_step2 rest)).
+
+Definition a_var3 := Eletvar (6%nat).
+Definition a_d_var3 := Eletvar (5%nat).
+Definition b_var3 := Eletvar (4%nat).
+Definition b_d_var3 := Eletvar (3%nat).
+Definition binv_d_var3 := Eletvar (2%nat).
+Definition step1_var3 := Eletvar (1%nat).
+Definition step2_var3 := Eletvar (0%nat).
+
+Definition e_step3 :=
  e_long_of_float_ne
-       (e_mulf (e_float_of_long step1_var2)
+       (e_mulf (e_float_of_long step2_var3)
           (e_fmaddf
-             binv_d_var2
+             binv_d_var3
              (e_fmsubf (e_float_const ExtFloat.one)
-                binv_d_var2
-                b_d_var2 )
-             binv_d_var2)).
+                binv_d_var3
+                b_d_var3 )
+             binv_d_var3)).
 
-Lemma e_step2_correct :
+Lemma e_step3_correct :
   forall (ge : genv)  (sp: val) cmenv memenv (le : letenv)
          (expr_a : expr) (a : int64) (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a))
          (expr_b : expr) (b : int64) (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)),
-         (eval_expr ge sp cmenv memenv le (e_setup2 expr_a expr_b (e_step2))
-           (step2_div_long' (step1_div_longu (Vlong a) (Vlong b)) (Vlong b))).
+         (eval_expr ge sp cmenv memenv le (e_setup3 expr_a expr_b (e_step3))
+           (step2_div_long' (Val.subl (Vlong a) (Val.mull (Vlong b) (step1_div_longu (Vlong a) (Vlong b)))) (Vlong b))).
 Proof.
 intros.  
 unfold e_setup2, e_setup1, e_step2, step2_div_long', step2_real_div_long, approx_inv_longu.
 repeat (econstructor + apply eval_lift + eassumption).
 Qed.
 
-Definition e_setup3 a b rest := (e_setup2 a b (Elet e_step2 rest)).
+Definition e_setup4 a b rest := (e_setup3 a b (Elet e_step3 rest)).
 
-Definition a_var3 := Eletvar (6%nat).
-Definition a_d_var3 := Eletvar (5%nat).
-Definition b_var3 := Eletvar (4%nat).
-Definition b_d_var3 := Eletvar (3%nat).
-Definition binv_d_var3 := Eletvar (2%nat).
-Definition step1_var3 := Eletvar (1%nat).
-Definition step2_var3 := Eletvar (0%nat).
+Definition a_var4 := Eletvar (7%nat).
+Definition a_d_var4 := Eletvar (6%nat).
+Definition b_var4 := Eletvar (5%nat).
+Definition b_d_var4 := Eletvar (4%nat).
+Definition binv_d_var4 := Eletvar (3%nat).
+Definition step1_var4 := Eletvar (2%nat).
+Definition step2_var4 := Eletvar (1%nat).
+Definition step3_var4 := Eletvar (0%nat).
 
-Definition e_step3 :=
-  e_ite Tlong (Ccompl0 Clt) (e_msubl step1_var3 step2_var3 b_var3)
-             (e_addlimm step2_var3 Int64.mone) step2_var3.
+Definition e_step4 :=
+  e_ite Tlong (Ccompl0 Clt) (e_msubl step2_var4 step3_var4 b_var4)
+             (e_addlimm step3_var4 Int64.mone) step3_var4.
 
-Lemma e_step3_correct :
+Lemma e_step4_correct :
   forall (ge : genv)  (sp: val) cmenv memenv (le : letenv)
          (expr_a : expr) (a : int64) (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a))
          (expr_b : expr) (b : int64) (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)),
-         (eval_expr ge sp cmenv memenv le (e_setup3 expr_a expr_b (e_step3))
-           (step2_div_long (step1_div_longu (Vlong a) (Vlong b)) (Vlong b))).
+         (eval_expr ge sp cmenv memenv le (e_setup4 expr_a expr_b (e_step4))
+           (step2_div_long (Val.subl (Vlong a) (Val.mull (Vlong b) (step1_div_longu (Vlong a) (Vlong b)))) (Vlong b))).
 Proof.
 intros.  
 unfold e_setup2, e_setup1, e_step2, step2_div_long, step2_div_long', step2_real_div_long, approx_inv_longu, step1_div_longu.
-- 
cgit 


From d8a3b4a525bcd3ea17d1585ec2e93f3d2dcacba6 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Thu, 10 Feb 2022 19:27:56 +0100
Subject: progress

---
 kvx/FPDivision64.v | 46 ++++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 44 insertions(+), 2 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 457b280b..a1ab09b8 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -2348,8 +2348,10 @@ Definition e_fmaddf a b c := Eop Ofmaddf (a ::: b ::: c ::: Enil).
 Definition e_fmsubf a b c := Eop Ofmsubf (a ::: b ::: c ::: Enil).
 Definition e_addlimm a b := Eop (Oaddlimm b) (a ::: Enil).
 Definition e_msubl a b c := Eop Omsubl (a ::: b ::: c ::: Enil).
-Definition e_ite ty c vc v1 v2 := Eop (Osel c ty) (v1 ::: v2 ::: vc ::: Enil). 
-  
+Definition e_ite ty c vc v1 v2 := Eop (Osel c ty) (v1 ::: v2 ::: vc ::: Enil).
+Definition e_cmplimm c v n := Eop (Ocmp (Ccomplimm c n)) (v ::: Enil).
+Definition e_addl a b := Eop Oaddl (a ::: b ::: Enil).
+
 Definition a_var1 := Eletvar (4%nat).
 Definition a_d_var1 := Eletvar (3%nat).
 Definition b_var1 := Eletvar (2%nat).
@@ -2446,3 +2448,43 @@ intros.
 unfold e_setup2, e_setup1, e_step2, step2_div_long, step2_div_long', step2_real_div_long, approx_inv_longu, step1_div_longu.
 repeat (econstructor + apply eval_lift + eassumption).
 Qed.
+
+Definition e_setup5 a b rest := (e_setup4 a b (Elet e_step4 rest)).
+
+
+Definition a_var5 := Eletvar (8%nat).
+Definition a_d_var5 := Eletvar (7%nat).
+Definition b_var5 := Eletvar (6%nat).
+Definition b_d_var5 := Eletvar (5%nat).
+Definition binv_d_var5 := Eletvar (4%nat).
+Definition step1_var5 := Eletvar (3%nat).
+Definition step2_var5 := Eletvar (2%nat).
+Definition step3_var5 := Eletvar (1%nat).
+Definition step4_var5 := Eletvar (0%nat).
+
+Definition e_step5 := e_addl step1_var5 step4_var5.
+
+Lemma e_step5_correct :
+  forall (ge : genv)  (sp: val) cmenv memenv (le : letenv)
+         (expr_a : expr) (a : int64) (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a))
+         (expr_b : expr) (b : int64) (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)),
+         (eval_expr ge sp cmenv memenv le (e_setup5 expr_a expr_b (e_step5))
+           (twostep_div_longu (Vlong a) (Vlong b))).
+Proof.
+  intros.
+  Local Transparent twostep_div_longu.
+  repeat unfold e_setup2, e_setup1, e_step2, step2_div_long, step2_div_long', step2_real_div_long, approx_inv_longu, step1_div_longu, twostep_div_longu.
+repeat (econstructor + apply eval_lift + eassumption).
+Qed.
+
+(*
+  let is_big := Val.cmpl_bool Clt b (Vlong Int64.zero) in
+  let is_one := Val.cmplu_bool (Mem.valid_pointer m) Cle b (Vlong Int64.one) in
+  let is_special := Val.or (Val.of_optbool is_big) (Val.of_optbool is_one) in
+  let if_big := Val.select (Val.cmplu_bool (Mem.valid_pointer m) Clt a b)
+                           (Vlong Int64.zero) (Vlong Int64.one) Tlong in
+  let if_special := Val.select is_big if_big a Tlong in
+  Val.select (Val.cmp_bool Cne is_special (Vint Int.zero))
+             if_special
+             (twostep_div_longu a b) Tlong.
+ *)
-- 
cgit 


From 6bd3c44a582d3d1f23fe4ed99dec8b59893e3a85 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Thu, 10 Feb 2022 22:35:22 +0100
Subject: finished proof but still why need for decomposition

---
 kvx/FPDivision64.v | 72 ++++++++++++++++++++++++++++++++++++++++++------------
 1 file changed, 56 insertions(+), 16 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index a1ab09b8..c08103b1 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -2075,8 +2075,7 @@ Definition full_div_longu a b m :=
   let is_big := Val.cmpl_bool Clt b (Vlong Int64.zero) in
   let is_one := Val.cmplu_bool (Mem.valid_pointer m) Cle b (Vlong Int64.one) in
   let is_special := Val.or (Val.of_optbool is_big) (Val.of_optbool is_one) in
-  let if_big := Val.select (Val.cmplu_bool (Mem.valid_pointer m) Clt a b)
-                           (Vlong Int64.zero) (Vlong Int64.one) Tlong in
+  let if_big := Val.longofintu (Val.of_optbool (Val.cmplu_bool (Mem.valid_pointer m) Cge a b)) in
   let if_special := Val.select is_big if_big a Tlong in
   Val.select (Val.cmp_bool Cne is_special (Vint Int.zero))
              if_special
@@ -2245,8 +2244,7 @@ Definition full2_div_longu a b m :=
   let is_big := Val.cmpl_bool Clt b (Vlong Int64.zero) in
   let is_one := Val.cmplu_bool (Mem.valid_pointer m) Cle b (Vlong Int64.one) in
   let is_special := Val.or (Val.of_optbool is_big) (Val.of_optbool is_one) in
-  let if_big := Val.select (Val.cmplu_bool (Mem.valid_pointer m) Clt a b)
-                           (Vlong Int64.zero) (Vlong Int64.one) Tlong in
+  let if_big := Val.longofintu (Val.of_optbool (Val.cmplu_bool (Mem.valid_pointer m) Cge a b)) in
   let if_special := Val.select is_big if_big a Tlong in
   Val.select (Val.cmp_bool Cne is_special (Vint Int.zero))
              if_special
@@ -2350,7 +2348,11 @@ Definition e_addlimm a b := Eop (Oaddlimm b) (a ::: Enil).
 Definition e_msubl a b c := Eop Omsubl (a ::: b ::: c ::: Enil).
 Definition e_ite ty c vc v1 v2 := Eop (Osel c ty) (v1 ::: v2 ::: vc ::: Enil).
 Definition e_cmplimm c v n := Eop (Ocmp (Ccomplimm c n)) (v ::: Enil).
+Definition e_cmpluimm c v n := Eop (Ocmp (Ccompluimm c n)) (v ::: Enil).
 Definition e_addl a b := Eop Oaddl (a ::: b ::: Enil).
+Definition e_or a b := Eop Oor (a ::: b ::: Enil).
+Definition e_cast32unsigned a := Eop Ocast32unsigned (a ::: Enil).
+Definition e_cmplu c a b := Eop (Ocmp (Ccomplu c)) (a ::: b ::: Enil).
 
