Source file ival.ml
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module Extlib = struct
let the = function
| None -> invalid_arg "the"
| Some x -> x
let may_map f ?dft x =
match x, dft with
| None, None -> assert false
| None, Some dft -> dft
| Some x, _ -> f x
let opt_map f = function
| None -> None
| Some x -> Some (f x)
end
open Abstract_interp
let small_cardinal = ref 8
let small_cardinal_Int = ref (Int.of_int !small_cardinal)
let small_cardinal_log = ref 3
let debug_cardinal = false
let set_small_cardinal i =
assert (2 <= i && i <= 1024);
let rec log j p =
if i <= p then j
else log (j+1) (2*p)
in
small_cardinal := i;
small_cardinal_Int := Int.of_int i;
small_cardinal_log := log 1 2
let get_small_cardinal () = !small_cardinal
let emitter = Codex_log.register_category "Ival";;
let log_imprecision s =
Codex_log.imprecision_warning s
;;
module Widen_Arithmetic_Value_Set = struct
include Set.Make(Integer)
let hash _ = assert false
let pretty fmt s =
if is_empty s then Format.fprintf fmt "{}"
else
Pretty_utils.pp_iter
~pre:"@[<hov 1>{"
~suf:"}@]"
~sep:";@ "
iter Int.pretty fmt s
let of_list l =
match l with
| [] -> empty
| [e] -> singleton e
| e :: q ->
List.fold_left (fun acc x -> add x acc) (singleton e) q
let default_widen_hints =
of_list (List.map Int.of_int [-1;0;1])
let nearest_elt_le elt l =
match l with
| [] -> raise Not_found
| hd::rest when Z.gt hd elt -> raise Not_found
| hd::rest ->
let rec loop candidate = function
| [] -> candidate
| hd::rest when Z.gt hd elt -> candidate
| hd::rest -> loop hd rest
in loop hd rest
;;
let nearest_elt_le elt l = nearest_elt_le elt @@ elements l
let nearest_elt_ge elt l =
let rec loop = function
| [] -> raise Not_found
| hd::rest when Z.geq hd elt -> hd
| _::rest -> loop rest
in loop l
;;
let nearest_elt_ge elt l = nearest_elt_ge elt @@ elements l
end
exception Infinity
let opt2 f m1 m2 =
match m1, m2 with
None, _ | _, None -> raise Infinity
| Some m1, Some m2 -> f m1 m2
let opt1 f m =
match m with
None -> None
| Some m -> Some (f m)
module O = Set.Make(Integer)
type pre_set =
Pre_set of O.t * int
| Pre_top of Int.t * Int.t * Int.t
type t =
| Set of Int.t array
| Float of Fval.t
| Top of Int.t option * Int.t option * Int.t * Int.t
module Widen_Hints = Widen_Arithmetic_Value_Set
type size_widen_hint = Integer.t
type generic_widen_hint = Widen_Hints.t
type widen_hint = size_widen_hint * generic_widen_hint
let some_zero = Some Int.zero
let bottom = Set (Array.make 0 Int.zero)
let top = Top(None, None, Int.zero, Int.one)
let hash_v_option v =
match v with None -> 97 | Some v -> Int.hash v
let hash v =
match v with
Set s -> Array.fold_left (fun acc v -> 1031 * acc + (Int.hash v)) 17 s
| Top(mn,mx,r,m) ->
hash_v_option mn + 5501 * (hash_v_option mx) +
59 * (Int.hash r) + 13031 * (Int.hash m)
| Float(f) ->
3 + 17 * Fval.hash f
let bound_compare x y =
match x,y with
None, None -> 0
| None, Some _ -> 1
| Some _, None -> -1
| Some x, Some y -> Int.compare x y
exception Unequal of int
let compare e1 e2 =
if e1==e2 then 0 else
match e1,e2 with
| Set e1,Set e2 ->
let l1 = Array.length e1 in
let l2 = Array.length e2 in
if l1 <> l2
then l1 - l2
else
(try
for i=0 to l1 -1 do
let r = Int.compare e1.(i) e2.(i) in
if r <> 0 then raise (Unequal r)
done;
0
with Unequal v -> v )
| _, Set _ -> 1
| Set _, _ -> -1
| Top(mn,mx,r,m), Top(mn',mx',r',m') ->
let r1 = bound_compare mn mn' in
if r1 <> 0 then r1
else let r2 = bound_compare mx mx' in
if r2 <> 0 then r2
else let r3 = Int.compare r r' in
if r3 <> 0 then r3
else Int.compare m m'
| _, Top _ -> 1
| Top _, _ -> -1
| Float(f1), Float(f2) ->
Fval.compare f1 f2
let equal e1 e2 = compare e1 e2 = 0
let pretty fmt t =
match t with
| Top(mn,mx,r,m) ->
let print_bound fmt =
function
None -> Format.fprintf fmt "--"
| Some v -> Int.pretty fmt v
in
Format.fprintf fmt "[%a..%a]%t"
print_bound mn
print_bound mx
(fun fmt ->
if Int.is_zero r && Int.is_one m then
Format.fprintf fmt ""
else Format.fprintf fmt ",%a%%%a"
Int.pretty r
Int.pretty m)
| Float (f) ->
Fval.pretty fmt f
| Set s ->
if Array.length s = 0 then Format.fprintf fmt "BottomMod"
else begin
Pretty_utils.pp_iter
~pre:"@[<hov 1>{"
~suf:"}@]"
~sep:";@ "
Array.iter Int.pretty fmt s
end
let min_le_elt min elt =
match min with
| None -> true
| Some m -> Int.le m elt
let max_ge_elt max elt =
match max with
| None -> true
| Some m -> Int.ge m elt
let all_positives min =
match min with
| None -> false
| Some m -> Int.ge m Int.zero
let all_negatives max =
match max with
| None -> false
| Some m -> Int.le m Int.zero
let fail min max r modu =
let bound fmt = function
| None -> Format.fprintf fmt "--"
| Some(x) -> Int.pretty fmt x
in
Codex_log.fatal "Ival: broken Top, min=%a max=%a r=%a modu=%a"
bound min bound max Int.pretty r Int.pretty modu
let is_safe_modulo r modu =
(Int.ge r Int.zero ) && (Int.ge modu Int.one) && (Int.lt r modu)
let is_safe_bound bound r modu = match bound with
| None -> true
| Some m -> Int.equal (Int.e_rem m modu) r
let check min max r modu =
if not (is_safe_modulo r modu
&& is_safe_bound min r modu
&& is_safe_bound max r modu)
then fail min max r modu
let cardinal_zero_or_one v =
match v with
| Top _ -> false
| Set s -> Array.length s <= 1
| Float f -> Fval.is_singleton f
let is_singleton_int v = match v with
| Float _ | Top _ -> false
| Set s -> Array.length s = 1
let is_bottom x = x == bottom
let o_zero = O.singleton Int.zero
let o_one = O.singleton Int.one
let o_zero_or_one = O.union o_zero o_one
let small_nums =
Array.init 33 (fun i -> Set [| (Integer.of_int i) |])
let zero = small_nums.(0)
let one = small_nums.(1)
let minus_one = Set [| Int.minus_one |]
let zero_or_one = Set [| Int.zero ; Int.one |]
let float_zeros = Float Fval.zeros
let positive_integers = Top(Some Int.zero, None, Int.zero, Int.one)
let negative_integers =
Top(None, Some Int.zero, Int.zero, Int.one)
let is_zero x = x == zero
let inject_singleton e =
if Int.le Int.zero e && Int.le e Int.thirtytwo
then small_nums.(Int.to_int e)
else Set [| e |]
let share_set o s =
if s = 0 then bottom
else if s = 1
then begin
let e = O.min_elt o in
inject_singleton e
end
else if O.equal o o_zero_or_one
then zero_or_one
else
let a = Array.make s Int.zero in
let i = ref 0 in
O.iter (fun e -> a.(!i) <- e; incr i) o;
assert (!i = s);
Set a
let share_array a s =
if s = 0 then bottom
else
let e = a.(0) in
if s = 1 && Int.le Int.zero e && Int.le e Int.thirtytwo
then small_nums.(Int.to_int e)
else if s = 2 && Int.is_zero e && Int.is_one a.(1)
then zero_or_one
else Set a
let inject_float f =
if Fval.(equal plus_zero f)
then zero
else Float f
let inject_float_interval flow fup =
let flow = Fval.F.of_float flow in
let fup = Fval.F.of_float fup in
if Fval.F.equal Fval.F.plus_zero flow && Fval.F.equal Fval.F.plus_zero fup
then zero
else Float (Fval.inject Fval.Double flow fup)
let is_one = equal one
let project_float v =
if is_zero v
then Fval.plus_zero
else
match v with
| Float f -> f
| Top _ | Set _ -> assert false
let in_interval x min max r modu =
Int.equal (Int.e_rem x modu) r && min_le_elt min x && max_ge_elt max x
let array_mem v a =
let l = Array.length a in
let rec c i =
if i = l then (-1)
else
let ae = a.(i) in
if Int.equal ae v
then i
else if Int.gt ae v
then (-1)
else c (succ i)
in
c 0
let contains_zero s =
match s with
| Top(mn,mx,r,m) -> in_interval Int.zero mn mx r m
| Set s -> (array_mem Int.zero s)>=0
| Float f -> Fval.contains_a_zero f
let contains_non_zero s =
match s with
| Top _ -> true
| Set _ -> not (is_zero s || is_bottom s)
| Float f -> Fval.contains_non_zero f
exception Not_Singleton_Int
let project_int v = match v with
| Set [| e |] -> e
| _ -> raise Not_Singleton_Int
let cardinal v =
match v with
| Top (None,_,_,_) | Top (_,None,_,_) -> None
| Top (Some mn, Some mx,_,m) ->
