5 files changed, 114 insertions, 131 deletions

diff --git a/kernel/byterun/coq_uint63_native.h b/kernel/byterun/coq_uint63_native.h
index 1fdafc9d8f..9fbd3f83d8 100644
--- a/kernel/byterun/coq_uint63_native.h
+++ b/kernel/byterun/coq_uint63_native.h
@@ -111,51 +111,26 @@ value uint63_mulc(value x, value y, value* h) {
 #define le128(xh,xl,yh,yl) (uint63_lt(xh,yh) || (uint63_eq(xh,yh) && uint63_leq(xl,yl)))
 
 #define maxuint63 ((uint64_t)0x7FFFFFFFFFFFFFFF)
-/* precondition: y <> 0 */
-/* outputs r and sets ql to q % 2^63 s.t. x = q * y + r, r < y */
+/* precondition: xh < y */
+/* outputs r and sets ql to q s.t. x = q * y + r, r < y */
 static value uint63_div21_aux(value xh, value xl, value y, value* ql) {
-  xh = uint63_of_value(xh);
-  xl = uint63_of_value(xl);
+  uint64_t nh = uint63_of_value(xh);
+  uint64_t nl = uint63_of_value(xl);
   y = uint63_of_value(y);
-  uint64_t maskh = 0;
-  uint64_t maskl = 1;
-  uint64_t dh = 0;
-  uint64_t dl = y;
-  int cmp = 1;
-  /* int n = 0 */
-  /* loop invariant: mask = 2^n, d = mask * y, (2 * d <= x -> cmp), n >= 0, d < 2^(2*63) */
-  while (!(dh >> (63 - 1)) && cmp) {
-    dh = (dh << 1) | (dl >> (63 - 1));
-    dl = (dl << 1) & maxuint63;
-    maskh = (maskh << 1) | (maskl >> (63 - 1));
-    maskl = (maskl << 1) & maxuint63;
-    /* ++n */
-    cmp = lt128(dh,dl,xh,xl);
+  uint64_t q = 0;
+  for (int i = 0; i < 63; ++i) {
+    // invariants: 0 <= nh < y, nl = (xl*2^i) % 2^64,
+    // (q*y + nh) * 2^(63-i) + (xl % 2^(63-i)) = (xh%y) * 2^63 + xl
+    nl <<= 1;
+    nh = (nh << 1) | (nl >> 63);
+    q <<= 1;
+    if (nh >= y) { q |= 1; nh -= y; }
   }
-  uint64_t remh = xh;
-  uint64_t reml = xl;
-  /* uint64_t quotienth = 0; */
-  uint64_t quotientl = 0;
-  /* loop invariant: x = quotient * y + rem, y * 2^(n+1) > r,
-     mask = floor(2^n), d = mask * y, n >= -1 */
-  while (maskh | maskl) {
-    if (le128(dh,dl,remh,reml)) { /* if rem >= d, add one bit and subtract d */
-      /* quotienth = quotienth | maskh */
-      quotientl = quotientl | maskl;
-      remh = (uint63_lt(reml,dl)) ? (remh - dh - 1) : (remh - dh);
-      reml = reml - dl;
-    }
-    maskl = (maskl >> 1) | ((maskh << (63 - 1)) & maxuint63);
-    maskh = maskh >> 1;
-    dl = (dl >> 1) | ((dh << (63 - 1)) & maxuint63);
-    dh = dh >> 1;
-    /* decr n */
-  }
-  *ql = Val_int(quotientl);
-  return Val_int(reml);
+  *ql = Val_int(q);
+  return Val_int(nh);
 }
 value uint63_div21(value xh, value xl, value y, value* ql) {
-  if (uint63_of_value(y) == 0) {
+  if (uint63_leq(y, xh)) {
     *ql = Val_int(0);
     return Val_int(0);
   } else {
diff --git a/kernel/cClosure.ml b/kernel/cClosure.ml
index 1cc3dc3975..3fd613e905 100644
--- a/kernel/cClosure.ml
+++ b/kernel/cClosure.ml
@@ -1075,7 +1075,7 @@ module FNativeEntries =
 