 Definition a_var1 := Eletvar (4%nat).
 Definition a_d_var1 := Eletvar (3%nat).
@@ -2451,7 +2453,6 @@ Qed.
 
 Definition e_setup5 a b rest := (e_setup4 a b (Elet e_step4 rest)).
 
-
 Definition a_var5 := Eletvar (8%nat).
 Definition a_d_var5 := Eletvar (7%nat).
 Definition b_var5 := Eletvar (6%nat).
@@ -2477,14 +2478,53 @@ Proof.
 repeat (econstructor + apply eval_lift + eassumption).
 Qed.
 
-(*
-  let is_big := Val.cmpl_bool Clt b (Vlong Int64.zero) in
-  let is_one := Val.cmplu_bool (Mem.valid_pointer m) Cle b (Vlong Int64.one) in
-  let is_special := Val.or (Val.of_optbool is_big) (Val.of_optbool is_one) in
-  let if_big := Val.select (Val.cmplu_bool (Mem.valid_pointer m) Clt a b)
-                           (Vlong Int64.zero) (Vlong Int64.one) Tlong in
-  let if_special := Val.select is_big if_big a Tlong in
-  Val.select (Val.cmp_bool Cne is_special (Vint Int.zero))
-             if_special
-             (twostep_div_longu a b) Tlong.
- *)
+Definition e_setup6 a b rest := (e_setup5 a b (Elet e_step5 rest)).
+
+Definition a_var6 := Eletvar (9%nat).
+Definition a_d_var6 := Eletvar (8%nat).
+Definition b_var6 := Eletvar (7%nat).
+Definition b_d_var6 := Eletvar (6%nat).
+Definition binv_d_var6 := Eletvar (5%nat).
+Definition step1_var6 := Eletvar (4%nat).
+Definition step2_var6 := Eletvar (3%nat).
+Definition step3_var6 := Eletvar (2%nat).
+Definition step4_var6 := Eletvar (1%nat).
+Definition twostep_var6 := Eletvar (0%nat).
+
+Definition e_step6 := e_cmplimm Clt b_var6 Int64.zero.
+
+Definition e_setup7 a b rest := e_setup6 a b (Elet e_step6 rest).
+
+Definition a_var7 := Eletvar (10%nat).
+Definition a_d_var7 := Eletvar (9%nat).
+Definition b_var7 := Eletvar (8%nat).
+Definition b_d_var7 := Eletvar (7%nat).
+Definition binv_d_var7 := Eletvar (6%nat).
+Definition step1_var7 := Eletvar (5%nat).
+Definition step2_var7 := Eletvar (5%nat).
+Definition step3_var7 := Eletvar (3%nat).
+Definition step4_var7 := Eletvar (2%nat).
+Definition twostep_var7 := Eletvar (1%nat).
+Definition is_big_var7 := Eletvar (0%nat).
+
+Definition e_is_one :=  e_cmpluimm Cle b_var7 Int64.one.
+Definition e_is_special := e_or is_big_var7 e_is_one.
+Definition e_if_big := e_cast32unsigned (e_cmplu Cge a_var7 b_var7).
+Definition e_if_special := e_ite Tlong (Ccompu0 Cne) is_big_var7 e_if_big a_var7.
+Definition e_step7 := e_ite Tlong (Ccompu0 Cne) e_is_special e_if_special twostep_var7.
+
+Lemma e_step7_correct :
+  forall (ge : genv)  (sp: val) cmenv memenv (le : letenv)
+         (expr_a : expr) (a : int64) (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a))
+         (expr_b : expr) (b : int64) (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)),
+         (eval_expr ge sp cmenv memenv le (e_setup7 expr_a expr_b (e_step7))
+           (full2_div_longu (Vlong a) (Vlong b) memenv)).
+Proof.
+  intros.
+  Local Transparent full2_div_longu.
+  repeat unfold e_setup2, e_setup1, e_step2, step2_div_long, step2_div_long', step2_real_div_long, approx_inv_longu, step1_div_longu, twostep_div_longu, full2_div_longu.
+  repeat (econstructor + apply eval_lift + eassumption).
+  cbn.
+  repeat f_equal.
+  destruct (Int64.lt b Int64.zero); cbn; change (Int.eq Int.one Int.zero) with false; change (Int.eq Int.zero Int.zero) with true; cbn; reflexivity.
+Qed.
-- 
cgit 


From b45d9ac4b98ed9ce095a1ad64a4192fea89f7b5f Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Thu, 10 Feb 2022 22:48:26 +0100
Subject: fp_divu64_correct

---
 kvx/FPDivision64.v | 17 +++++++++++++++++
 1 file changed, 17 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index c08103b1..411865ba 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -2528,3 +2528,20 @@ Proof.
   repeat f_equal.
   destruct (Int64.lt b Int64.zero); cbn; change (Int.eq Int.one Int.zero) with false; change (Int.eq Int.zero Int.zero) with true; cbn; reflexivity.
 Qed.
+
+Definition fp_divu64 a b := e_setup7 a b e_step7.
+
+Theorem fp_divu64_correct :
+  forall (ge : genv)  (sp: val) cmenv memenv
+         (le : letenv) (expr_a expr_b : expr) (a b : int64)
+         (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a))
+         (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b))
+         (b_nz : (Int64.unsigned b > 0)%Z),
+  eval_expr ge sp cmenv memenv le (fp_divu64 expr_a expr_b)
+            (Vlong (Int64.divu a b)).
+Proof.
+  intros.
+  unfold Int64.divu.
+  rewrite <- full2_div_longu_correct with (m := memenv) by lia.
+  apply e_step7_correct; assumption.
+Qed.
-- 
cgit 


From f8d32a19caf88733a9bbeee976f5c2fc549d4f92 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 11 Feb 2022 10:32:15 +0100
Subject: rewrite with longu -> double -> single conversions

---
 kvx/FPDivision64.v | 61 ++++++++++++++++++++++++++++++++++--------------------
 1 file changed, 38 insertions(+), 23 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 411865ba..a4d609f0 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -532,7 +532,7 @@ Proof.
 Qed.
 
 Definition step1_real_inv_longu b :=
-  let invb_s := ExtValues.invfs (Val.maketotal (Val.singleoflongu b)) in
+  let invb_s := ExtValues.invfs (Val.singleoffloat (Val.maketotal (Val.floatoflongu b))) in
   Val.floatofsingle invb_s.
 
 Definition step1_real_inv_thresh := (3/33554432)%R.
@@ -543,7 +543,7 @@ Theorem step1_real_inv_longu_correct :
     (0 < Int64.unsigned b)%Z ->
     exists (f : float),
       (step1_real_inv_longu (Vlong b)) = Vfloat f /\
-        (B2R _ _ f) = (rd (rs (1 / rs (IZR (Int64.unsigned b))))) /\
+        (B2R _ _ f) = (rd (rs (1 / rs (rd (IZR (Int64.unsigned b)))))) /\
         is_finite _ _ f = true /\
         Bsign _ _ f = false.
 Proof.
@@ -553,8 +553,8 @@ Proof.
   econstructor.
   split.
   reflexivity.
-  Local Transparent Float.neg Float.of_single Float32.of_longu Float32.div Float.of_longu Float32.of_int Float.of_int.
-  unfold Float.fma, Float.neg, Float.of_single, Float32.of_longu, ExtFloat32.inv, Float32.div, Float.of_longu, ExtFloat32.one, Float32.of_int, ExtFloat.one, Float.of_int.
+  Local Transparent Float.neg Float.of_single Float32.of_longu Float32.div Float.of_longu Float32.of_int Float.of_int Float.to_single.
+  unfold Float.fma, Float.neg, Float.of_single, Float32.of_longu, ExtFloat32.inv, Float32.div, Float.of_longu, ExtFloat32.one, Float32.of_int, ExtFloat.one, Float.of_int, Float.to_single.
   set (re := (@eq_refl Datatypes.comparison Lt)).
   change (Int.signed (Int.repr 1)) with 1%Z.
   set (b' := Int64.unsigned b) in *.
@@ -565,25 +565,37 @@ Proof.
   }
 
   assert (-16777216 <= 1 <= 16777216)%Z as SILLY by lia.
-  destruct (BofZ_exact 24 128 re re 1 SILLY) as (C0R & C0F & _).
+  destruct (BofZ_exact 24 128 re re 1 SILLY) as (C0R & C0F & C0S).
   clear SILLY.
   set (one_s := (BofZ 24 128 re re 1)) in *.
-  
-  pose proof (BofZ_correct 24 128 re re b') as C1.
-  cbn in C1.
+
+  pose proof (BofZ_correct 53 1024 re re b') as C0'.
+  rewrite Rlt_bool_true in C0'; cycle 1.
+  { apply (Rabs_relax (bpow radix2 64)).
+    { apply bpow_lt. lia. }
+    cbn.
+    gappa.
+  }
+  cbn in C0'.
+  destruct C0' as (C0'R & C0'F & C0'S).
+  set (b_d :=  (BofZ 53 1024 re re b')) in *.
+
+  pose proof (Bconv_correct 53 1024 24 128 re re Float.to_single_nan mode_NE b_d C0'F) as C1.
+  rewrite C0'R in C1.
+  rewrite C0'S in C1.
   rewrite Rlt_bool_true in C1; cycle 1.
   { clear C1.
-    eapply (Rabs_relax (IZR 18446744073709551616)).
-    lra.
-    set (b'' := IZR b') in *.
+    eapply (Rabs_relax (bpow radix2 64)).
+    { apply bpow_lt. lia. }
+    cbn.
     gappa.
   }
-  rewrite (Zlt_bool_false b' 0) in C1 by lia.
   destruct C1 as (C1R & C1F & C1S).
-  set (b_s :=  (BofZ 24 128 re re b')) in *.
+  set (b_s := (Bconv 53 1024 24 128 re re Float.to_single_nan mode_NE b_d)) in *.
 