Some (Int.succ ((Int.e_div (Int.sub mx mn) m)))
| Set s -> Some (Int.of_int (Array.length s))
| Float f -> if Fval.is_singleton f then Some Int.one else None
let cardinal_estimate v ~size =
match v with
| Set s -> Int.of_int (Array.length s)
| Top (None, _, _, _)
| Top (_, None, _, _) -> Int.two_power size
| Top (Some mn, Some mx, _, d) -> Int.(succ (e_div (sub mx mn) d))
| Float f ->
if Fval.is_singleton f
then Int.one
else
let bits_of_float =
if Integer.(equal size (of_int 32))
then Fval.bits_of_float32_list
else if Integer.(equal size (of_int 64))
then Fval.bits_of_float64_list
else (fun _ -> [Int.zero, Int.pred (Int.two_power size)])
in
let bits_list = bits_of_float f in
let count acc (min, max) = Int.add acc (Int.length min max) in
List.fold_left count Int.zero bits_list
let cardinal_less_than v n =
let c =
match v with
| Top (None,_,_,_) | Top (_,None,_,_) -> raise Not_less_than
| Top (Some mn, Some mx,_,m) ->
Int.succ ((Int.e_div (Int.sub mx mn) m))
| Set s -> Int.of_int (Array.length s)
| Float f ->
if Fval.is_singleton f then Int.one else raise Not_less_than
in
if Int.le c (Int.of_int n)
then Int.to_int c
else raise Not_less_than
let cardinal_is_less_than v n =
match cardinal v with
| None -> false
| Some c -> Int.le c (Int.of_int n)
let share_top min max r modu =
let r = Top (min, max, r, modu) in
if equal r top then top else r
let make ~min ~max ~rem ~modu =
match min, max with
| Some mn, Some mx ->
if Int.gt mx mn then
let l = Int.succ (Int.e_div (Int.sub mx mn) modu) in
if Int.le l !small_cardinal_Int
then
let l = Int.to_int l in
let s = Array.make l Int.zero in
let v = ref mn in
let i = ref 0 in
while (!i < l)
do
s.(!i) <- !v;
v := Int.add modu !v;
incr i
done;
assert (Int.equal !v (Int.add modu mx));
share_array s l
else Top (min, max, rem, modu)
else if Int.equal mx mn
then inject_singleton mn
else bottom
| _ ->
share_top min max rem modu
let inject_top min max rem modu =
check min max rem modu;
make ~min ~max ~rem ~modu
let inject_interval ~min ~max ~rem:r ~modu =
assert (is_safe_modulo r modu);
let fix_bound fix bound = match bound with
| None -> None
| Some b -> Some (if Int.equal b (Int.e_rem r modu) then b else fix b)
in
let min = fix_bound (fun min -> Int.round_up_to_r ~min ~r ~modu) min
and max = fix_bound (fun max -> Int.round_down_to_r ~max ~r ~modu) max in
make ~min ~max ~rem:r ~modu
let subdiv_int v =
match v with
| Float _ -> raise Can_not_subdiv
| Set arr ->
let len = Array.length arr in
assert (len > 0 );
if len <= 1 then raise Can_not_subdiv;
let m = len lsr 1 in
let lenhi = len - m in
let lo = Array.sub arr 0 m in
let hi = Array.sub arr m lenhi in
share_array lo m,
share_array hi lenhi
| Top (Some lo, Some hi, rem, modu) ->
let mean = Int.e_div (Int.add lo hi) Int.two in
let succmean = Int.succ mean in
inject_interval ~min:(Some lo) ~max:(Some mean) ~rem ~modu,
inject_interval ~min:(Some succmean) ~max:(Some hi) ~rem ~modu
| Top _ -> raise Can_not_subdiv
let subdivide ~size = function
| Float fval ->
let fkind = match Integer.to_int size with
| 32 -> Fval.Single
| 64 -> Fval.Double
| _ -> raise Can_not_subdiv
in
let f1, f2 = Fval.subdiv_float_interval fkind fval in
inject_float f1, inject_float f2
| ival -> subdiv_int ival
let inject_range min max = inject_top min max Int.zero Int.one
let top_float = Float Fval.top
let top_single_precision_float = Float Fval.top
let unsafe_make_top_from_set_4 s =
if debug_cardinal then assert (O.cardinal s >= 2);
let m = O.min_elt s in
let modu = O.fold
(fun x acc ->
if Int.equal x m
then acc
else Int.pgcd (Int.sub x m) acc)
s
Int.zero
in
let r = Int.e_rem m modu in
let max = O.max_elt s in
let min = m in
(min,max,r,modu)
let unsafe_make_top_from_array_4 s =
let l = Array.length s in
assert (l >= 2);
let m = s.(0) in
let modu =
Array.fold_left
(fun acc x ->
if Int.equal x m
then acc
else Int.pgcd (Int.sub x m) acc)
Int.zero
s
in
let r = Int.e_rem m modu in
let max = Some s.(pred l) in
let min = Some m in
check min max r modu;
(min,max,r,modu)
let unsafe_make_top_from_array s =
let min, max, r, modu = unsafe_make_top_from_array_4 s in
share_top min max r modu
let empty_ps = Pre_set (O.empty, 0)
let add_ps ps x =
match ps with
| Pre_set(o,s) ->
if debug_cardinal then assert (O.cardinal o = s);
if (O.mem x o)
then ps
else
let no = O.add x o in
if s < !small_cardinal
then begin
if debug_cardinal then assert (O.cardinal no = succ s);
Pre_set (no, succ s)
end
else
let min, max, _r, modu = unsafe_make_top_from_set_4 no in
Pre_top (min, max, modu)
| Pre_top (min, max, modu) ->
let new_modu =
if Int.equal x min
then modu
else Int.pgcd (Int.sub x min) modu
in
let new_min = Int.min min x
in
let new_max = Int.max max x
in
Pre_top (new_min, new_max, new_modu)
let inject_ps ps =
match ps with
Pre_set(o, s) -> share_set o s
| Pre_top (min, max, modu) ->
Top(Some min, Some max, Int.e_rem min modu, modu)
let min_max_r_mod t =
match t with
| Set s ->
assert (Array.length s >= 2);
unsafe_make_top_from_array_4 s
| Top (a,b,c,d) -> a,b,c,d
| Float _ -> None, None, Int.zero, Int.one
let min_and_max t =
match t with
| Set s ->
let l = Array.length s in
if l = 0
then raise Error_Bottom
else Some s.(0), Some s.(pred l)
| Top (a,b,_,_) -> a, b
| Float _ -> None, None
let min_and_max_float t =
match t with
| Set _ when is_zero t -> Some (Fval.F.plus_zero, Fval.F.plus_zero), false
| Float f -> Fval.min_and_max f
| _ -> assert false
let is_float = function
| Float _ -> true
| Set _ | Top _ -> false
let has_greater_min_bound t1 t2 =
if is_float t1 || is_float t2
then Fval.has_greater_min_bound (project_float t1) (project_float t2)
else
let m1, _ = min_and_max t1 in
let m2, _ = min_and_max t2 in
match m1, m2 with
| None, None -> 0
| None, Some _ -> -1
| Some _, None -> 1
| Some m1, Some m2 -> Int.compare m1 m2
let has_smaller_max_bound t1 t2 =
if is_float t1 || is_float t2
then Fval.has_smaller_max_bound (project_float t1) (project_float t2)
else
let _, m1 = min_and_max t1 in
let _, m2 = min_and_max t2 in
match m1, m2 with
| None, None -> 0
| None, Some _ -> -1
| Some _, None -> 1
| Some m1, Some m2 -> Int.compare m2 m1
let widen (bitsize,wh) t1 t2 =
if equal t1 t2 || cardinal_zero_or_one t1 then t2
else
match t2 with
| Float f2 ->
let f1 = project_float t1 in
Float (Fval.widen f1 f2)
| Top _ | Set _ ->
let wh =
if Integer.gt bitsize (Integer.of_int 128)
then Widen_Hints.empty
else if Integer.is_zero bitsize
then wh
else
let limits = [
Integer.neg (Integer.two_power (Integer.pred bitsize));
Integer.pred (Integer.two_power (Integer.pred bitsize));
Integer.pred (Integer.two_power bitsize);
] in
Widen_Hints.(union wh (of_list limits))
in
let (mn2,mx2,r2,m2) = min_max_r_mod t2 in
let (mn1,mx1,r1,m1) = min_max_r_mod t1 in
let new_mod = Int.pgcd (Int.pgcd m1 m2) (Int.abs (Int.sub r1 r2)) in
let new_rem = Int.e_rem r1 new_mod in
let new_min = if bound_compare mn1 mn2 = 0 then mn2 else
match mn2 with
| None -> None
| Some mn2 ->
try
let v = Widen_Hints.nearest_elt_le mn2 wh
in Some (Int.round_up_to_r ~r:new_rem ~modu:new_mod ~min:v)
with Not_found -> None
in
let new_max = if bound_compare mx1 mx2 = 0 then mx2 else
match mx2 with None -> None
| Some mx2 ->
try
let v = Widen_Hints.nearest_elt_ge mx2 wh
in Some (Int.round_down_to_r ~r:new_rem ~modu:new_mod ~max:v)
with Not_found -> None
in
let result = inject_top new_min new_max new_rem new_mod in
result
let compute_first_common mn1 mn2 r modu =
if mn1 = None && mn2 = None
then None
else
let m =
match (mn1, mn2) with
| Some m, None | None, Some m -> m
| Some m1, Some m2 ->
Int.max m1 m2
| None, None -> assert false
in
Some (Int.round_up_to_r m r modu)
let compute_last_common mx1 mx2 r modu =
if mx1 = None && mx2 = None
then None
else
let m =
match (mx1, mx2) with
| Some m, None | None, Some m -> m
| Some m1, Some m2 ->
Int.min m1 m2