     let mkInt env i =
       check_int env;
-      { mark = mark Norm KnownR; term = FInt i }
+      { mark = mark Cstr KnownR; term = FInt i }
 
     let mkBool env b =
       check_bool env;
diff --git a/kernel/uint63.mli b/kernel/uint63.mli
index 93632da110..5542716af2 100644
--- a/kernel/uint63.mli
+++ b/kernel/uint63.mli
@@ -37,6 +37,8 @@ val mul     : t -> t -> t
 val div     : t -> t -> t
 val rem     : t -> t -> t
 
+val diveucl : t -> t -> t * t
+
       (* Specific arithmetic operations *)
 val mulc    : t -> t -> t * t
 val addmuldiv : t -> t -> t -> t
@@ -57,3 +59,13 @@ val head0   : t -> t
 val tail0   : t -> t
 
 val is_uint63 : Obj.t -> bool
+
+(* Arithmetic with explicit carries *)
+
+(* Analog of Numbers.Abstract.Cyclic.carry *)
+type 'a carry = C0 of 'a | C1 of 'a
+
+val addc : t -> t -> t carry
+val addcarryc : t -> t -> t carry
+val subc : t -> t -> t carry
+val subcarryc : t -> t -> t carry
diff --git a/kernel/uint63_amd64_63.ml b/kernel/uint63_amd64_63.ml
index 20b2f58496..5c4028e1c8 100644
--- a/kernel/uint63_amd64_63.ml
+++ b/kernel/uint63_amd64_63.ml
@@ -82,6 +82,8 @@ let div (x : int) (y : int) =
 let rem (x : int) (y : int) =
   if y = 0 then 0 else Int64.to_int (Int64.rem (to_uint64 x) (to_uint64 y))
 
+let diveucl x y = (div x y, rem x y)
+
 let addmuldiv p x y =
   l_or (l_sl x p) (l_sr y (uint_size - p))
 
@@ -94,55 +96,32 @@ let le (x : int) (y : int) =
   (x lxor 0x4000000000000000) <= (y lxor 0x4000000000000000)
 [@@ocaml.inline always]
 
-(* A few helper functions on 128 bits *)
-let lt128 xh xl yh yl =
-  lt xh yh || (xh = yh && lt xl yl)
-
-let le128 xh xl yh yl =
-  lt xh yh || (xh = yh && le xl yl)
-
     (* division of two numbers by one *)
-(* precondition: y <> 0 *)
-(* outputs: q % 2^63, r s.t. x = q * y + r, r < y *)
+(* precondition: xh < y *)
+(* outputs: q, r s.t. x = q * y + r, r < y *)
 let div21 xh xl y =
-  let maskh = ref 0 in
-  let maskl = ref 1 in
-  let dh = ref 0 in
-  let dl = ref y in
-  let cmp = ref true in
-  (* n = ref 0 *)
-  (* loop invariant: mask = 2^n, d = mask * y, (2 * d <= x -> cmp), n >= 0 *)
-  while !dh >= 0 && !cmp do (* dh >= 0 tests that dh highest bit is zero *)
-    (* We don't use addmuldiv below to avoid checks on 1 *)
-    dh := (!dh lsl 1) lor (!dl lsr (uint_size - 1));
-    dl := !dl lsl 1;
-    maskh := (!maskh lsl 1) lor (!maskl lsr (uint_size - 1));
-    maskl := !maskl lsl 1;
-    (* incr n *)
-    cmp := lt128 !dh !dl xh xl;
-  done; (* mask = 2^n, d = 2^n * y, 2 * d > x *)
-  let remh = ref xh in
-  let reml = ref xl in
-  (* quotienth = ref 0 *)
-  let quotientl = ref 0 in
-  (* loop invariant: x = quotient * y + rem, y * 2^(n+1) > r,
-     mask = floor(2^n), d = mask * y, n >= -1 *)
-  while !maskh lor !maskl <> 0 do
-    if le128 !dh !dl !remh !reml then begin (* if rem >= d, add one bit and subtract d *)
-      (* quotienth := !quotienth lor !maskh *)
-      quotientl := !quotientl lor !maskl;
-      remh := if lt !reml !dl then !remh - !dh - 1 else !remh - !dh;
-      reml := !reml - !dl;
-    end;
-    maskl := (!maskl lsr 1) lor (!maskh lsl (uint_size - 1));
-    maskh := !maskh lsr 1;
-    dl := (!dl lsr 1) lor (!dh lsl (uint_size - 1));
-    dh := !dh lsr 1;
-    (* decr n *)
+  (* nh might temporarily grow as large as 2*y - 1 in the loop body,
+     so we store it as a 64-bit unsigned integer *)
+  let nh = ref xh in
+  let nl = ref xl in
+  let q = ref 0 in
+  for _i = 0 to 62 do
+    (* invariants: 0 <= nh < y, nl = (xl*2^i) % 2^63,
+       (q*y + nh) * 2^(63-i) + (xl % 2^(63-i)) = (xh%y) * 2^63 + xl *)
+    nh := Int64.logor (Int64.shift_left !nh 1) (Int64.of_int (!nl lsr 62));
+    nl := !nl lsl 1;
+    q := !q lsl 1;
+    (* TODO: use "Int64.unsigned_compare !nh y >= 0",
+       once OCaml 4.08 becomes the minimal required version *)
+    if Int64.compare !nh 0L < 0 || Int64.compare !nh y >= 0 then
+      begin q := !q lor 1; nh := Int64.sub !nh y; end
   done;
-  !quotientl, !reml
+  !q, Int64.to_int !nh
 