   assert(1 <= B2R 24 128 b_s <= 18446744073709551616)%R as b_s_RANGE.
   { rewrite C1R.
+    cbn.
     gappa.
   }
   assert(B2R 24 128 b_s <> 0)%R as b_s_NONZ by lra.
@@ -599,9 +611,12 @@ Proof.
     gappa.
   }
   rewrite C1R in C2.
+  rewrite C1S in C2.
+  rewrite C0S in C2.
   destruct C2 as (C2R & C2F & C2Sz).
-  rewrite C1S in C2Sz.
-  change (xorb _ _) with false in C2Sz.
+  change (1 <? 0)%Z with false in C2Sz.
+  replace (b' <? 0)%Z with false in C2Sz by lia.
+  change (xorb false false) with false in C2Sz.
   set (invb_s := (Bdiv 24 128 re re Float32.binop_nan mode_NE one_s b_s)) in *.
   rewrite C0F in C2F.
   assert (is_nan 24 128 invb_s = false) as NAN.
@@ -709,7 +724,7 @@ Definition step1_div_longu a b :=
   Val.maketotal (Val.longuoffloat_ne (step1_real_div_longu a b)).
 
 Definition step1_real_quotient (a b : R) :=
-             rd ((rd (a)) * (rd (rs (1 / rs (b))))).
+             rd ((rd (a)) * (rd (rs (1 / rs (rd (b)))))).
   
 Theorem step1_real_div_longu_correct:
   forall a b,
@@ -794,10 +809,10 @@ Proof.
   unfold smallb_thresh in b_RANGE.
   unfold smallb_approx_real_range.
   unfold step1_real_quotient.
-  set (q := ((rd (a)) * (rd (rs (1 / rs (b)))))%R) in *.
+  set (q := ((rd (a)) * (rd (rs (1 / rs (rd b)))))%R) in *.
   replace ((rd q) *b - a)%R with
-     ((rd(q)-q)/q * rd(a) * (1 + (rd (rs (1 / rs (b))) - 1/b)/(1/b)) +
-  (rd (a)) * ((rd (rs (1 / rs (b))) - 1 / b) / (1/b)) +
+     ((rd(q)-q)/q * rd(a) * (1 + (rd (rs (1 / rs (rd b))) - 1/b)/(1/b)) +
+  (rd (a)) * ((rd (rs (1 / rs (rd b))) - 1 / b) / (1/b)) +
         (rd(a) - a))%R; cycle 1.
   { unfold q.
     field.
@@ -1140,7 +1155,7 @@ Proof.
   }
   cbn.
   econstructor; split. reflexivity.
-  set (q_r := (rd (rd (IZR a') * rd (rs (1 / rs (IZR b')))))%R).
+  set (q_r := (rd (rd (IZR a') * rd (rs (1 / rs ( rd (IZR b'))))))%R).
   assert (Rabs (IZR (ZnearestE q_r) - q_r) <= /2)%R as NEAR by apply Znearest_imp2.
   set (delta1 := (IZR (ZnearestE q_r) - q_r)%R) in NEAR.
   apply le_IZR3.
@@ -1159,11 +1174,11 @@ Proof.
   unfold q_r.
   set (a1 := IZR a') in *. 
   set (b1 := IZR b') in *.
-  replace (rd (rd a1 * rd (rs (1 / rs b1))))%R with
-    ((((rd (rd a1 * rd (rs (1 / rs b1)))-(a1 * (1 / b1))) / (a1 * (1 / b1)))+1) * (a1 / b1))%R;
+  replace (rd (rd a1 * rd (rs (1 / rs (rd b1)))))%R with
+    ((((rd (rd a1 * rd (rs (1 / rs (rd b1))))-(a1 * (1 / b1))) / (a1 * (1 / b1)))+1) * (a1 / b1))%R;
     cycle 1.
   { field. lra. }
-  set (delta2 := ((rd (rd a1 * rd (rs (1 / rs b1)))-(a1 * (1 / b1))) / (a1 * (1 / b1)))%R) in *.
+  set (delta2 := ((rd (rd a1 * rd (rs (1 / rs (rd b1))))-(a1 * (1 / b1))) / (a1 * (1 / b1)))%R) in *.
   (* assert (Rabs (delta2) <= 1/4194304)%R.
   {  unfold delta2. gappa. } *)
   replace (a1 - b1 * (delta1 + (delta2 + 1) * (a1 / b1)))%R with
-- 
cgit 


From 54dea0fc061f5daee5976450ec84ddb7e15c5db9 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 11 Feb 2022 11:12:45 +0100
Subject: remove singleoflongu (does not exist)

---
 kvx/FPDivision64.v | 65 +++++++++++++++++++++++++++++-------------------------
 1 file changed, 35 insertions(+), 30 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index a4d609f0..bbd94321 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -50,7 +50,7 @@ Proof.
 Qed.
 
 Definition approx_inv_longu b :=
-  let invb_s := ExtValues.invfs (Val.maketotal (Val.singleoflongu b)) in
+  let invb_s := ExtValues.invfs (Val.singleoffloat (Val.maketotal (Val.floatoflongu b))) in
   let invb_d := Val.floatofsingle invb_s in
   let b_d := Val.maketotal (Val.floatoflongu b) in
   let one := Vfloat (ExtFloat.one) in
@@ -68,7 +68,8 @@ Qed.
 
 Definition approx_inv_thresh := (25/2251799813685248)%R.
 (* 1.11022302462516e-14 *)
-    
+
+(*
 Theorem approx_inv_longu_correct_abs :
   forall b,
     (0 < Int64.unsigned b)%Z ->
@@ -302,6 +303,7 @@ Proof.
   unfold approx_inv_thresh.
   gappa.
 Qed.
+ *)
 
 Local Notation "'rd'" := (round radix2 (FLT_exp (-1074) 53) ZnearestE).
 Local Notation "'rs'" := (round radix2 (FLT_exp (-149) 24) ZnearestE).
@@ -320,8 +322,8 @@ Proof.
   econstructor.
   split.
   reflexivity.
-  Local Transparent Float.neg Float.of_single Float32.of_longu Float32.div Float.of_longu Float32.of_int Float.of_int.
-  unfold Float.fma, Float.neg, Float.of_single, Float32.of_longu, ExtFloat32.inv, Float32.div, Float.of_longu, ExtFloat32.one, Float32.of_int, ExtFloat.one, Float.of_int.
+  Local Transparent Float.neg Float.of_single Float32.of_longu Float32.div Float.of_longu Float32.of_int Float.of_int Float.to_single.
+  unfold Float.fma, Float.neg, Float.of_single, Float32.of_longu, ExtFloat32.inv, Float32.div, Float.of_longu, ExtFloat32.one, Float32.of_int, ExtFloat.one, Float.of_int, Float.to_single.
   set (re := (@eq_refl Datatypes.comparison Lt)).
   change (Int.signed (Int.repr 1)) with 1%Z.
   set (b' := Int64.unsigned b) in *.
@@ -336,20 +338,34 @@ Proof.
   clear SILLY.
   set (one_s := (BofZ 24 128 re re 1)) in *.
   