| None, None -> assert false
in
Some (Int.round_down_to_r m r modu)
let min_min x y =
match x,y with
| None,_ | _,None -> None
| Some x, Some y -> Some (Int.min x y)
let max_max x y =
match x,y with
| None,_ | _,None -> None
| Some x, Some y -> Some (Int.max x y)
let extended_euclidian_algorithm a b =
assert (Int.gt a Int.zero);
assert (Int.gt b Int.zero);
let a = ref a and b = ref b in
let x = ref Int.zero and lastx = ref Int.one in
let y = ref Int.one and lasty = ref Int.zero in
while not (Int.is_zero !b) do
let (q,r) = Int.e_div_rem !a !b in
a := !b;
b := r;
let tmpx = !x in
(x:= Int.sub !lastx (Int.mul q !x); lastx := tmpx);
let tmpy = !y in
(y:= Int.sub !lasty (Int.mul q !y); lasty := tmpy);
done;
(!lastx,!lasty,!a)
let _modular_inverse a m =
let (x,_,gcd) = extended_euclidian_algorithm a m in
assert (Int.equal Int.one gcd);
x
let compute_r_common r1 m1 r2 m2 =
let (x1,_,pgcd) = extended_euclidian_algorithm m1 m2 in
let r = Int.sub r2 r1 in
let r_div,r_rem = Int.e_div_rem r pgcd in
if not (Int.equal r_rem Int.zero)
then raise Error_Bottom
else let k1 = Int.mul x1 r_div in
let x = Int.add r1 (Int.mul k1 m1) in
let ppcm = Integer.ppcm m1 m2 in
(Int.e_rem x ppcm, ppcm)
;;
let array_truncate r i =
if i = 0
then bottom
else if i = 1
then inject_singleton r.(0)
else begin
let r = Array.sub r 0 i in
assert (Array.length r = i);
Set r
end
let array_filter (f : Int.t -> bool) (a : Int.t array) : t =
let l = Array.length a in
let r = Array.make l Int.zero in
let j = ref 0 in
for i = 0 to l - 1 do
let x = a.(i) in
if f x then begin
r.(!j) <- x;
incr j;
end
done;
array_truncate r !j
let array_map_reduce (f : 'a -> 'b) (g : 'b -> 'b -> 'b) (set : 'a array) : 'b =
if Array.length set <= 0 then
raise Error_Bottom
else
let acc = ref (f set.(0)) in
for i = 1 to Array.length set - 1 do
acc := g !acc (f set.(i))
done;
!acc
let array_inter a1 a2 =
let l1 = Array.length a1 in
let l2 = Array.length a2 in
let lr_max = min l1 l2 in
let r = Array.make lr_max Int.zero in
let rec c i i1 i2 =
if i1 = l1 || i2 = l2
then array_truncate r i
else
let e1 = a1.(i1) in
let e2 = a2.(i2) in
if Int.equal e1 e2
then begin
r.(i) <- e1;
c (succ i) (succ i1) (succ i2)
end
else if Int.lt e1 e2
then c i (succ i1) i2
else c i i1 (succ i2)
in
c 0 0 0
let arrays_intersect a1 a2 =
let l1 = Array.length a1 in
let l2 = Array.length a2 in
let rec aux i1 i2 =
if i1 = l1 || i2 = l2 then false
else
let e1 = a1.(i1) in
let e2 = a2.(i2) in
if Int.equal e1 e2 then true
else if Int.lt e1 e2 then aux (succ i1) i2
else aux i1 (succ i2)
in
aux 0 0
let meet v1 v2 =
if v1 == v2 then v1 else
let result =
match v1,v2 with
| Top(min1,max1,r1,modu1), Top(min2,max2,r2,modu2) ->
begin
try
let r,modu = compute_r_common r1 modu1 r2 modu2 in
inject_top
(compute_first_common min1 min2 r modu)
(compute_last_common max1 max2 r modu)
r
modu
with Error_Bottom ->
bottom
end
| Set s1 , Set s2 -> array_inter s1 s2
| Set s, Top(min, max, rm, modu)
| Top(min, max, rm, modu), Set s ->
let l = Array.length s in
let r = Array.make l Int.zero in
let rec c i j =
if i = l
then
array_truncate r j
else
let si = succ i in
let x = s.(i) in
if in_interval x min max rm modu
then begin
r.(j) <- x;
c si (succ j)
end
else
c si j
in
c 0 0
| Float(f1), Float(f2) -> begin
match Fval.meet f1 f2 with
| `Value f -> inject_float f
| `Bottom -> bottom
end
| (Float f as ff), (Top _ | Set _ as o)
| (Top _ | Set _ as o), (Float f as ff) ->
if equal o top then ff
else if contains_zero o && Fval.contains_plus_zero f then zero
else bottom
in
result
let intersects v1 v2 =
v1 == v2 ||
match v1, v2 with
| Top _, Top _ -> not (is_bottom (meet v1 v2))
| Set s1 , Set s2 -> arrays_intersect s1 s2
| Set s, Top (min, max, rm, modu) | Top (min, max, rm, modu), Set s ->
Array.exists (fun x -> in_interval x min max rm modu) s
| Float f1, Float f2 -> begin
match Fval.forward_comp Comp.Eq f1 f2 with
| Comp.False -> false
| Comp.True | Comp.Unknown -> true
end
| Float f, other | other, Float f ->
equal top other || (Fval.contains_plus_zero f && contains_zero other)
let narrow v1 v2 =
match v1, v2 with
| _, Set [||] | Set [||], _ -> bottom
| Float _, Float _ | (Top _| Set _), (Top _ | Set _) ->
meet v1 v2
| v, (Top _ as t) when equal t top -> v
| (Top _ as t), v when equal t top -> v
| Float f, (Set _ as s) | (Set _ as s), Float f when is_zero s -> begin
match Fval.narrow f Fval.zeros with
| `Value f -> inject_float f
| `Bottom -> bottom
end
| Float _, (Set _ | Top _) | (Set _ | Top _), Float _ ->
top
let set_to_ival_under set =
let card = Int.Set.cardinal set in
if card <= !small_cardinal
then
(let a = Array.make card Int.zero in
ignore(Int.Set.fold (fun elt i ->
Array.set a i elt;
i + 1) set 0);
share_array a card)
else
if (Int.equal
(Int.sub (Int.Set.max_elt set) (Int.Set.min_elt set))
(Int.of_int (card - 1)))
then Top( Some(Int.Set.min_elt set),
Some(Int.Set.max_elt set),
Int.one,
Int.zero)
else
let a = Array.make !small_cardinal Int.zero in
log_imprecision "Ival.set_to_ival_under";
try
ignore(Int.Set.fold (fun elt i ->
if i = !small_cardinal then raise Exit;
Array.set a i elt;
i + 1) set 0);
assert false
with Exit -> Set a
;;
let link v1 v2 = match v1, v2 with
| Set a1, Set a2 ->
let s1 = Array.fold_right Int.Set.add a1 Int.Set.empty in
let s2 = Array.fold_right Int.Set.add a2 s1 in
set_to_ival_under s2
| Top(mn1,mx1,r1,m1), Top(mn2,mx2,r2,m2) ->
if Int.equal r1 r2 && Int.equal m1 m2
then
let min = match mn1,mn2 with
| Some(a), Some(b) -> Some(Int.min a b)
| _ -> None in
let max = match mx1,mx2 with
| Some(a), Some(b) -> Some(Int.max a b)
| _ -> None in
inject_top min max r1 m1
else v1
| Top(mn,mx,r,m), Set s | Set s, Top(mn,mx,r,m) ->
let max = match mx with
| None -> None
| Some(max) ->
let curmax = ref max in
for i = 0 to (Array.length s) - 1 do
let elt = s.(i) in
if Int.equal elt (Int.add !curmax m)
then curmax := elt
done;
Some(!curmax) in
let min = match mn with
| None -> None
| Some(min) ->
let curmin = ref min in
for i = (Array.length s) - 1 downto 0 do
let elt = s.(i) in
if Int.equal elt (Int.sub !curmin m)
then curmin := elt
done;
Some(!curmin) in
inject_top min max r m
| _ -> bottom
;;
let join v1 v2 =
let result =
if v1 == v2 then v1 else
match v1,v2 with
| Top(mn1,mx1,r1,m1), Top(mn2,mx2,r2,m2) ->
check mn1 mx1 r1 m1;
check mn2 mx2 r2 m2;
let modu = Int.pgcd (Int.pgcd m1 m2) (Int.abs(Int.sub r1 r2)) in
let r = Int.e_rem r1 modu in
let min = min_min mn1 mn2 in
let max = max_max mx1 mx2 in
let r = inject_top min max r modu in
r
| Set s, (Top(min, max, r, modu) as t)
| (Top(min, max, r, modu) as t), Set s ->
let l = Array.length s in
if l = 0 then t
else
let f modu elt = Int.pgcd modu (Int.abs(Int.sub r elt)) in
let new_modu = Array.fold_left f modu s in
let new_r = Int.e_rem r new_modu in
let new_min = match min with
None -> None
| Some m -> Some (Int.min m s.(0))
in
let new_max = match max with
None -> None
| Some m -> Some (Int.max m s.(pred l))
in
check new_min new_max new_r new_modu;
share_top new_min new_max new_r new_modu
| Set s1 , Set s2 ->
let l1 = Array.length s1 in
if l1 = 0
then v2
else
let l2 = Array.length s2 in
if l2 = 0
then v1
else
let second uniq =
if uniq <= !small_cardinal
then
let r = Array.make uniq Int.zero in
let rec c i i1 i2 =
if i1 = l1
then begin
Array.blit s2 i2 r i (l2 - i2);
share_array r uniq
end
else if i2 = l2
then begin
Array.blit s1 i1 r i (l1 - i1);
share_array r uniq
end
else
let si = succ i in
let e1 = s1.(i1) in
let e2 = s2.(i2) in
if Int.lt e2 e1
then begin
r.(i) <- e2;
c si i1 (succ i2)
end
else begin
r.(i) <- e1;
let si1 = succ i1 in
if Int.equal e1 e2
then begin
c si si1 (succ i2)
end
else begin
c si si1 i2
end
end
in
c 0 0 0
else begin
let m = Int.min s1.(0) s2.(0) in
let accum acc x =
if Int.equal x m
then acc
else Int.pgcd (Int.sub x m) acc
in
let modu = ref Int.zero in