-let div21 xh xl y = if y = 0 then 0, 0 else div21 xh xl y
+let div21 xh xl y =
+  let xh = to_uint64 xh in
+  let y = to_uint64 y in
+  if Int64.compare y xh <= 0 then 0, 0 else div21 xh xl y
 
      (* exact multiplication *)
 (* TODO: check that none of these additions could be a logical or *)
@@ -225,3 +204,24 @@ let tail0 x =
 let is_uint63 t =
   Obj.is_int t
 [@@ocaml.inline always]
+
+(* Arithmetic with explicit carries *)
+
+(* Analog of Numbers.Abstract.Cyclic.carry *)
+type 'a carry = C0 of 'a | C1 of 'a
+
+let addc x y =
+  let r = x + y in
+  if lt r x then C1 r else C0 r
+
+let addcarryc x y =
+  let r = x + y + 1 in
+  if le r x then C1 r else C0 r
+
+let subc x y =
+  let r = x - y in
+  if le y x then C0 r else C1 r
+
+let subcarryc x y =
+  let r = x - y - 1 in
+  if lt y x then C0 r else C1 r
diff --git a/kernel/uint63_i386_31.ml b/kernel/uint63_i386_31.ml
index c3279779e1..b8eccd19fb 100644
--- a/kernel/uint63_i386_31.ml
+++ b/kernel/uint63_i386_31.ml
@@ -83,58 +83,33 @@ let div x y =
 let rem x y =
   if y = 0L then 0L else Int64.rem x y
 
+let diveucl x y = (div x y, rem x y)
+
 let addmuldiv p x y =
   l_or (l_sl x p) (l_sr y Int64.(sub (of_int uint_size) p))
 