-  pose proof (BofZ_correct 24 128 re re b') as C1.
-  cbn in C1.
+  pose proof (BofZ_correct 53 1024 re re b') as C5.
+  rewrite Rlt_bool_true in C5; cycle 1.
+  { clear C5.
+    eapply (Rabs_relax (bpow radix2 64)).
+    { apply bpow_lt. lia. }
+    cbn.
+    gappa.
+  }
+  cbn in C5.
+  destruct C5 as (C5R & C5F & C5S).
+  set (b_d :=  (BofZ 53 1024 re re b')) in *.
+
+  pose proof (Bconv_correct 53 1024 24 128 re re Float.to_single_nan mode_NE b_d C5F) as C1.
   rewrite Rlt_bool_true in C1; cycle 1.
   { clear C1.
-    eapply (Rabs_relax (IZR 18446744073709551616)).
-    lra.
-    set (b'' := IZR b') in *.
+    apply (Rabs_relax (bpow radix2 64)).
+    { apply bpow_lt. lia. }
+    rewrite C5R.
+    cbn.
     gappa.
   }
-  destruct C1 as (C1R & C1F & _).
-  set (b_s :=  (BofZ 24 128 re re b')) in *.
-
+  cbn in C1.
+  destruct C1 as (C1R & C1F & C1S).
+  set (b_s := (Bconv 53 1024 24 128 re re Float.to_single_nan mode_NE b_d)) in *.
   assert(1 <= B2R 24 128 b_s <= 18446744073709551616)%R as b_s_RANGE.
   { rewrite C1R.
+    rewrite C5R.
+    cbn.
     gappa.
   }
   assert(B2R 24 128 b_s <> 0)%R as b_s_NONZ by lra.
@@ -413,19 +429,7 @@ Proof.
     gappa.
   }
   destruct C4 as (C4R & C4F & _).
-  
-  pose proof (BofZ_correct 53 1024 re re b') as C5.
-  cbn in C5.
-  rewrite Rlt_bool_true in C5; cycle 1.
-  { clear C5.
-    eapply (Rabs_relax (IZR 18446744073709551616)).
-    lra.
-    set (b'' := IZR b') in *.
-    gappa.
-  }
-  destruct C5 as (C5R & C5F & _).
-  set (b_d :=  (BofZ 53 1024 re re b')) in *.
-  
+    
   assert(1 <= B2R 53 1024 b_d <= 18446744073709551616)%R as b_d_RANGE.
   { rewrite C5R.
     gappa.
@@ -477,15 +481,16 @@ Proof.
   rewrite C3R.
   rewrite C2R.
   rewrite C1R.
+  rewrite C5R.
   rewrite C0R.
   cbn.
   set(b1 := IZR b') in *.
 
   replace (rd 1) with 1%R by gappa.
-  replace (rd (rs (1 / rs b1))) with
-    ((((rd (rs (1 / rs b1)) - (/b1))/(/b1))+1)*(/ b1))%R ; cycle 1.
+  replace (rd (rs (1 / rs (rd b1)))) with
+    ((((rd (rs (1 / rs (rd b1))) - (/b1))/(/b1))+1)*(/ b1))%R ; cycle 1.
   { field. lra. }
-  set (er0 := ((rd (rs (1 / rs b1)) - (/b1))/(/b1))%R).
+  set (er0 := ((rd (rs (1 / rs (rd b1))) - (/b1))/(/b1))%R).
   replace (rd b1) with ((((rd b1) - b1)/b1 + 1) * b1)%R; cycle 1.
   { field. lra. }
   set (er1 := (((rd b1) - b1)/b1)%R).
@@ -2351,8 +2356,8 @@ Open Scope cminorsel_scope.
 Definition e_invfs a := Eop Oinvfs (a ::: Enil).
 Definition e_float_of_longu a := Eop Ofloatoflongu (a ::: Enil).
 Definition e_float_of_long a := Eop Ofloatoflong (a ::: Enil).
-Definition e_single_of_longu a := Eop Osingleoflongu (a ::: Enil).
 Definition e_float_of_single a := Eop Ofloatofsingle (a ::: Enil).
+Definition e_single_of_float a := Eop Osingleoffloat (a ::: Enil).
 Definition e_long_of_float_ne a := Eop Olongoffloat_ne (a ::: Enil).
 Definition e_longu_of_float_ne a := Eop Olonguoffloat_ne (a ::: Enil).
 Definition e_mulf a b := Eop Omulf (a ::: b ::: Enil).
@@ -2378,7 +2383,7 @@ Definition binv_d_var1 := Eletvar (0%nat).
 Definition e_setup1 a b rest :=
   Elet a (Elet (e_float_of_longu (Eletvar 0%nat))
     (Elet (lift (lift b)) (Elet (e_float_of_longu (Eletvar 0%nat))
-      (Elet (e_float_of_single (e_invfs (e_single_of_longu (Eletvar 1%nat))))
+      (Elet (e_float_of_single (e_invfs (e_single_of_float (Eletvar 0%nat))))
       rest)))).                  
 Definition e_step1 := e_longu_of_float_ne (e_mulf a_d_var1 binv_d_var1).
 
-- 
cgit 


From f6e1cd79c17461d50f5119818e788d1e07451ef6 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 11 Feb 2022 17:50:40 +0100
Subject: rm useless stuff

---
 kvx/FPDivision64.v | 123 -----------------------------------------------------
 1 file changed, 123 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index bbd94321..8cab70b8 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -2053,54 +2053,6 @@ Proof.
   reflexivity.
 Qed.
 
-Definition anystep_div_longu a b m :=
-  Val.select (Val.cmplu_bool (Mem.valid_pointer m) Cgt b (Vlong (Int64.repr smallb_thresh)))
-             (step2_div_longu a b)
-             (twostep_div_longu a b) Tlong.
-
-Lemma anystep_div_longu_correct :
-  forall a b m
-           (b_RANGE : (1 < Int64.unsigned b <= Int64.max_signed)%Z),
-      anystep_div_longu (Vlong a) (Vlong b) m = Vlong (Int64.repr (Int64.unsigned a / Int64.unsigned b))%Z.
-Proof.
-  intros.
-  unfold anystep_div_longu.
-  set (a' := Int64.unsigned a).
-  set (b' := Int64.unsigned b).
-  assert (0 < b')%Z as b_NOT0 by lia.
-  cbn.
-  unfold Int64.ltu.
-  fold b'.
-  rewrite Int64.unsigned_repr; cycle 1.
-  { unfold smallb_thresh.
-    change Int64.max_unsigned with 18446744073709551615%Z.
-    lia.
-  }
-  destruct zlt.
-  { rewrite step2_div_longu_bigb_correct by lia.
-    reflexivity.
-  }
-  rewrite twostep_div_longu_smallb_correct by lia.
-  cbn.
-  rewrite Int64.eq_false ; cycle 1.
-  { intro Z. unfold b' in b_NOT0. rewrite Z in b_NOT0.
-    rewrite Int64.unsigned_zero in b_NOT0.
-    lia.
-  }
-  reflexivity.
-Qed.
-
-
-Definition full_div_longu a b m :=
-  let is_big := Val.cmpl_bool Clt b (Vlong Int64.zero) in
-  let is_one := Val.cmplu_bool (Mem.valid_pointer m) Cle b (Vlong Int64.one) in
-  let is_special := Val.or (Val.of_optbool is_big) (Val.of_optbool is_one) in
-  let if_big := Val.longofintu (Val.of_optbool (Val.cmplu_bool (Mem.valid_pointer m) Cge a b)) in
-  let if_special := Val.select is_big if_big a Tlong in
-  Val.select (Val.cmp_bool Cne is_special (Vint Int.zero))
-             if_special
-             (anystep_div_longu a b m) Tlong.
-
 Lemma one_bigb_div :
   forall a b
      (b_RANGE : (9223372036854775808 <= b < 18446744073709551616)%Z)
@@ -2117,81 +2069,6 @@ Proof.
   ring.
 Qed.
 
-Lemma full_div_longu_correct :
-  forall a b m
-           (b_NOT0 : (0 < Int64.unsigned b)%Z),
-      full_div_longu (Vlong a) (Vlong b) m = Vlong (Int64.repr (Int64.unsigned a / Int64.unsigned b))%Z.
-Proof.
-
-  Local Opaque anystep_div_longu.
-  intros.
-  unfold full_div_longu.
-  cbn.
-  unfold Int64.lt, Int64.ltu.
-  pose proof (Int64.unsigned_range a).
-  pose proof (Int64.unsigned_range b).
-  set (a' := Int64.unsigned a) in *.
-  set (b' := Int64.unsigned b) in *.
-  rewrite Int64.signed_zero.
-  rewrite Int64.unsigned_one.
-  destruct zlt.
-  { replace  (Val.cmp_bool Cne
-       (Val.or Vtrue
-          (if negb (if zlt 1 b' then true else false) then Vtrue else Vfalse))
-       (Vint Int.zero)) with (Some true) ; cycle 1.
-    { destruct zlt; reflexivity.
-    }
-    cbn.
-    destruct zlt; cbn.
-    { rewrite Zdiv_small by lia.
-      reflexivity.
-    }
-    rewrite one_bigb_div.
-    reflexivity.
-    {
-      change Int64.modulus with 18446744073709551616%Z in *.
-      split. 2: lia.
-      unfold b'.
-      rewrite Int64.unsigned_signed.
-      unfold Int64.lt.
-      rewrite Int64.signed_zero.
-      rewrite zlt_true by lia.
-      pose proof (Int64.signed_range b).
-      change Int64.min_signed with (-9223372036854775808)%Z in *.
-      change Int64.max_signed with (9223372036854775807)%Z in *.
-      change Int64.modulus with 18446744073709551616%Z in *.
-      lia.
-    }
-    change Int64.modulus with 18446744073709551616%Z in *.
-    lia.
-  }
-  destruct zlt; cbn.
-  { change (negb (Int.eq (Int.or Int.zero Int.zero) Int.zero)) with false.
-    cbn.
-    rewrite anystep_div_longu_correct.
-    reflexivity.
-    
-    change Int64.modulus with 18446744073709551616%Z in *.
-    split. lia.
-    rewrite Int64.unsigned_signed.
-    unfold Int64.lt.
-    rewrite Int64.signed_zero.
-    rewrite zlt_false by lia.
-    pose proof (Int64.signed_range b).
-    change Int64.min_signed with (-9223372036854775808)%Z in *.
-    change Int64.max_signed with (9223372036854775807)%Z in *.
-    change Int64.modulus with 18446744073709551616%Z in *.
-    lia.
-  }
-  change (negb (Int.eq (Int.or Int.zero Int.one) Int.zero)) with true.
-  cbn.
-  replace b' with 1%Z by lia.
-  rewrite Z.div_1_r.
-  unfold a'.
-  rewrite Int64.repr_unsigned.
-  reflexivity.
-Qed.
-
 Lemma repr_unsigned_sub2:
   forall a b,
     (Int64.repr (a - Int64.unsigned (Int64.repr b))) = Int64.repr (a - b).
-- 
cgit 


From df6918762b58dcda4913d7f15b61bd64673af567 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 11 Feb 2022 17:56:31 +0100
Subject: moved stuff to appropriate places and removed irrelevant content

---
 kvx/FPDivision64.v | 258 -----------------------------------------------------
 1 file changed, 258 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 8cab70b8..0d584283 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -30,25 +30,6 @@ Require Import Memory.
 