for j = 0 to pred l1 do
modu := accum !modu s1.(j)
done;
for j = 0 to pred l2 do
modu := accum !modu s2.(j)
done;
inject_ps
(Pre_top (m, Int.max s1.(pred l1) s2.(pred l2), !modu))
end
in
let rec first i1 i2 uniq inc1 inc2 =
let finished1 = i1 = l1 in
if finished1
then begin
if inc2
then v2
else second (uniq + l2 - i2)
end
else
let finished2 = i2 = l2 in
if finished2
then begin
if inc1
then v1
else second (uniq + l1 - i1)
end
else
let e1 = s1.(i1) in
let e2 = s2.(i2) in
if Int.lt e2 e1
then begin
first i1 (succ i2) (succ uniq) false inc2
end
else if Int.gt e2 e1
then begin
first (succ i1) i2 (succ uniq) inc1 false
end
else first (succ i1) (succ i2) (succ uniq) inc1 inc2
in
first 0 0 0 true true
| Float(f1), Float(f2) ->
inject_float (Fval.join f1 f2)
| Float (f) as ff, other | other, (Float (f) as ff) ->
if is_zero other
then inject_float (Fval.join Fval.plus_zero f)
else if is_bottom other then ff
else top
in
result
let fold_int f v acc =
match v with
Top(None,_,_,_) | Top(_,None,_,_) | Float _ ->
raise Error_Top
| Top(Some inf, Some sup, _, step) ->
Int.fold f ~inf ~sup ~step acc
| Set s ->
Array.fold_left (fun acc x -> f x acc) acc s
let fold_int_decrease f v acc =
match v with
Top(None,_,_,_) | Top(_,None,_,_) | Float _ ->
raise Error_Top
| Top(Some inf, Some sup, _, step) ->
Int.fold f ~inf ~sup ~step:(Int.neg step) acc
| Set s ->
Array.fold_right (fun x acc -> f x acc) s acc
let fold_enum f v acc =
match v with
| Float fl when Fval.is_singleton fl -> f v acc
| Float _ -> raise Error_Top
| Set _ | Top _ -> fold_int (fun x acc -> f (inject_singleton x) acc) v acc
(** [min_is_lower mn1 mn2] is true iff mn1 is a lower min than mn2 *)
let min_is_lower mn1 mn2 =
match mn1, mn2 with
None, _ -> true
| _, None -> false
| Some m1, Some m2 ->
Int.le m1 m2
(** [max_is_greater mx1 mx2] is true iff mx1 is a greater max than mx2 *)
let max_is_greater mx1 mx2 =
match mx1, mx2 with
None, _ -> true
| _, None -> false
| Some m1, Some m2 ->
Int.ge m1 m2
let rem_is_included r1 m1 r2 m2 =
(Int.is_zero (Int.e_rem m1 m2)) && (Int.equal (Int.e_rem r1 m2) r2)
let array_for_all f (a : Integer.t array) =
let l = Array.length a in
let rec c i =
i = l ||
((f a.(i)) && c (succ i))
in
c 0
let array_subset a1 a2 =
let l1 = Array.length a1 in
let l2 = Array.length a2 in
if l1 > l2 then false
else
let rec c i1 i2 =
if i1 = l1 then true
else if i2 = l2 then false
else
let e1 = a1.(i1) in
let e2 = a2.(i2) in
let si2 = succ i2 in
if Int.equal e1 e2
then c (succ i1) si2
else if Int.lt e1 e2
then false
else c i1 si2
in
c 0 0
let is_included t1 t2 =
(t1 == t2) ||
match t1,t2 with
| Set [||], _ -> true
| Top(mn1,mx1,r1,m1), Top(mn2,mx2,r2,m2) ->
(min_is_lower mn2 mn1) &&
(max_is_greater mx2 mx1) &&
rem_is_included r1 m1 r2 m2
| Top _, Set _ -> false
| Set s, Top(min, max, r, modu) ->
min_le_elt min s.(0) && max_ge_elt max s.(Array.length s-1)
&& (Int.equal Int.one modu ||
array_for_all (fun x -> Int.equal (Int.e_rem x modu) r) s)
| Set s1, Set s2 -> array_subset s1 s2
| Float f1, Float f2 -> Fval.is_included f1 f2
| Float _, _ -> equal t2 top
| Set _, Float f -> is_zero t1 && Fval.contains_plus_zero f
| Top _, Float _ -> false
let map_set_exnsafe_acc f acc (s : Integer.t array) =
Array.fold_left
(fun acc v -> add_ps acc (f v))
acc
s
let map_set_exnsafe f (s : Integer.t array) =
inject_ps (map_set_exnsafe_acc f empty_ps s)
let apply2_notzero f (s1 : Integer.t array) s2 =
inject_ps
(Array.fold_left
(fun acc v1 ->
Array.fold_left
(fun acc v2 ->
if Int.is_zero v2
then acc
else add_ps acc (f v1 v2))
acc
s2)
empty_ps
s1)
let apply2_n f (s1 : Integer.t array) (s2 : Integer.t array) =
let ps = ref empty_ps in
let l1 = Array.length s1 in
let l2 = Array.length s2 in
for i1 = 0 to pred l1 do
let e1 = s1.(i1) in
for i2 = 0 to pred l2 do
ps := add_ps !ps (f e1 s2.(i2))
done
done;
inject_ps !ps
let apply_set f v1 v2 =
match v1,v2 with
| Set s1, Set s2 -> apply2_n f s1 s2
| _ -> top
let apply_set_unary f v =
match v with
| Set s -> map_set_exnsafe f s
| _ -> top
let apply_bin_1_strict_incr f x (s : Integer.t array) =
let l = Array.length s in
let r = Array.make l Int.zero in
let rec c i =
if i = l
then share_array r l
else
let v = f x s.(i) in
r.(i) <- v;
c (succ i)
in
c 0
let apply_bin_1_strict_decr f x (s : Integer.t array) =
let l = Array.length s in
let r = Array.make l Int.zero in
let rec c i =
if i = l
then share_array r l
else
let v = f x s.(i) in
r.(l - i - 1) <- v;
c (succ i)
in
c 0
let map_set_strict_decr f (s : Integer.t array) =
let l = Array.length s in
let r = Array.make l Int.zero in
let rec c i =
if i = l
then share_array r l
else
let v = f s.(i) in
r.(l - i - 1) <- v;
c (succ i)
in
c 0
let map_set_decr f (s : Integer.t array) =
let l = Array.length s in
if l = 0
then bottom
else
let r = Array.make l Int.zero in
let rec c srcindex dstindex last =
if srcindex < 0
then begin
r.(dstindex) <- last;
array_truncate r (succ dstindex)
end
else
let v = f s.(srcindex) in
if Int.equal v last
then
c (pred srcindex) dstindex last
else begin
r.(dstindex) <- last;
c (pred srcindex) (succ dstindex) v
end
in
c (l-2) 0 (f s.(pred l))
let map_set_incr f (s : Integer.t array) =
let l = Array.length s in
if l = 0
then bottom
else
let r = Array.make l Int.zero in
let rec c srcindex dstindex last =
if srcindex = l
then begin
r.(dstindex) <- last;
array_truncate r (succ dstindex)
end
else
let v = f s.(srcindex) in
if Int.equal v last
then
c (succ srcindex) dstindex last
else begin
r.(dstindex) <- last;
c (succ srcindex) (succ dstindex) v
end
in
c 1 0 (f s.(0))
let add_singleton_int i v = match v with
| Float _ -> assert false
| Set s -> apply_bin_1_strict_incr Int.add i s
| Top (mn, mx, r, m) ->
let incr v = Int.add i v in
let new_mn = opt1 incr mn in
let new_mx = opt1 incr mx in
let new_r = Int.e_rem (incr r) m in
share_top new_mn new_mx new_r m
let rec add_int v1 v2 =
match v1,v2 with
| Float _, _ | _, Float _ -> assert false
| Set [| x |], Set s | Set s, Set [| x |]->
apply_bin_1_strict_incr Int.add x s
| Set s1, Set s2 ->
apply2_n Int.add s1 s2
| Top(mn1,mx1,r1,m1), Top(mn2,mx2,r2,m2) ->
let m = Int.pgcd m1 m2 in
let r = Int.e_rem (Int.add r1 r2) m in
let mn =
try
Some (Int.round_up_to_r (opt2 Int.add mn1 mn2) r m)
with Infinity -> None
in
let mx =
try
Some (Int.round_down_to_r (opt2 Int.add mx1 mx2) r m)
with Infinity -> None
in
inject_top mn mx r m
| Set s, (Top _ as t) | (Top _ as t), Set s ->
let l = Array.length s in
if l = 0
then bottom
else if l = 1
then
add_singleton_int s.(0) t
else
add_int t (unsafe_make_top_from_array s)
let add_int_under v1 v2 = match v1,v2 with
| Float _, _ | _, Float _ -> assert false
| Set [| x |], Set s | Set s, Set [| x |]->
apply_bin_1_strict_incr Int.add x s
| Set s1, Set s2 ->
let set =
Array.fold_left (fun acc i1 ->
Array.fold_left (fun acc i2 ->
Int.Set.add (Int.add i1 i2) acc) acc s2)
Int.Set.empty s1
in set_to_ival_under set
| Top(min1,max1,r1,modu1) , Top(min2,max2,r2,modu2)
when Int.equal modu1 modu2 ->
let r = Int.e_rem (Int.add r1 r2) modu1 in
let min = match min1, min2 with
| Some min1, Some min2 -> Some (Int.add min1 min2)
| _ -> None in
let max = match max1, max2 with
| Some max1, Some max2 -> Some (Int.add max1 max2)
| _ -> None in
inject_top min max r modu1
| Top _, Top _ -> bottom
| Set s, (Top _ as t) | (Top _ as t), Set s ->
let l = Array.length s in
if l = 0
then bottom
else if l = 1
then
add_singleton_int s.(0) t
else begin
log_imprecision "Ival.add_int_under";
add_singleton_int s.(0) t
end
;;
let neg_int v =
match v with
| Float _ -> assert false
| Set s -> map_set_strict_decr Int.neg s
| Top(mn,mx,r,m) ->
share_top
(opt1 Int.neg mx)
(opt1 Int.neg mn)
(Int.e_rem (Int.neg r) m)
m
let sub_int v1 v2 = add_int v1 (neg_int v2)
let sub_int_under v1 v2 = add_int_under v1 (neg_int v2)