-(* A few helper functions on 128 bits *)
-let lt128 xh xl yh yl =
-  lt xh yh || (xh = yh && lt xl yl)
-
-let le128 xh xl yh yl =
-  lt xh yh || (xh = yh && le xl yl)
-
     (* division of two numbers by one *)
-(* precondition: y <> 0 *)
-(* outputs: q % 2^63, r s.t. x = q * y + r, r < y *)
+(* precondition: xh < y *)
+(* outputs: q, r s.t. x = q * y + r, r < y *)
 let div21 xh xl y =
-  let maskh = ref zero in
-  let maskl = ref one in
-  let dh = ref zero in
-  let dl = ref y in
-  let cmp = ref true in
-  (* n = ref 0 *)
-  (* loop invariant: mask = 2^n, d = mask * y, (2 * d <= x -> cmp), n >= 0 *)
-  while Int64.equal (l_sr !dh (of_int (uint_size - 1))) zero && !cmp do
-    (* We don't use addmuldiv below to avoid checks on 1 *)
-    dh := l_or (l_sl !dh one) (l_sr !dl (of_int (uint_size - 1)));
-    dl := l_sl !dl one;
-    maskh := l_or (l_sl !maskh one) (l_sr !maskl (of_int (uint_size - 1)));
-    maskl := l_sl !maskl one;
-    (* incr n *)
-    cmp := lt128 !dh !dl xh xl;
-  done; (* mask = 2^n, d = 2^n * d, 2 * d > x *)
-  let remh = ref xh in
-  let reml = ref xl in
-  (* quotienth = ref 0 *)
-  let quotientl = ref zero in
-  (* loop invariant: x = quotient * y + rem, y * 2^(n+1) > r,
-     mask = floor(2^n), d = mask * y, n >= -1 *)
-  while not (Int64.equal (l_or !maskh !maskl) zero) do
-    if le128 !dh !dl !remh !reml then begin (* if rem >= d, add one bit and subtract d *)
-      (* quotienth := !quotienth lor !maskh *)
-      quotientl := l_or !quotientl !maskl;
-      remh := if lt !reml !dl then sub (sub !remh !dh) one else sub !remh !dh;
-      reml := sub !reml !dl
-    end;
-    maskl := l_or (l_sr !maskl one) (l_sl !maskh (of_int (uint_size - 1)));
-    maskh := l_sr !maskh one;
-    dl := l_or (l_sr !dl one) (l_sl !dh (of_int (uint_size - 1)));
-    dh := l_sr !dh one
-    (* decr n *)
+  let nh = ref xh in
+  let nl = ref xl in
+  let q = ref 0L in
+  for _i = 0 to 62 do
+    (* invariants: 0 <= nh < y, nl = (xl*2^i) % 2^64,
+       (q*y + nh) * 2^(63-i) + (xl % 2^(63-i)) = (xh%y) * 2^63 + xl *)
+    nl := Int64.shift_left !nl 1;
+    nh := Int64.logor (Int64.shift_left !nh 1) (Int64.shift_right_logical !nl 63);
+    q := Int64.shift_left !q 1;
+    (* TODO: use "Int64.unsigned_compare !nh y >= 0",
+       once OCaml 4.08 becomes the minimal required version *)
+    if Int64.compare !nh 0L < 0 || Int64.compare !nh y >= 0 then
+      begin q := Int64.logor !q 1L; nh := Int64.sub !nh y; end
   done;
-  !quotientl, !reml
+  !q, !nh
 
-let div21 xh xl y = if Int64.equal y zero then zero, zero else div21 xh xl y
+let div21 xh xl y =
+  if Int64.compare y xh <= 0 then zero, zero else div21 xh xl y
 
      (* exact multiplication *)
 let mulc x y =
@@ -191,6 +166,27 @@ let is_uint63 t =
   Obj.is_block t && Int.equal (Obj.tag t) Obj.custom_tag
   && le (Obj.magic t) maxuint63
 
+(* Arithmetic with explicit carries *)
+
+(* Analog of Numbers.Abstract.Cyclic.carry *)
+type 'a carry = C0 of 'a | C1 of 'a
+
+let addc x y =
+  let r = add x y in
+  if lt r x then C1 r else C0 r
+
+let addcarryc x y =
+  let r = addcarry x y in
+  if le r x then C1 r else C0 r
+
+let subc x y =
+  let r = sub x y in
+  if le y x then C0 r else C1 r
+
+let subcarryc x y =
+  let r = subcarry x y in
+  if lt y x then C0 r else C1 r
+
 (* Register all exported functions so that they can be called from C code *)
 
 let () =