 From Gappa Require Import Gappa_tactic.
 
-
-Lemma Znearest_lub :
-  forall choice (n : Z) (x : R), (IZR n <= x)%R -> (n <= Znearest choice x)%Z.
-Proof.
-  intros until x. intro BND.
-  pose proof (Zfloor_lub n x BND).
-  pose proof (Znearest_ge_floor choice x).
-  lia.
-Qed.
-
-Lemma Znearest_glb :
-  forall choice (n : Z) (x : R), (x <= IZR n)%R -> (Znearest choice x <= n)%Z.
-Proof.
-  intros until x. intro BND.
-  pose proof (Zceil_glb n x BND).
-  pose proof (Znearest_le_ceil choice x).
-  lia.
-Qed.
-
 Definition approx_inv_longu b :=
   let invb_s := ExtValues.invfs (Val.singleoffloat (Val.maketotal (Val.floatoflongu b))) in
   let invb_d := Val.floatofsingle invb_s in
@@ -66,245 +47,6 @@ Proof.
   lra.
 Qed.
 
-Definition approx_inv_thresh := (25/2251799813685248)%R.
-(* 1.11022302462516e-14 *)
-
-(*
-Theorem approx_inv_longu_correct_abs :
-  forall b,
-    (0 < Int64.unsigned b)%Z ->
-    exists (f : float),
-      (approx_inv_longu (Vlong b)) = Vfloat f /\
-      is_finite _ _ f = true /\ (Rabs((B2R _ _ f) - (1 / IZR (Int64.unsigned b))) <= approx_inv_thresh)%R.
-Proof.
-  intros b NONZ.
-  unfold approx_inv_longu.
-  cbn.
-  econstructor.
-  split.
-  reflexivity.
-  Local Transparent Float.neg Float.of_single Float32.of_longu Float32.div Float.of_longu Float32.of_int Float.of_int.
-  unfold Float.fma, Float.neg, Float.of_single, Float32.of_longu, ExtFloat32.inv, Float32.div, Float.of_longu, ExtFloat32.one, Float32.of_int, ExtFloat.one, Float.of_int.
-  set (re := (@eq_refl Datatypes.comparison Lt)).
-  change (Int.signed (Int.repr 1)) with 1%Z.
-  set (b' := Int64.unsigned b) in *.
-  pose proof (Int64.unsigned_range b) as RANGE.
-  change Int64.modulus with 18446744073709551616%Z in RANGE.                                              
-  assert(1 <= IZR b' <= 18446744073709551616)%R as RANGE'.
-  { split; apply IZR_le; lia.
-  }
-
-  assert (-16777216 <= 1 <= 16777216)%Z as SILLY by lia.
-  destruct (BofZ_exact 24 128 re re 1 SILLY) as (C0R & C0F & _).
-  clear SILLY.
-  set (one_s := (BofZ 24 128 re re 1)) in *.
-  
-  pose proof (BofZ_correct 24 128 re re b') as C1.
-  cbn in C1.
-  rewrite Rlt_bool_true in C1; cycle 1.
-  { clear C1.
-    eapply (Rabs_relax (IZR 18446744073709551616)).
-    lra.
-    set (b'' := IZR b') in *.
-    gappa.
-  }
-  destruct C1 as (C1R & C1F & _).
-  set (b_s :=  (BofZ 24 128 re re b')) in *.
-
-  assert(1 <= B2R 24 128 b_s <= 18446744073709551616)%R as b_s_RANGE.
-  { rewrite C1R.
-    gappa.
-  }
-  assert(B2R 24 128 b_s <> 0)%R as b_s_NONZ by lra.
-  
-  pose proof (Bdiv_correct 24 128 re re Float32.binop_nan mode_NE one_s b_s b_s_NONZ) as C2.
-  rewrite Rlt_bool_true in C2; cycle 1.
-  { clear C2.
-    apply Rabs_relax with (b := 1%R).
-    { cbn; lra. }
-    rewrite C0R.
-    set (r_b_s := B2R 24 128 b_s) in *.
-    cbn.
-    gappa.
-  }
-  
-  destruct C2 as (C2R & C2F & _).
-  set (invb_s := (Bdiv 24 128 re re Float32.binop_nan mode_NE one_s b_s)) in *.
-  rewrite C0F in C2F.
-
-  assert ((1/18446744073709551616 <= B2R 24 128 invb_s <= 1)%R) as invb_s_RANGE.
-  { rewrite C2R.
-    set (r_b_s := B2R 24 128 b_s) in *.
-    rewrite C0R.
-    cbn.
-    gappa.
-  }
-  
-  pose proof (Bconv_correct 24 128 53 1024 re re Float.of_single_nan mode_NE invb_s C2F) as C3.
-  rewrite Rlt_bool_true in C3; cycle 1.
-  { clear C3.
-    set (r_invb_s := (B2R 24 128 invb_s)) in *.
-    apply Rabs_relax with (b := 1%R).
-    { replace 1%R with (bpow radix2 0)%R by reflexivity.
-      apply bpow_lt.
-      lia.
-    }
-    cbn.
-    gappa.
-  }
-  
-  destruct C3 as (C3R & C3F & _).
-  set (invb_d :=  (Bconv 24 128 53 1024 re re Float.of_single_nan mode_NE invb_s)) in *.
-  assert ((1/18446744073709551616 <= B2R 53 1024 invb_d <= 1)%R) as invb_d_RANGE.
-  { 
-    rewrite C3R.
-    set (r_invb_s := B2R 24 128 invb_s) in *.
-    cbn.
-    gappa.
-  }
-
-  pose proof (is_finite_Bopp 53 1024 Float.neg_nan invb_d) as opp_finite.
-  rewrite C3F in opp_finite.
-
-  pose proof (BofZ_correct 53 1024 re re 1) as C4.
-  rewrite Rlt_bool_true in C4; cycle 1.
-  { clear C4.
-    cbn.
-    eapply (Rabs_relax (IZR 18446744073709551616)).
-    lra.
-    set (b'' := IZR b') in *.
-    gappa.
-  }
-  destruct C4 as (C4R & C4F & _).
-  
-  pose proof (BofZ_correct 53 1024 re re b') as C5.
-  cbn in C5.
-  rewrite Rlt_bool_true in C5; cycle 1.
-  { clear C5.
-    eapply (Rabs_relax (IZR 18446744073709551616)).
-    lra.
-    set (b'' := IZR b') in *.
-    gappa.
-  }
-  destruct C5 as (C5R & C5F & _).
-  set (b_d :=  (BofZ 53 1024 re re b')) in *.
-  
-  assert(1 <= B2R 53 1024 b_d <= 18446744073709551616)%R as b_d_RANGE.
-  { rewrite C5R.
-    gappa.
-  }
-
-  pose proof (Bfma_correct 53 1024 re re Float.fma_nan mode_NE
-          (Bopp 53 1024 Float.neg_nan invb_d) (BofZ 53 1024 re re b')
-          (BofZ 53 1024 re re 1) opp_finite C5F C4F) as C6.
-  rewrite Rlt_bool_true in C6; cycle 1.
-  { clear C6.
-    rewrite C4R.
-    rewrite B2R_Bopp.
-    cbn.
-    eapply (Rabs_relax (IZR 18446744073709551616)).
-    { lra. }
-    fold invb_d.
-    fold b_d.
-    set (r_invb_d := B2R 53 1024 invb_d) in *.
-    set (r_b_d := B2R 53 1024 b_d) in *.
-    gappa.
-  }
-  fold b_d in C6.
-  destruct C6 as (C6R & C6F & _).
-
-  pose proof (Bfma_correct 53 1024 re re Float.fma_nan mode_NE
-       (Bfma 53 1024 re re Float.fma_nan mode_NE
-          (Bopp 53 1024 Float.neg_nan invb_d) b_d (BofZ 53 1024 re re 1))
-       invb_d invb_d C6F C3F C3F) as C7.
-  rewrite Rlt_bool_true in C7; cycle 1.
-  { clear C7.
-    rewrite C6R.
-    rewrite B2R_Bopp.
-    eapply (Rabs_relax (bpow radix2 64)).
-    { apply bpow_lt. lia. }
-    rewrite C4R.
-    cbn.
-    set (r_invb_d := B2R 53 1024 invb_d) in *.
-    set (r_b_d := B2R 53 1024 b_d) in *.
-    gappa.
-  }
-  destruct C7 as (C7R & C7F & _).
-
-  split. assumption.
-  rewrite C7R.
-  rewrite C6R.
-  rewrite C5R.
-  rewrite C4R.
-  rewrite B2R_Bopp.
-  rewrite C3R.
-  rewrite C2R.
-  rewrite C1R.
-  rewrite C0R.
-  cbn.
-  set(b1 := IZR b') in *.
-  replace (round radix2 (FLT_exp (-1074) 53) ZnearestE 1) with 1%R by gappa.
-  set (bd := round radix2 (FLT_exp (-1074) 53) ZnearestE b1).
-  set (x0 := round radix2 (FLT_exp (-1074) 53) ZnearestE
-         (round radix2 (FLT_exp (-149) 24) ZnearestE
-                    (1 / round radix2 (FLT_exp (-149) 24) ZnearestE b1))).
-  set (alpha0 := (- x0 * bd + 1)%R).
-  set (y1 := (round radix2 (FLT_exp (-1074) 53) ZnearestE alpha0 * x0 + x0)%R).
-  set (x1 := round radix2 (FLT_exp (-1074) 53) ZnearestE y1).
-  replace (x1 - 1/b1)%R with ((y1-1/b1)+(x1-y1))%R by ring.
-
-  assert(alpha0 = -((x0-1/bd)/(1/bd)))%R as alpha0_EQ.
-  { unfold alpha0.
-    field.
-    unfold bd.
-    gappa.
-  }
-  assert(y1-1/b1 = ((round radix2 (FLT_exp (-1074) 53) ZnearestE alpha0)
-                   - alpha0) * x0
-                   + alpha0*(x0-1/b1) - ((bd-b1)/b1) * x0)%R as y1_EQ.
-  { unfold y1, alpha0.
-    field.
-    lra.
-  }
-  assert(Rabs alpha0 <= 257/2147483648)%R as alpha0_ABS.
-  { rewrite alpha0_EQ.
-    unfold x0, bd.
-    gappa.
-  }
-  assert (Rabs (x0 - 1 / b1) <= 3/33554432)%R as x0_delta_ABS.
-  { unfold x0.
-    gappa.
-  }
-  set (x0_delta := (x0 - 1 / b1)%R) in *.
-  assert (Rabs ((bd - b1) / b1) <= 1/9007199254740992)%R as bd_delta_ABS.
-  { unfold bd.
-    gappa.
-  }
-  set (bd_delta := ((bd - b1) / b1)%R) in *.
-  set (rnd_alpha0_delta := (round radix2 (FLT_exp (-1074) 53) ZnearestE alpha0 - alpha0)%R) in *.
-  assert (Rabs rnd_alpha0_delta <= 1/75557863725914323419136)%R as rnd_alpha0_delta_ABS.
-  { unfold rnd_alpha0_delta.
-    gappa.
-  }
-  assert (1/18446744073709551616 <= x0 <= 1)%R as x0_RANGE.
-  { unfold x0.
-    gappa.
-  }
-  assert (Rabs (y1 - 1 / b1) <= 49/4503599627370496)%R as y1_delta_ABS.
-  { rewrite y1_EQ.
-    gappa.
-  }
-  assert (Rabs(x1 - y1) <= 1/9007199254740992)%R as x1_delta_ABS.
-  { unfold x1.
-    gappa.
-  }
-  set (y1_delta := (y1 - 1 / b1)%R) in *.
-  set (x1_delta := (x1-y1)%R) in *.
-  unfold approx_inv_thresh.
-  gappa.
-Qed.
- *)
-
 Local Notation "'rd'" := (round radix2 (FLT_exp (-1074) 53) ZnearestE).
 Local Notation "'rs'" := (round radix2 (FLT_exp (-149) 24) ZnearestE).
 