type ext_value = Ninf | Pinf | Val of Int.t
let inject_min = function None -> Ninf | Some m -> Val m
let inject_max = function None -> Pinf | Some m -> Val m
let ext_neg = function Ninf -> Pinf | Pinf -> Ninf | Val v -> Val (Int.neg v)
let ext_mul x y =
match x, y with
| Ninf, Ninf | Pinf, Pinf -> Pinf
| Ninf, Pinf | Pinf, Ninf -> Ninf
| Val v1, Val v2 -> Val (Int.mul v1 v2)
| (Ninf | Pinf as x), Val v | Val v, (Ninf | Pinf as x) ->
if Int.gt v Int.zero then x
else if Int.lt v Int.zero then ext_neg x
else Val Int.zero
let ext_min x y =
match x,y with
Ninf, _ | _, Ninf -> Ninf
| Pinf, x | x, Pinf -> x
| Val x, Val y -> Val(Int.min x y)
let ext_max x y =
match x,y with
Pinf, _ | _, Pinf -> Pinf
| Ninf, x | x, Ninf -> x
| Val x, Val y -> Val(Int.max x y)
let ext_proj = function Val x -> Some x | _ -> None
let min_int s =
match s with
| Top (min,_,_,_) -> min
| Set s ->
if Array.length s = 0
then raise Error_Bottom
else
Some s.(0)
| Float _ -> None
let max_int s =
match s with
| Top (_,max,_,_) -> max
| Set s ->
let l = Array.length s in
if l = 0
then raise Error_Bottom
else
Some s.(pred l)
| Float _ -> None
let scale f v =
if Int.is_zero f
then zero
else
match v with
| Float _ -> top
| Top(mn1,mx1,r1,m1) ->
let incr = Int.mul f in
if Int.gt f Int.zero
then
let modu = incr m1 in
share_top
(opt1 incr mn1) (opt1 incr mx1)
(Int.e_rem (incr r1) modu) modu
else
let modu = Int.neg (incr m1) in
share_top
(opt1 incr mx1) (opt1 incr mn1)
(Int.e_rem (incr r1) modu) modu
| Set s ->
if Int.ge f Int.zero
then apply_bin_1_strict_incr Int.mul f s
else apply_bin_1_strict_decr Int.mul f s
let scale_div_common ~pos f v degenerate_ival degenerate_float =
assert (not (Int.is_zero f));
let div_f =
if pos
then fun a -> Int.e_div a f
else fun a -> Int.c_div a f
in
match v with
| Top(mn1,mx1,r1,m1) ->
let r, modu =
let negative = max_is_greater (some_zero) mx1 in
if (negative ||
pos ||
(min_is_lower (some_zero) mn1) ||
(Int.is_zero (Int.e_rem r1 f)) )
&& (Int.is_zero (Int.e_rem m1 f))
then
let modu = Int.abs (div_f m1) in
let r = if negative then Int.sub r1 m1 else r1 in
(Int.e_rem (div_f r) modu), modu
else
degenerate_ival r1 m1
in
let divf_mn1 = opt1 div_f mn1 in
let divf_mx1 = opt1 div_f mx1 in
let mn, mx =
if Int.gt f Int.zero
then divf_mn1, divf_mx1
else divf_mx1, divf_mn1
in
inject_top mn mx r modu
| Set s ->
if Int.lt f Int.zero
then
map_set_decr div_f s
else
map_set_incr div_f s
| Float _ -> degenerate_float
let scale_div ~pos f v =
scale_div_common ~pos f v (fun _ _ -> Int.zero, Int.one) top
;;
let scale_div_under ~pos f v =
try
scale_div_common ~pos f v (fun _r _m -> raise Exit) bottom
with Exit -> bottom
;;
let div_set x sy =
Array.fold_left
(fun acc elt ->
if Int.is_zero elt
then acc
else join acc (scale_div ~pos:false elt x))
bottom
sy
let div_range x ymn ymx =
match min_and_max x with
| Some xmn, Some xmx ->
let c1 = Int.c_div xmn ymn in
let c2 = Int.c_div xmx ymn in
let c3 = Int.c_div xmn ymx in
let c4 = Int.c_div xmx ymx in
let min = Int.min (Int.min c1 c2) (Int.min c3 c4) in
let max = Int.max (Int.max c1 c2) (Int.max c3 c4) in
inject_range (Some min) (Some max)
| _ ->
log_imprecision "Ival.div_range";
top
let div x y =
match y with
Set sy ->
div_set x sy
| Top (Some mn,Some mx, r, modu) ->
let result_pos =
if Int.gt mx Int.zero
then
let lpos =
if Int.gt mn Int.zero
then mn
else
Int.round_up_to_r ~min:Int.one ~r ~modu
in
div_range x lpos mx
else
bottom
in
let result_neg =
if Int.lt mn Int.zero
then
let gneg =
if Int.lt mx Int.zero
then mx
else
Int.round_down_to_r ~max:Int.minus_one ~r ~modu
in
div_range x mn gneg
else
bottom
in
join result_neg result_pos
| Float _ -> assert false
| Top (None, _, _, _) | Top (_, None, _, _) ->
log_imprecision "Ival.div";
top
let scale_rem ~pos f v =
if Int.is_zero f then bottom
else
let f = if Int.lt f Int.zero then Int.neg f else f in
let rem_f a =
if pos then Int.e_rem a f else Int.c_rem a f
in
match v with
| Top(mn,mx,r,m) ->
let modu = Int.pgcd f m in
let rr = Int.e_rem r modu in
let binf,bsup =
if pos
then (Int.round_up_to_r ~min:Int.zero ~r:rr ~modu),
(Int.round_down_to_r ~max:(Int.pred f) ~r:rr ~modu)
else
let min =
if all_positives mn then Int.zero else Int.neg (Int.pred f)
in
let max =
if all_negatives mx then Int.zero else Int.pred f
in
(Int.round_up_to_r ~min ~r:rr ~modu,
Int.round_down_to_r ~max ~r:rr ~modu)
in
let mn_rem,mx_rem =
match mn,mx with
| Some mn,Some mx ->
let div_f a =
if pos then Int.e_div a f else Int.c_div a f
in
if Int.equal (div_f mn) (div_f mx) then
rem_f mn, rem_f mx
else binf,bsup
| _ -> binf,bsup
in
inject_top (Some mn_rem) (Some mx_rem) rr modu
| Set s -> map_set_exnsafe rem_f s
| Float _ -> top
let c_rem x y =
match y with
| Top (None, _, _, _) | Top (_, None, _, _)
| Float _ -> top
| Top (Some mn, Some mx, _, _) ->
if Int.equal mx Int.zero then
bottom
else
let neg, pos, max_x = match x with
| Float _ -> true, true, None
| Set set ->
let s = Array.length set in
if s = 0 then false, false, None
else
Int.le set.(0) Int.minus_one,
Int.ge set.(s-1) Int.one,
Some (Int.max (Int.abs set.(0)) (Int.abs set.(s-1)))
| Top (mn, mx, _, _) ->
min_le_elt mn Int.minus_one,
max_ge_elt mx Int.one,
(match mn, mx with
| Some mn, Some mx -> Some (Int.max (Int.abs mn) (Int.abs mx))
| _ -> None)
in
let pos_rem = Integer.max (Int.abs mn) (Int.abs mx) in
let bound = Int.pred pos_rem in
let bound = Extlib.may_map (Int.min bound) ~dft:bound max_x in
let mn = if neg then Some (Int.neg bound) else Some Int.zero in
let mx = if pos then Some bound else Some Int.zero in
inject_top mn mx Int.zero Int.one
| Set yy ->
( match x with
Set xx -> apply2_notzero Int.c_rem xx yy
| Float _ -> top
| Top _ ->
let f acc y =
join (scale_rem ~pos:false y x) acc
in
Array.fold_left f bottom yy)
module AllValueHashtbl =
Hashtbl.Make
(struct
type t = Int.t * bool * int
let equal (a,b,c:t) (d,e,f:t) = b=e && c=f && Int.equal a d
let hash (a,b,c:t) =
257 * (Hashtbl.hash b) + 17 * (Hashtbl.hash c) + Int.hash a
end)
let all_values_table = AllValueHashtbl.create 7
let create_all_values_modu ~modu ~signed ~size =
let t = modu, signed, size in
try
AllValueHashtbl.find all_values_table t
with Not_found ->
let mn, mx =
if signed then
let b = Int.two_power_of_int (size-1) in
(Int.round_up_to_r ~min:(Int.neg b) ~modu ~r:Int.zero,
Int.round_down_to_r ~max:(Int.pred b) ~modu ~r:Int.zero)
else
let b = Int.two_power_of_int size in
Int.zero,
Int.round_down_to_r ~max:(Int.pred b) ~modu ~r:Int.zero
in
let r = inject_top (Some mn) (Some mx) Int.zero modu in
AllValueHashtbl.add all_values_table t r;
r
let create_all_values ~signed ~size =
if size <= !small_cardinal_log then
create_all_values_modu ~signed ~size ~modu:Int.one
else
if signed then
let b = Int.two_power_of_int (size-1) in
Top (Some (Int.neg b), Some (Int.pred b), Int.zero, Int.one)
else
let b = Int.two_power_of_int size in
Top (Some Int.zero, Some (Int.pred b), Int.zero, Int.one)
let big_int_64 = Int.of_int 64
let big_int_32 = Int.thirtytwo
let cast_int_to_int ~size ~signed value =
if equal top value
then create_all_values ~size:(Int.to_int size) ~signed
else
let result =
let factor = Int.two_power size in
let mask = Int.two_power (Int.pred size) in
let rem_f value = Int.cast ~size ~signed ~value
in
let not_p_factor = Int.neg factor in
let best_effort r m =
let modu = Int.pgcd factor m in
let rr = Int.e_rem r modu in
let min_val = Some (if signed then
Int.round_up_to_r ~min:(Int.neg mask) ~r:rr ~modu
else
Int.round_up_to_r ~min:Int.zero ~r:rr ~modu)
in
let max_val = Some (if signed then
Int.round_down_to_r ~max:(Int.pred mask) ~r:rr ~modu
else
Int.round_down_to_r ~max:(Int.pred factor)
~r:rr
~modu)
in
inject_top min_val max_val rr modu
in
match value with
| Top(Some mn,Some mx,r,m) ->
let highbits_mn,highbits_mx =
if signed then
Int.logand (Int.add mn mask) not_p_factor,
Int.logand (Int.add mx mask) not_p_factor
else
Int.logand mn not_p_factor, Int.logand mx not_p_factor
in
if Int.equal highbits_mn highbits_mx
then
if Int.is_zero highbits_mn
then value
else
let new_min = rem_f mn in
let new_r = Int.e_rem new_min m in
inject_top (Some new_min) (Some (rem_f mx)) new_r m
else best_effort r m
| Top (_,_,r,m) ->
best_effort r m
| Set s -> begin
let all =
create_all_values ~size:(Int.to_int size) ~signed
in
if is_included value all then value else map_set_exnsafe rem_f s
end
| Float _ -> assert false
in
if equal result value then value else result
let reinterpret_float_as_int ~signed ~size f =
let reinterpret_list l =
let reinterpret_one (b, e) =
let i = inject_range (Some b) (Some e) in
cast_int_to_int ~size ~signed i
in
let l = List.map reinterpret_one l in
List.fold_left join bottom l
in
if Int.equal size big_int_64
then
let itvs = Fval.bits_of_float64_list f in
reinterpret_list itvs
else
if Int.equal size big_int_32
then
let itvs = Fval.bits_of_float32_list f in
reinterpret_list itvs
else top
let reinterpret_as_int ~size ~signed i =
match i with
| Set _ | Top _ ->
cast_int_to_int ~signed ~size i
| Float f -> reinterpret_float_as_int ~signed ~size f
let cast_float_to_float fkind v =
match v with
| Float f ->
begin match fkind with
| Fval.Real | Fval.Long_Double | Fval.Double -> v
| Fval.Single ->
inject_float (Fval.round_to_single_precision_float f)
end
| Set _ when is_zero v -> zero
| Set _ | Top _ -> top_float
let rec mul v1 v2 =
let result =
if is_one v1 then v2
else if is_zero v2 || is_zero v1 then zero
else if is_one v2 then v1
else
match v1,v2 with
| Float _, _ | _, Float _ ->
top
| Set s1, Set [| x |] | Set [| x |], Set s1 ->
if Int.ge x Int.zero
then apply_bin_1_strict_incr Int.mul x s1
else apply_bin_1_strict_decr Int.mul x s1
| Set s1, Set s2 ->
apply2_n Int.mul s1 s2
| Top(mn1,mx1,r1,m1), Top(mn2,mx2,r2,m2) ->
check mn1 mx1 r1 m1;
check mn2 mx2 r2 m2;
let mn1 = inject_min mn1 in
let mx1 = inject_max mx1 in
let mn2 = inject_min mn2 in
let mx2 = inject_max mx2 in
let a = ext_mul mn1 mn2 in
let b = ext_mul mn1 mx2 in
let c = ext_mul mx1 mn2 in
let d = ext_mul mx1 mx2 in
let min = ext_min (ext_min a b) (ext_min c d) in
let max = ext_max (ext_max a b) (ext_max c d) in
let modu = Int.(pgcd (pgcd (mul m1 m2) (mul r1 m2)) (mul r2 m1))
in
let r = Int.e_rem (Int.mul r1 r2) modu in
inject_top (ext_proj min) (ext_proj max) r modu
| Set s, (Top(_,_,_,_) as t) | (Top(_,_,_,_) as t), Set s ->
let l = Array.length s in
if l = 0
then bottom
else if l = 1
then
scale s.(0) t
else mul t (unsafe_make_top_from_array s)
in
result
(** Computes [x (op) ({y >= 0} * 2^n)], as an auxiliary function for
[shift_left] and [shift_right]. [op] and [scale] must verify
[scale a b == op (inject_singleton a) b] *)
let shift_aux scale op (x: t) (y: t) =
let y = narrow (inject_range (Some Int.zero) None) y in
try
match y with
| Set s ->
Array.fold_left (fun acc n -> join acc (scale (Int.two_power n) x)) bottom s
| _ ->
let min_factor = Extlib.opt_map Int.two_power (min_int y) in
let max_factor = Extlib.opt_map Int.two_power (max_int y) in
let modu = match min_factor with None -> Int.one | Some m -> m in
let factor = inject_top min_factor max_factor Int.zero modu in
op x factor
with Z.Overflow ->
Codex_log.imprecision_warning "Ival.shift_aux";
if is_included x positive_integers then positive_integers
else
if is_included x negative_integers then negative_integers
else top
let shift_right x y = shift_aux (scale_div ~pos:true) div x y
let shift_left x y = shift_aux scale mul x y
let interp_boolean ~contains_zero ~contains_non_zero =
match contains_zero, contains_non_zero with
| true, true -> zero_or_one
| true, false -> zero
| false, true -> one
| false, false -> bottom
module Infty = struct
let lt0 = function
| None -> true
| Some a -> Int.lt a Int.zero
let div a b = match a with
| None -> None
| Some a -> match b with
| None -> Some Int.zero
| Some b -> Some (Int.e_div a b)
let neg = function
| Some a -> Some (Int.neg a)
| None -> None
end
let backward_mult_pos_left min_right max_right result =
let min_res, max_res = min_and_max result in
let min_left =
Infty.div min_res (if Infty.lt0 min_res then Some min_right else max_right)
and max_left =
Infty.div max_res (if Infty.lt0 max_res then max_right else Some min_right)
in
inject_range min_left max_left
let backward_mult_neg_left min_right max_right result =
backward_mult_pos_left (Integer.neg max_right) (Infty.neg min_right) (neg_int result)
let backward_mult_int_left ~right ~result =
match min_and_max right with
| None, None -> `Value None
| Some a, Some b when a > b -> `Bottom
| Some a, Some b when a = Int.zero && b = Int.zero ->
if contains_zero result then `Value None else `Bottom
| Some a, max when a > Int.zero ->
`Value (Some (backward_mult_pos_left a max result))
| Some a, max when a >= Int.zero ->
if contains_zero result
then `Value None
else `Value (Some (backward_mult_pos_left Int.one max result))
| min, Some b when b < Int.zero ->
`Value (Some (backward_mult_neg_left min b result))
| min, Some b when b = Int.zero ->
if contains_zero result
then `Value None
else `Value (Some (backward_mult_neg_left min Int.minus_one result))
| min, max ->
if contains_zero result
then `Value None
else
`Value (Some (join
(backward_mult_pos_left Int.one max result)
(backward_mult_neg_left min Int.one result)))
let backward_le_int max v =
match v with
| Float _ -> v
| Set _ | Top _ ->
narrow v (Top(None,max,Int.zero,Int.one))
let backward_ge_int min v =
match v with
| Float _ -> v
| Set _ | Top _ ->
narrow v (Top(min,None,Int.zero,Int.one))
let backward_lt_int max v = backward_le_int (opt1 Int.pred max) v
let backward_gt_int min v = backward_ge_int (opt1 Int.succ min) v
let diff_if_one value rem =
match rem, value with
| Set [| v |], Set a ->
let index = array_mem v a in
if index >= 0
then
let l = Array.length a in
let pl = pred l in
let r = Array.make pl Int.zero in
Array.blit a 0 r 0 index;
Array.blit a (succ index) r index (pl-index);
share_array r pl
else value
| Set [| v |], Top (Some mn, mx, r, m) when Int.equal v mn ->
inject_top (Some (Int.add mn m)) mx r m
| Set [| v |], Top (mn, Some mx, r, m) when Int.equal v mx ->
inject_top mn (Some (Int.sub mx m)) r m
| Set [| v |], Top ((Some mn as min), (Some mx as max), r, m) when
Int.equal (Int.sub mx mn) (Int.mul m !small_cardinal_Int) &&
in_interval v min max r m ->
let r = ref mn in
Set
(Array.init
!small_cardinal
(fun _ ->
let c = !r in
let corrected_c =
if Int.equal c v then Int.add c m else c
in
r := Int.add corrected_c m;
corrected_c))
| _ -> value
let diff value rem =
log_imprecision "Ival.diff";
diff_if_one value rem
let fold_int_bounds f v acc =
match v with
| Float _ -> f v acc
| Set _ | Top _ ->
if cardinal_zero_or_one v then f v acc
else
let on_bound b v acc = match b with
| None -> v, acc
| Some b ->
let b = inject_singleton b in
diff_if_one v b, f b acc
in
let min, max = min_and_max v in
let v, acc = on_bound min v acc in
let v, acc = on_bound max v acc in
if equal v bottom then acc else f v acc
let backward_comp_int_left op l r =
let open Comp in
try
match op with
| Le -> backward_le_int (max_int r) l
| Ge -> backward_ge_int (min_int r) l
| Lt -> backward_lt_int (max_int r) l
| Gt -> backward_gt_int (min_int r) l
| Eq -> narrow l r
| Ne -> diff_if_one l r
with Error_Bottom -> bottom
let backward_comp_float_left_true op fkind f1 f2 =
let f1 = project_float f1 in
let f2 = project_float f2 in
begin match Fval.backward_comp_left_true op fkind f1 f2 with
| `Value f -> inject_float f
| `Bottom -> bottom
end
let backward_comp_float_left_false op fkind f1 f2 =
let f1 = project_float f1 in
let f2 = project_float f2 in
begin match Fval.backward_comp_left_false op fkind f1 f2 with
| `Value f -> inject_float f