-- 
cgit 


From 884cbaab11e294c9c5224b2fc98e68074d6a1584 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 11 Feb 2022 20:42:32 +0100
Subject: twostep_mod_longu_mostb_correct

---
 kvx/FPDivision64.v | 94 ++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 94 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 0d584283..73dba8eb 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -1971,6 +1971,100 @@ Proof.
   reflexivity.
 Qed.
 
+Definition step2_mod_long a b :=
+  let q := step2_div_long' a b in
+  let r := Val.subl a (Val.mull q b) in
+  Val.select (Val.cmpl_bool Clt r (Vlong Int64.zero))
+             (Val.addl r b) r Tlong.
+
+Definition twostep_mod_longu a b :=
+  let q1 := step1_div_longu a b in
+  step2_mod_long (Val.subl a (Val.mull b q1)) b.
+
+Lemma vlong_eq: forall a b, (Vlong a) = (Vlong b) -> a = b.
+Proof.
+  intros a b EQ.
+  congruence.
+Qed.
+
+Lemma move_repr_in_mod :
+  forall a b c,
+    Int64.repr (a - b * c)%Z =
+      Int64.repr (a - b * Int64.unsigned (Int64.repr c))%Z.
+Proof.
+  intros.
+  apply Int64.eqm_samerepr.
+  auto 10 with ints.
+Qed.
+
+Lemma twostep_mod_longu_mostb_correct :
+    forall a b
+      (b_RANGE : (1 < Int64.unsigned b <= Int64.max_signed)%Z),
+          (twostep_mod_longu (Vlong a) (Vlong b)) =
+            (Val.maketotal (Val.modlu (Vlong a) (Vlong b))).
+Proof.
+  intros.
+  Local Transparent twostep_div_longu.
+  pose proof (twostep_div_longu_mostb_correct a b b_RANGE) as div_correct.
+  unfold twostep_div_longu, twostep_mod_longu, step2_div_long, step2_mod_long in *.
+  set (q1 := (step1_div_longu (Vlong a) (Vlong b))) in *.
+  set (q2 := (step2_div_long' (Val.subl (Vlong a) (Val.mull (Vlong b) q1)) (Vlong b))) in *.
+  cbn in div_correct.
+  cbn.
+  unfold Int64.eq in *.
+  change (Int64.unsigned Int64.zero) with 0%Z in *.
+  rewrite zeq_false by lia.
+  rewrite zeq_false in div_correct by lia.
+  cbn in div_correct.
+  cbn.
+  destruct q1 as [  |  | q1l |  |  |  ] ; cbn in *; try discriminate.
+  destruct q2 as [  |  | q2l |  |  |  ] ; cbn in *; try discriminate.
+  rewrite <- (function_ite Vlong).
+  rewrite <- (function_ite Vlong) in div_correct.
+  cbn. cbn in div_correct.
+  unfold Int64.lt, Int64.sub, Int64.mul, Int64.add, Int64.divu, Int64.modu in *.
+  set (a' := Int64.unsigned a) in *.
+  set (b' := Int64.unsigned b) in *.
+  set (q1' := Int64.unsigned q1l) in *.
+  set (q2' := Int64.unsigned q2l) in *.
+  change (Int64.signed Int64.zero) with 0%Z in *.
+  replace (Int64.repr
+               (Int64.unsigned
+                  (Int64.repr (a' - Int64.unsigned (Int64.repr (b' * q1')))) -
+                  Int64.unsigned (Int64.repr (q2' * b'))))
+    with (Int64.repr (a' - (b' * q1') - (q2' * b')))%Z in * ; cycle 1.
+  {
+    apply Int64.eqm_samerepr.
+    auto 10 with ints.
+  }
+  replace (a' - b' * q1' - q2' * b')%Z with (a' - b' * (q1' + q2'))%Z in * by ring.
+  f_equal.
+  apply vlong_eq in div_correct.
+  rewrite Z.mod_eq by lia.
+  rewrite (move_repr_in_mod a' b' (a' / b'))%Z.
+  rewrite <- div_correct.
+  clear div_correct.
+  rewrite <- (move_repr_in_mod a' b')%Z.
+ 
+  destruct zlt as [NEG | POS].
+  2: reflexivity.
+  rewrite repr_unsigned_add.
+  replace (a' - b' * (q1' + q2') + b')%Z with (a' - b' * (q1' + q2' - 1))%Z by ring.
+  apply Int64.eqm_samerepr.
+  assert (Int64.eqm (Int64.unsigned (Int64.repr (q2' + Int64.unsigned Int64.mone)))
+                    (q2' -1))%Z as EQM.
+  { apply Int64.eqm_trans with (y :=  (q2' + Int64.unsigned Int64.mone)%Z).
+    apply Int64.eqm_sym.
+    apply Int64.eqm_unsigned_repr.
+    apply Int64.eqm_add.
+    apply Int64.eqm_refl.
+    exists (1)%Z.
+    reflexivity.
+  }
+  replace  (q1' + q2' - 1)%Z with (q1' + (q2' - 1))%Z by ring.
+  auto with ints.
+Qed.
+    
 Open Scope cminorsel_scope.
 Definition e_invfs a := Eop Oinvfs (a ::: Enil).
 Definition e_float_of_longu a := Eop Ofloatoflongu (a ::: Enil).
-- 
cgit 


From 87bbbd23d6681bfdc29ac5f8bd24b28ba89da4aa Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 11 Feb 2022 21:01:17 +0100
Subject: full2 mod correct

---
 kvx/FPDivision64.v | 98 ++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 98 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 73dba8eb..56170419 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -2065,6 +2065,104 @@ Proof.
   auto with ints.
 Qed.
     