| `Bottom -> bottom
end
let rec ~start ~stop ~size v =
match v with
| Set s ->
inject_ps
(Array.fold_left
(fun acc elt -> add_ps acc (Int.extract_bits ~start ~stop elt))
empty_ps
s)
| Float f ->
let inject (b, e) = inject_range (Some b) (Some e) in
let itvs =
if Int.equal size big_int_64 then
List.map inject (Fval.bits_of_float64_list f)
else if Int.equal size big_int_32 then
List.map inject (Fval.bits_of_float32_list f)
else
[top]
in
let bits = List.map (extract_bits ~start ~stop ~size) itvs in
List.fold_left join bottom bits
| Top(_,_,_,_) as d ->
try
let dived = scale_div ~pos:true (Int.two_power start) d in
scale_rem ~pos:true (Int.two_power (Int.length start stop)) dived
with Z.Overflow ->
Codex_log.imprecision_warning "Ival.extract_bits";
top
;;
let all_values ~size v =
if Int.lt big_int_64 size then false
else
match v with
| Float _ -> false
| Top (None,_,_,modu) | Top (_,None,_,modu) ->
Int.is_one modu
| Top (Some mn, Some mx,_,modu) ->
Int.is_one modu &&
Int.le
(Int.two_power size)
(Int.length mn mx)
| Set s ->
let siz = Int.to_int size in
Array.length s >= 1 lsl siz &&
equal
(cast_int_to_int ~size ~signed:false v)
(create_all_values ~size:siz ~signed:false)
let compare_min_max min max =
match min, max with
| None,_ -> -1
| _,None -> -1
| Some min, Some max -> Int.compare min max
let compare_max_min max min =
match max, min with
| None,_ -> 1
| _,None -> 1
| Some max, Some min -> Int.compare max min
let forward_le_int i1 i2 =
if compare_max_min (max_int i1) (min_int i2) <= 0 then Comp.True
else if compare_min_max (min_int i1) (max_int i2) > 0 then Comp.False
else Comp.Unknown
let forward_lt_int i1 i2 =
if compare_max_min (max_int i1) (min_int i2) < 0 then Comp.True
else if compare_min_max (min_int i1) (max_int i2) >= 0 then Comp.False
else Comp.Unknown
let forward_eq_int i1 i2 =
if cardinal_zero_or_one i1 && equal i1 i2 then Comp.True
else if intersects i2 i2 then Comp.Unknown
else Comp.False
let forward_comp_int op i1 i2 =
let open Abstract_interp.Comp in
match op with
| Le -> forward_le_int i1 i2
| Ge -> forward_le_int i2 i1
| Lt -> forward_lt_int i1 i2
| Gt -> forward_lt_int i2 i1
| Eq -> forward_eq_int i1 i2
| Ne -> inv_truth (forward_eq_int i1 i2)
let rehash x =
match x with
| Set a -> share_array a (Array.length a)
| _ -> x
type overflow_float_to_int =
| FtI_Ok of Int.t
| FtI_Overflow of Floating_point.sign
let cast_float_to_int_non_nan ~signed ~size (min, max) =
let all = create_all_values ~size ~signed in
let min_all = Extlib.the (min_int all) in
let max_all = Extlib.the (max_int all) in
let conv f =
try
let i = Floating_point.truncate_to_integer f in
if Int.ge i min_all then
if Int.le i max_all then FtI_Ok i
else FtI_Overflow Floating_point.Pos
else FtI_Overflow Floating_point.Neg
with Floating_point.Float_Non_representable_as_Int64 sign ->
FtI_Overflow sign
in
let min_int = conv (Fval.F.to_float min) in
let max_int = conv (Fval.F.to_float max) in
match min_int, max_int with
| FtI_Ok min_int, FtI_Ok max_int ->
inject_range (Some min_int) (Some max_int)
| FtI_Overflow Floating_point.Neg, FtI_Ok max_int ->
inject_range (Some min_all) (Some max_int)
| FtI_Ok min_int, FtI_Overflow Floating_point.Pos ->
inject_range (Some min_int) (Some max_all)
| FtI_Overflow Floating_point.Neg, FtI_Overflow Floating_point.Pos ->
inject_range (Some min_all) (Some max_all)
| FtI_Overflow Floating_point.Pos, FtI_Overflow Floating_point.Pos
| FtI_Overflow Floating_point.Neg, FtI_Overflow Floating_point.Neg ->
bottom
| FtI_Overflow Floating_point.Pos, FtI_Overflow Floating_point.Neg
| FtI_Overflow Floating_point.Pos, FtI_Ok _
| FtI_Ok _, FtI_Overflow Floating_point.Neg ->
assert false
let cast_float_to_int ~signed ~size iv =
match Fval.min_and_max (project_float iv) with
| Some (min, max), _nan -> cast_float_to_int_non_nan ~signed ~size (min, max)
| None, _ -> bottom
let double_min_exact_integer = Int.neg (Int.two_power_of_int 53)
let double_max_exact_integer = Int.two_power_of_int 53
let single_min_exact_integer = Int.neg (Int.two_power_of_int 24)
let single_max_exact_integer = Int.two_power_of_int 24
let double_min_exact_integer_d = -. (2. ** 53.)
let double_max_exact_integer_d = 2. ** 53.
let single_min_exact_integer_d = -. (2. ** 24.)
let single_max_exact_integer_d = 2. ** 24.
let cast_float_to_int_inverse ~single_precision i =
let exact_min, exact_max =
if single_precision
then single_min_exact_integer, single_max_exact_integer
else double_min_exact_integer, double_max_exact_integer
in
let fkind = if single_precision then Fval.Single else Fval.Double in
match min_and_max i with
| Some min, Some max when Int.lt exact_min min && Int.lt max exact_max ->
let minf =
if Int.le min Int.zero then
Fval.F.next_float fkind (Int.to_float (Int.pred min))
else
Int.to_float min
in
let maxf =
if Int.le Int.zero max
then
Fval.F.prev_float fkind (Int.to_float (Int.succ max))
else Int.to_float max
in
assert (Fval.F.is_finite (Fval.F.of_float minf));
assert (Fval.F.is_finite (Fval.F.of_float maxf));
Float (Fval.inject fkind (Fval.F.of_float minf) (Fval.F.of_float maxf))
| _ -> if single_precision then top_single_precision_float else top_float
let cast_int_to_float_inverse_not_nan ~single_precision (min, max) =
let exact_min, exact_max =
if single_precision
then single_min_exact_integer_d, single_max_exact_integer_d
else double_min_exact_integer_d, double_max_exact_integer_d
in
let min = Fval.F.to_float min in
let max = Fval.F.to_float max in
if exact_min <= min && max <= exact_max then
let min = ceil min in
let max = floor max in
let conv f = try Some (Integer.of_float f) with Z.Overflow -> None in
let r = inject_range (conv min) (conv max) in
r
else top
let cast_int_to_float_inverse ~single_precision f =
match min_and_max_float f with
| None, _ -> bottom
| Some (min, max), _
->
cast_int_to_float_inverse_not_nan ~single_precision (min, max)
let of_int i = inject_singleton (Int.of_int i)
let of_int64 i = inject_singleton (Int.of_int64 i)
let cast_int_to_float fkind v =
let min,max = min_and_max v in
inject_float (Fval.cast_int_to_float fkind min max)
let reinterpret_as_float kind i =
match i with
| Float _ -> i
| Set _ when is_zero i || is_bottom i -> i
| Top _ | Set _ ->
let reinterpret size kind conv min_f max_f =
let size = Integer.of_int size in
let i = cast_int_to_int ~size ~signed:true i in
let bounds_narrow i' =
let r = narrow i i' in
if is_bottom r then `Bottom
else
match min_and_max r with
| None, _ | _, None -> assert false
| Some b, Some e -> `Value (b, e)
in
let s_max_f = Int.succ max_f in
let s_min_f = Int.succ min_f in
let s_s_max_f = Int.succ s_max_f in
let s_s_min_f = Int.succ s_min_f in
let f_pos = inject_range (Some Integer.zero) (Some s_max_f) in
let f_neg = inject_range None (Some s_min_f) in
let nan_pos = inject_range (Some s_s_max_f) None in
let nan_neg = inject_range (Some s_s_min_f) (Some Int.minus_one) in
let nan =
if intersects i nan_pos || intersects i nan_neg
then [`Value Fval.nan]
else []
in
let open Bottom in
let range mn mx = Fval.inject kind (conv mn) (conv mx) in
let pos = bounds_narrow f_pos >>-: (fun (b, e) -> range b e) in
let neg = bounds_narrow f_neg >>-: (fun (b, e) -> range e b) in
let f = Bottom.join_list Fval.join (pos :: neg :: nan) in
inject_float (Bottom.non_bottom f)
in
let open Floating_point in
match kind with
| _ -> top_float
let overlaps ~partial ~size t1 t2 =
let diff = sub_int t1 t2 in
match diff with
| Set array ->
not (array_for_all
(fun i -> Int.ge (Int.abs i) size || (partial && Int.is_zero i))
array)
| Top (min, max, _r, _modu) ->
let pred_size = Int.pred size in
min_le_elt min pred_size && max_ge_elt max (Int.neg pred_size)
| Float _ -> assert false
type bit_value = On | Off | Both
module Bit =
struct
type t = bit_value
let to_string = function
| Off -> "0"