+Definition full2_mod_longu a b m :=
+  let is_big := Val.cmpl_bool Clt b (Vlong Int64.zero) in
+  let is_one := Val.cmplu_bool (Mem.valid_pointer m) Cle b (Vlong Int64.one) in
+  let is_special := Val.or (Val.of_optbool is_big) (Val.of_optbool is_one) in
+  let if_big := Val.select (Val.cmplu_bool (Mem.valid_pointer m) Cge a b) (Val.subl a b) a Tlong in
+  let if_special := Val.select is_big if_big (Vlong Int64.zero) Tlong in
+  Val.select (Val.cmp_bool Cne is_special (Vint Int.zero))
+             if_special
+             (twostep_mod_longu a b) Tlong.
+
+Lemma full2_mod_longu_correct :
+  forall a b m
+           (b_NOT0 : (0 < Int64.unsigned b)%Z),
+      full2_mod_longu (Vlong a) (Vlong b) m = Vlong (Int64.repr ((Int64.unsigned a) mod (Int64.unsigned b)))%Z.
+Proof.
+
+  Local Opaque twostep_mod_longu.
+  intros.
+  unfold full2_mod_longu.
+  cbn.
+  unfold Int64.lt, Int64.ltu.
+  pose proof (Int64.unsigned_range a).
+  pose proof (Int64.unsigned_range b).
+  set (a' := Int64.unsigned a) in *.
+  set (b' := Int64.unsigned b) in *.
+  rewrite Int64.signed_zero.
+  rewrite Int64.unsigned_one.
+  destruct zlt.
+  { replace  (Val.cmp_bool Cne
+       (Val.or Vtrue
+          (if negb (if zlt 1 b' then true else false) then Vtrue else Vfalse))
+       (Vint Int.zero)) with (Some true) ; cycle 1.
+    { destruct zlt; reflexivity.
+    }
+    cbn.
+    rewrite Z.mod_eq by lia.
+    
+    destruct zlt; cbn.
+    { rewrite Zdiv_small by lia.
+      replace (a' - b' * 0)%Z with a' by ring.
+      unfold a'.
+      rewrite Int64.repr_unsigned.
+      reflexivity.
+    }
+    rewrite one_bigb_div.
+    { unfold Int64.sub.
+      fold a' b'.
+      repeat f_equal. ring.
+    }
+    {
+      change Int64.modulus with 18446744073709551616%Z in *.
+      split. 2: lia.
+      unfold b'.
+      rewrite Int64.unsigned_signed.
+      unfold Int64.lt.
+      rewrite Int64.signed_zero.
+      rewrite zlt_true by lia.
+      pose proof (Int64.signed_range b).
+      change Int64.min_signed with (-9223372036854775808)%Z in *.
+      change Int64.max_signed with (9223372036854775807)%Z in *.
+      change Int64.modulus with 18446744073709551616%Z in *.
+      lia.
+    }
+    change Int64.modulus with 18446744073709551616%Z in *.
+    lia.
+  }
+  destruct zlt; cbn.
+  { change (negb (Int.eq (Int.or Int.zero Int.zero) Int.zero)) with false.
+    cbn.
+    rewrite twostep_mod_longu_mostb_correct.
+    {
+      cbn.
+      unfold Int64.eq.
+      fold b'.
+      rewrite Int64.unsigned_zero.
+      rewrite zeq_false by lia.
+      reflexivity.
+    }
+    
+    change Int64.modulus with 18446744073709551616%Z in *.
+    split. lia.
+    rewrite Int64.unsigned_signed.
+    unfold Int64.lt.
+    rewrite Int64.signed_zero.
+    rewrite zlt_false by lia.
+    pose proof (Int64.signed_range b).
+    change Int64.min_signed with (-9223372036854775808)%Z in *.
+    change Int64.max_signed with (9223372036854775807)%Z in *.
+    change Int64.modulus with 18446744073709551616%Z in *.
+    lia.
+  }
+  change (negb (Int.eq (Int.or Int.zero Int.one) Int.zero)) with true.
+  cbn.
+  replace b' with 1%Z by lia.
+  rewrite Z.mod_1_r.
+  reflexivity.
+Qed.
+
 Open Scope cminorsel_scope.
 Definition e_invfs a := Eop Oinvfs (a ::: Enil).
 Definition e_float_of_longu a := Eop Ofloatoflongu (a ::: Enil).
-- 
cgit 


From f651db91393a3df1b3c65e4249423a5c744761e5 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Fri, 11 Feb 2022 21:19:55 +0100
Subject: fp_modu64_correct

---
 kvx/FPDivision64.v | 30 ++++++++++++++++++++++++++++++
 1 file changed, 30 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 56170419..1d0fd2ee 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -2376,3 +2376,33 @@ Proof.
   rewrite <- full2_div_longu_correct with (m := memenv) by lia.
   apply e_step7_correct; assumption.
 Qed.
+
+Definition fp_modu64 a b := Elet a (Elet (lift b) (e_msubl (Eletvar 1%nat) (Eletvar 0%nat)
+                                                    (fp_divu64 (Eletvar 1%nat) (Eletvar 0%nat)))).
+
+Theorem fp_modu64_correct :
+  forall (ge : genv)  (sp: val) cmenv memenv
+         (le : letenv) (expr_a expr_b : expr) (a b : int64)
+         (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a))
+         (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b))
+         (b_nz : (Int64.unsigned b > 0)%Z),
+  eval_expr ge sp cmenv memenv le (fp_modu64 expr_a expr_b)
+            (Vlong (Int64.modu a b)).
+Proof.
+  intros.
+  rewrite Int64.modu_divu; cycle 1.
+  { intro Z.
+    subst.
+    rewrite Int64.unsigned_zero in b_nz.
+    lia.
+  }
+  unfold fp_modu64.
+  Local Opaque fp_divu64.
+  repeat (econstructor + apply eval_lift + eassumption).
+  { apply fp_divu64_correct;
+    repeat (econstructor + apply eval_lift + eassumption).
+  }
+  cbn.
+  rewrite Int64.mul_commut.
+  reflexivity.
+Qed.
-- 
cgit 


From 191af63b4e3db42f203a5a2a62c5f924d7f37d81 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Tue, 15 Feb 2022 20:01:24 +0100
Subject: mods 64

---
 kvx/FPDivision64.v | 235 +++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 235 insertions(+)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 1d0fd2ee..113bf07a 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -2406,3 +2406,238 @@ Proof.
   rewrite Int64.mul_commut.
   reflexivity.
 Qed.
+
+Definition e_is_negl a := Eop (Ocmp (Ccomplimm Clt Int64.zero)) (a ::: Enil).
+Definition e_xorw a b := Eop Oxor (a ::: b ::: Enil).
+Definition e_negl a := Eop Onegl (a ::: Enil).
+Definition e_absl a := Eop (Oabsdifflimm Int64.zero) (a ::: Enil).
+
+Definition fp_divs64 a b :=
+  Elet a (Elet (lift b)
+    (Elet (fp_divu64 (e_absl (Eletvar (1%nat))) (e_absl (Eletvar (0%nat))))
+          (e_ite Tlong (Ccompu0 Cne) (e_xorw (e_is_negl (Eletvar 2%nat))
+                                            (e_is_negl (Eletvar 1%nat)))
+                 (e_negl (Eletvar 0%nat)) (Eletvar 0%nat)))).
+
+Lemma nonneg_signed_unsigned:
+  forall x (x_NONNEG : (Int64.signed x >= 0)%Z),
+    (Int64.signed x) = (Int64.unsigned x).
+Proof.
+  intros.
+  pose proof (Int64.unsigned_range x).
+  unfold Int64.signed in *.
+  destruct zlt. reflexivity.
+  change Int64.modulus with 18446744073709551616%Z in *.
+  change Int64.half_modulus with 9223372036854775808%Z in *.
+  lia.
+Qed.
+
+Lemma long_min_signed_unsigned :
+  (- Int64.min_signed < Int64.max_unsigned)%Z.
+Proof.
+  reflexivity.
+Qed.
+
+Lemma long_divs_divu :
+  forall a b
+    (b_NOT0 : (Int64.signed b <> 0)%Z),
+    Int64.divs a b = if xorb (Int64.lt a Int64.zero)
+                           (Int64.lt b Int64.zero)
+                   then Int64.neg (Int64.divu (ExtValues.long_abs a)
+                                          (ExtValues.long_abs b))
+                   else Int64.divu (ExtValues.long_abs a) (ExtValues.long_abs b).
+Proof.
+  intros.
+  unfold Int64.divs, Int64.divu, Int64.lt, ExtValues.long_abs.
+  pose proof (Int64.signed_range a) as a_RANGE.
+  pose proof (Int64.signed_range b) as b_RANGE.
+  change (Int64.signed Int64.zero) with 0%Z.
+  destruct zlt.
+  - cbn. rewrite (Z.abs_neq (Int64.signed a)) by lia.
+    rewrite (Int64.unsigned_repr (- Int64.signed a)); cycle 1.
+    { pose proof long_min_signed_unsigned. lia. }
+
+    destruct zlt.
+    + rewrite (Z.abs_neq (Int64.signed b)) by lia.
+      rewrite Int64.unsigned_repr ; cycle 1.
+      { pose proof long_min_signed_unsigned. lia. }
+      rewrite <- (Z.opp_involutive (Int64.signed b)) at 1.
+      rewrite Z.quot_opp_r by lia.
+      rewrite <- (Z.opp_involutive (Int64.signed a)) at 1.
+      rewrite Z.quot_opp_l by lia.
+      rewrite Z.quot_div_nonneg by lia.
+      rewrite Z.opp_involutive.
+      reflexivity.
+      
+    + rewrite (Z.abs_eq (Int64.signed b)) by lia.
+      rewrite Int64.unsigned_repr ; cycle 1.
+      { pose proof Int64.max_signed_unsigned. lia. }
+      rewrite <- (Z.opp_involutive (Int64.signed a)) at 1.
+      rewrite Z.quot_opp_l by lia.
+      rewrite Z.quot_div_nonneg by lia.
+      rewrite Int64.neg_repr.
+      reflexivity.
+    
+  - cbn. rewrite (Z.abs_eq (Int64.signed a)) by lia.
+    rewrite (Int64.unsigned_repr (Int64.signed a)); cycle 1.
+    { pose proof Int64.max_signed_unsigned. lia. }
+    destruct zlt.
+    + rewrite (Z.abs_neq (Int64.signed b)) by lia.
+      rewrite Int64.unsigned_repr ; cycle 1.
+      { pose proof long_min_signed_unsigned. lia. }
+      rewrite Int64.neg_repr.
+      rewrite <- (Z.opp_involutive (Int64.signed b)) at 1.
+      rewrite Z.quot_opp_r by lia.
+      rewrite Z.quot_div_nonneg by lia.
+      reflexivity.
+      
+    + rewrite (Z.abs_eq (Int64.signed b)) by lia.
+      rewrite Int64.unsigned_repr ; cycle 1.
+      { pose proof Int64.max_signed_unsigned. lia. }
+      rewrite Z.quot_div_nonneg by lia.
+      reflexivity.
+Qed.
+
+Lemma nonzero_unsigned_signed :
+  forall b, (Int64.unsigned b > 0 -> Int64.signed b <> 0)%Z.
+Proof.
+  intros b GT EQ.
+  rewrite Int64.unsigned_signed in GT.
+  unfold Int64.lt in GT.
+  rewrite Int64.signed_zero in GT.
+  destruct zlt in GT; lia.
+Qed.
+
+Theorem fp_divs64_correct :
+  forall (ge : genv)  (sp: val) cmenv memenv
+         (le : letenv) (expr_a expr_b : expr) (a b : int64)
+         (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a))
+         (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b))
+         (b_nz : (Int64.unsigned b > 0)%Z),
+  eval_expr ge sp cmenv memenv le (fp_divs64 expr_a expr_b)
+            (Vlong (Int64.divs a b)).
+Proof.
+  intros.
+  unfold fp_divs64.
+  Local Opaque fp_divu64.
+  repeat (econstructor + apply eval_lift + eassumption).
+  apply fp_divu64_correct.
+  all: repeat (econstructor + apply eval_lift + eassumption).
+  { unfold ExtValues.long_absdiff, ExtValues.Z_abs_diff.
+    rewrite Int64.signed_zero. rewrite Z.sub_0_r.
+    rewrite Int64.unsigned_repr.
+    { pose proof (nonzero_unsigned_signed b b_nz).
+      lia.
+    }
+    pose proof Int64.max_signed_unsigned.
+    pose proof long_min_signed_unsigned.
+    pose proof (Int64.signed_range b).
+    lia.
+  }
+  cbn.
+  rewrite long_divs_divu ; cycle 1.
+  { apply nonzero_unsigned_signed. assumption. }
+  unfold Int64.lt, ExtValues.long_abs, ExtValues.long_absdiff, ExtValues.Z_abs_diff.
+  change (Int64.signed Int64.zero) with 0%Z.
+  repeat rewrite Z.sub_0_r.
+  destruct zlt; destruct zlt; reflexivity.
+Qed.
+
+Lemma long_mods_modu :
+  forall a b
+    (b_NOT0 : (Int64.signed b <> 0)%Z),
+    Int64.mods a b = if Int64.lt a Int64.zero
+                   then Int64.neg (Int64.modu (ExtValues.long_abs a)
+                                          (ExtValues.long_abs b))
+                   else Int64.modu (ExtValues.long_abs a) (ExtValues.long_abs b).
+Proof.
+  intros.
+  unfold Int64.mods, Int64.modu, Int64.lt, ExtValues.long_abs.
+  pose proof (Int64.signed_range a) as a_RANGE.
+  pose proof (Int64.signed_range b) as b_RANGE.
+  change (Int64.signed Int64.zero) with 0%Z.
+  destruct zlt.
+  - cbn. rewrite (Z.abs_neq (Int64.signed a)) by lia.
+    rewrite (Int64.unsigned_repr (- Int64.signed a)); cycle 1.
+    { pose proof long_min_signed_unsigned. lia. }
+
+    destruct (zlt (Int64.signed b) 0%Z).
+    + rewrite (Z.abs_neq (Int64.signed b)) by lia.
+      rewrite Int64.unsigned_repr ; cycle 1.
+      { pose proof long_min_signed_unsigned. lia. }
+      rewrite <- (Z.opp_involutive (Int64.signed b)) at 1.
+      rewrite Z.rem_opp_r by lia.
+      rewrite <- (Z.opp_involutive (Int64.signed a)) at 1.
+      rewrite Z.rem_opp_l by lia.
+      rewrite Z.rem_mod_nonneg by lia.
+      rewrite Int64.neg_repr.
+      reflexivity.
+      
+    + rewrite (Z.abs_eq (Int64.signed b)) by lia.
+      rewrite Int64.unsigned_repr ; cycle 1.
+      { pose proof Int64.max_signed_unsigned. lia. }
+      rewrite <- (Z.opp_involutive (Int64.signed a)) at 1.
+      rewrite Z.rem_opp_l by lia.
+      rewrite Z.rem_mod_nonneg by lia.
+      rewrite Int64.neg_repr.
+      reflexivity.
+    
+  - cbn. rewrite (Z.abs_eq (Int64.signed a)) by lia.
+    rewrite (Int64.unsigned_repr (Int64.signed a)); cycle 1.
+    { pose proof Int64.max_signed_unsigned. lia. }
+    destruct (zlt (Int64.signed b) 0%Z).
+    + rewrite (Z.abs_neq (Int64.signed b)) by lia.
+      rewrite Int64.unsigned_repr ; cycle 1.
+      { pose proof long_min_signed_unsigned. lia. }
+      rewrite <- (Z.opp_involutive (Int64.signed b)) at 1.
+      rewrite Z.rem_opp_r by lia.
+      rewrite Z.rem_mod_nonneg by lia.
+      reflexivity.
+      
+    + rewrite (Z.abs_eq (Int64.signed b)) by lia.
+      rewrite Int64.unsigned_repr ; cycle 1.
+      { pose proof Int64.max_signed_unsigned. lia. }
+      rewrite Z.rem_mod_nonneg by lia.
+      reflexivity.
+Qed.
+
+Definition fp_mods64 a b :=
+  Elet a (Elet (lift b)
+    (Elet (fp_modu64 (e_absl (Eletvar (1%nat))) (e_absl (Eletvar (0%nat))))
+          (e_ite Tlong (Ccompl0 Clt) (Eletvar 2%nat)
+                 (e_negl (Eletvar 0%nat)) (Eletvar 0%nat)))).
+
+Theorem fp_mods64_correct :
+  forall (ge : genv)  (sp: val) cmenv memenv
+         (le : letenv) (expr_a expr_b : expr) (a b : int64)
+         (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a))
+         (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b))
+         (b_nz : (Int64.unsigned b > 0)%Z),
+  eval_expr ge sp cmenv memenv le (fp_mods64 expr_a expr_b)
+            (Vlong (Int64.mods a b)).
+Proof.
+  intros.
+  unfold fp_mods64.
+  Local Opaque fp_modu64.
+  repeat (econstructor + apply eval_lift + eassumption).
+  apply fp_modu64_correct.
+  all: repeat (econstructor + apply eval_lift + eassumption).
+  { unfold ExtValues.long_absdiff, ExtValues.Z_abs_diff.
+    rewrite Int64.signed_zero. rewrite Z.sub_0_r.
+    rewrite Int64.unsigned_repr.
+    { pose proof (nonzero_unsigned_signed b b_nz).
+      lia.
+    }
+    pose proof Int64.max_signed_unsigned.
+    pose proof long_min_signed_unsigned.
+    pose proof (Int64.signed_range b).
+    lia.
+  }
+  cbn.
+  rewrite long_mods_modu ; cycle 1.
+  { apply nonzero_unsigned_signed. assumption. }
+  unfold Int64.lt, ExtValues.long_abs, ExtValues.long_absdiff, ExtValues.Z_abs_diff.
+  change (Int64.signed Int64.zero) with 0%Z.
+  repeat rewrite Z.sub_0_r.
+  destruct zlt; reflexivity. 
+Qed.
-- 
cgit 


From 867b74d57fc2b834cc19176b650dc62a1e4e0fd2 Mon Sep 17 00:00:00 2001
From: David Monniaux <David.Monniaux@univ-grenoble-alpes.fr>
Date: Wed, 16 Feb 2022 09:14:01 +0100
Subject: simpler mod

---
 kvx/FPDivision64.v | 35 +++++++++++++++++++++++++++++++----
 1 file changed, 31 insertions(+), 4 deletions(-)

(limited to 'kvx/FPDivision64.v')

diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v
index 113bf07a..298831a0 100644
--- a/kvx/FPDivision64.v
+++ b/kvx/FPDivision64.v
@@ -2601,23 +2601,23 @@ Proof.
       reflexivity.
 Qed.
 
-Definition fp_mods64 a b :=
+Definition fp_mods64z a b :=
   Elet a (Elet (lift b)
     (Elet (fp_modu64 (e_absl (Eletvar (1%nat))) (e_absl (Eletvar (0%nat))))
           (e_ite Tlong (Ccompl0 Clt) (Eletvar 2%nat)
                  (e_negl (Eletvar 0%nat)) (Eletvar 0%nat)))).
 
-Theorem fp_mods64_correct :
+Theorem fp_mods64z_correct :
   forall (ge : genv)  (sp: val) cmenv memenv
          (le : letenv) (expr_a expr_b : expr) (a b : int64)
          (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a))
          (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b))
          (b_nz : (Int64.unsigned b > 0)%Z),
-  eval_expr ge sp cmenv memenv le (fp_mods64 expr_a expr_b)
+  eval_expr ge sp cmenv memenv le (fp_mods64z expr_a expr_b)
             (Vlong (Int64.mods a b)).
 Proof.
   intros.
-  unfold fp_mods64.
+  unfold fp_mods64z.
   Local Opaque fp_modu64.
   repeat (econstructor + apply eval_lift + eassumption).
   apply fp_modu64_correct.
@@ -2641,3 +2641,30 @@ Proof.
   repeat rewrite Z.sub_0_r.
   destruct zlt; reflexivity. 
 Qed.
+
+Definition fp_mods64 a b :=
+  Elet a (Elet (lift b)
+    (Elet (fp_divs64 (Eletvar (1%nat)) (Eletvar (0%nat)))
+          (e_msubl (Eletvar 2%nat) (Eletvar 1%nat) (Eletvar 0%nat)))).
+
+Theorem fp_mods64_correct :
+  forall (ge : genv)  (sp: val) cmenv memenv
+         (le : letenv) (expr_a expr_b : expr) (a b : int64)
+         (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a))
+         (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b))
+         (b_nz : (Int64.unsigned b > 0)%Z),
+  eval_expr ge sp cmenv memenv le (fp_mods64 expr_a expr_b)
+            (Vlong (Int64.mods a b)).
+Proof.
+  intros.
+  rewrite Int64.mods_divs.
+  unfold fp_mods64.
+  Local Opaque fp_divs64.
+  repeat (econstructor + apply eval_lift + eassumption).
+  { apply fp_divs64_correct;
+    repeat (econstructor + apply eval_lift + eassumption).
+  }
+  cbn.
+  rewrite Int64.mul_commut.
+  reflexivity.
+Qed.
-- 
cgit