| On -> "1"
| Both -> "T"
let _pretty (fmt : Format.formatter) (b :t) =
Format.pp_print_string fmt (to_string b)
let union (b1 : t) (b2 : t) : t =
if b1 = b2 then b1 else Both
let not : t -> t = function
| On -> Off
| Off -> On
| Both -> Both
end
module type BitOperator =
sig
val representation : string
val forward : bit_value -> bit_value -> bit_value
val backward_off : bit_value -> bit_value
val backward_on : bit_value -> bit_value
end
module And : BitOperator =
struct
let representation = "&"
let forward v1 v2 =
match v1 with
| Off -> Off
| On -> v2
| Both -> if v2 = Off then Off else Both
let backward_off = function
| (Off | Both) -> Both
| On -> Off
let backward_on = function
| Off -> assert false
| (On | Both) -> On
end
module Or : BitOperator =
struct
let representation = "|"
let forward v1 v2 =
match v1 with
| On -> On
| Off -> v2
| Both -> if v2 = On then On else Both
let backward_off = function
| On -> assert false
| (Off | Both) -> Off
let backward_on = function
| (On | Both) -> Both
| Off -> On
end
module Xor : BitOperator =
struct
let representation = "^"
let forward v1 v2 =
match v1 with
| Both -> Both
| Off -> v2
| On -> Bit.not v2
let backward_on v = Bit.not v
let backward_off v = v
end
let significant_bits (v : t) : int option =
match min_and_max v with
| None, _ | _, None -> None
| Some l, Some u -> Some (max (Z.numbits l) (Z.numbits u))
let (v : t) : bit_value =
match min_and_max v with
| _, Some u when Int.(lt u zero) -> On
| Some l, _ when Int.(ge l zero) -> Off
| _, _ -> Both
let extract_bit (i : int) (v : t) : bit_value =
let bit_value x = if Z.testbit x i then On else Off in
match v with
| Float _ -> Both
| Set s -> array_map_reduce bit_value Bit.union s
| Top (None, _, _r, _m) | Top (_, None, _r, _m) -> Both
| Top (Some l, Some u, _r, _m) ->
if Int.(ge (sub u l) (two_power_of_int i))
then Both
else Bit.union (bit_value l) (bit_value u)
let reduce_sign (v : t) (b : bit_value) : t =
match b with
| Both -> v
| On ->
begin match v with
| Float _ -> v
| Set s -> array_filter Int.(gt zero) s
| Top (_l, Some u, _r, _modu) when Int.(lt u zero) -> v
| Top (l, _u, r, modu) ->
let u = Some Int.(round_down_to_r ~max:minus_one ~r ~modu) in
inject_top l u r modu
end
| Off ->
begin match v with
| Float _ -> v
| Set s -> array_filter Int.(le zero) s
| Top (Some l, _u, _r, _modu) when Int.(ge l zero) -> v
| Top (_l, u, r, modu) ->
let l = Some Int.(round_up_to_r ~min:zero ~r ~modu) in
inject_top l u r modu
end
let reduce_bit (i : int) (v : t) (b : bit_value) : t =
let bit_value x = if Z.testbit x i then On else Off in
if b = Both
then v
else match v with
| Float _ -> v
| Set s -> array_filter (fun x -> bit_value x = b) s
| Top (l, u, r, modu) ->
let power = Int.(two_power_of_int i) in
let mask = Int.(pred (two_power_of_int (i+1))) in
let l' = match l with
| Some l when bit_value l <> b ->
let min = match b with
| On -> Int.(logor (logand l (lognot mask)) power)
| Off -> Int.(succ (logor l mask))
| Both -> assert false
in
Some (Int.round_up_to_r ~min ~r ~modu)
| _ -> l
and u' = match u with
| Some u when bit_value u <> b ->
let max = match b with
| On -> Int.(pred (logand u (lognot mask)))
| Off -> Int.(logand (logor u mask) (lognot power))
| Both -> assert false
in
Some (Int.round_down_to_r ~max ~r ~modu)
| _ -> u
in
inject_top l' u' r modu
type bit = Sign | Bit of int
let = function
| Sign -> extract_sign
| Bit i -> extract_bit i
let set_bit_on ~size bit =
let mask = match bit with
| Sign -> Int.(neg (two_power_of_int size))
| Bit i -> Int.(two_power_of_int i)
in
fun v -> Int.logor mask v
let reduce_bit = function
| Sign -> reduce_sign
| Bit i -> reduce_bit i
module BitwiseOperator (Op : BitOperator) =
struct
let backward (b : bit_value) = function
| On -> Op.backward_on b
| Off -> Op.backward_off b
| Both -> assert false
(** Bit masks are composed of an array of significant bit values where index 0
represents the lowest bit, and a single bit_value to represent the
possible leading bits. *)
type bit_mask = bit_value array * bit_value
let int_to_bit_array n (x : Int.t) =
let make i = if Z.testbit x i then On else Off in
Array.init n make
let low_bit_mask : t -> bit_mask = function
| Set [| |] -> raise Error_Bottom
| Set [| x |] ->
let n = Z.numbits x in
int_to_bit_array n x, if Int.(ge x zero) then Off else On
| v ->
let _,_,r,modu = min_max_r_mod v in
let n = Z.trailing_zeros modu in
int_to_bit_array n r, Both
let compute_modulo v1 v2 =
let b1, s1 = low_bit_mask v1
and b2, s2 = low_bit_mask v2 in
let size = max (Array.length b1) (Array.length b2) in
let rec step i rem =
let b1 = try b1.(i) with _ -> s1
and b2 = try b2.(i) with _ -> s2 in
let b = Op.forward b1 b2 in
if i >= size || b = Both
then rem, Int.two_power_of_int i
else
let rem = if b = On then set_bit_on ~size (Bit i) rem else rem in
step (i+1) rem
in
step 0 Int.zero
let result_size (v1 : t) (v2 : t) : int option =
let n1 = significant_bits v1 and n2 = significant_bits v2 in
let n1_greater =
match n1, n2 with
| None, _ -> true
| _, None -> false
| Some n1, Some n2 -> n1 >= n2
in
if n1_greater
then if Op.forward Both (extract_sign v2) = Both then n1 else n2
else if Op.forward (extract_sign v1) Both = Both then n2 else n1
exception Do_not_fit_small_sets
let compute_small_set ~size (v1 : t) (v2 : t) (r : Int.t) (modu : Int.t) =
let set_bit i acc (r, v1, v2) =
let b1 = extract_bit i v1
and b2 = extract_bit i v2 in
match Op.forward b1 b2 with
| On -> (set_bit_on ~size i r, v1, v2) :: acc
| Off -> (r, v1, v2) :: acc
| Both ->
let v1_off = reduce_bit i v1 (Op.backward_off b2)
and v2_off = reduce_bit i v2 (Op.backward_off b1) in
let v1_on = reduce_bit i v1 (Op.backward_on b2)
and v2_on = reduce_bit i v2 (Op.backward_on b1) in
(set_bit_on ~size i r, v1_on, v2_on) :: (r, v1_off, v2_off) :: acc
in
let acc = ref (set_bit Sign [] (r, v1, v2)) in
for i = size - 1 downto Z.numbits modu - 1 do
acc := List.fold_left (set_bit (Bit i)) [] !acc;
if List.length !acc > !small_cardinal then raise Do_not_fit_small_sets
done;
let o = List.fold_left (fun o (r,_,_) -> O.add r o) O.empty !acc in
share_set o (O.cardinal o)
let compute_bound ~size v1 v2 lower =
let set_bit i (r, v1, v2) =
let b1 = extract_bit i v1
and b2 = extract_bit i v2 in
let b, v1, v2 =
match Op.forward b1 b2 with
| On | Off as b -> b, v1, v2
| Both ->
let b = match i with
| Sign -> if lower then On else Off
| Bit _ -> if lower then Off else On
in
let v1 = reduce_bit i v1 (backward b2 b)
and v2 = reduce_bit i v2 (backward b1 b) in
b, v1, v2
in
let r = if b = On then set_bit_on ~size i r else r in
r, v1, v2
in
let r = ref (Int.zero, v1, v2) in
r := set_bit Sign !r;
for i = (size - 1) downto 0 do
r := set_bit (Bit i) !r;
done;
let bound, _v1, _v2 = !r in
bound
let bitwise_forward (v1 : t) (v2 : t) : t =
try
let r, modu = compute_modulo v1 v2 in
match result_size v1 v2 with
| None ->
inject_interval None None r modu
| Some size ->
try compute_small_set ~size v1 v2 r modu
with Do_not_fit_small_sets ->
let min = compute_bound ~size v1 v2 true
and max = compute_bound ~size v1 v2 false in
inject_interval (Some min) (Some max) r modu
with Error_Bottom -> bottom
end
let bitwise_or = let module M = BitwiseOperator (Or) in M.bitwise_forward
let bitwise_and = let module M = BitwiseOperator (And) in M.bitwise_forward
let bitwise_xor = let module M = BitwiseOperator (Xor) in M.bitwise_forward
let bitwise_signed_not v =
match v with
| Float _ -> assert false
| Top _ -> add_int (neg_int v) minus_one
| Set s -> map_set_strict_decr Int.lognot s
let bitwise_unsigned_not ~size v =
let size = Int.of_int size in
cast_int_to_int ~size ~signed:false (bitwise_signed_not v)
let bitwise_not ~size ~signed v =
if signed then
bitwise_signed_not v
else
bitwise_unsigned_not ~size v
let pretty_debug = pretty
let name = "ival"