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+(************************************************************************)
+(* v * The Coq Proof Assistant / The Coq Development Team *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(* \VV/ **************************************************************)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(************************************************************************)
+
+(* $Id: bigint.ml 7305 2005-08-19 19:51:02Z letouzey $ *)
+
+(*i*)
+open Pp
+(*i*)
+
+(***************************************************)
+(* Basic operations on (unbounded) integer numbers *)
+(***************************************************)
+
+(* An integer is canonically represented as an array of k-digits blocs.
+
+ 0 is represented by the empty array and -1 by the singleton [|-1|].
+ The first bloc is in the range ]0;10^k[ for positive numbers.
+ The first bloc is in the range ]-10^k;-1[ for negative ones.
+ All other blocs are numbers in the range [0;10^k[.
+
+ Negative numbers are represented using 2's complementation. For instance,
+ with 4-digits blocs, [-9655;6789] denotes -96543211
+*)
+
+(* The base is a power of 10 in order to facilitate the parsing and printing
+ of numbers in digital notation.
+
+ All functions, to the exception of to_string and of_string should work
+ with an arbitrary base, even if not a power of 10.
+
+ In practice, we set k=4 so that no overflow in ocaml machine words
+ (i.e. the interval [-2^30;2^30-1]) occur when multiplying two
+ numbers less than (10^k)
+*)
+
+(* The main parameters *)
+
+let size =
+ let rec log10 n = if n < 10 then 0 else 1 + log10 (n / 10) in
+ (log10 max_int) / 2
+
+let format_size =
+ (* How to parametrize a printf format *)
+ if size = 4 then Printf.sprintf "%04d"
+ else fun n ->
+ let rec aux j l n =
+ if j=size then l else aux (j+1) (string_of_int (n mod 10) :: l) (n/10)
+ in String.concat "" (aux 0 [] n)
+
+(* The base is 10^size *)
+let base =
+ let rec exp10 = function 0 -> 1 | n -> 10 * exp10 (n-1) in exp10 size
+
+(* Basic numbers *)
+let zero = [||]
+let neg_one = [|-1|]
+
+(* Sign of an integer *)
+let is_strictly_neg n = n<>[||] && n.(0) < 0
+let is_strictly_pos n = n<>[||] && n.(0) > 0
+let is_neg_or_zero n = n=[||] or n.(0) < 0
+let is_pos_or_zero n = n=[||] or n.(0) > 0
+
+let normalize_pos n =
+ let k = ref 0 in
+ while !k < Array.length n & n.(!k) = 0 do incr k done;
+ Array.sub n !k (Array.length n - !k)
+
+let normalize_neg n =
+ let k = ref 1 in
+ while !k < Array.length n & n.(!k) = base - 1 do incr k done;
+ let n' = Array.sub n !k (Array.length n - !k) in
+ if Array.length n' = 0 then [|-1|] else (n'.(0) <- n'.(0) - base; n')
+
+let rec normalize n =
+ if Array.length n = 0 then n else
+ if n.(0) = -1 then normalize_neg n else normalize_pos n
+
+let neg m =
+ if m = zero then zero else
+ let n = Array.copy m in
+ let i = ref (Array.length m - 1) in
+ while !i > 0 & n.(!i) = 0 do decr i done;
+ if !i > 0 then begin
+ n.(!i) <- base - n.(!i); decr i;
+ while !i > 0 do n.(!i) <- base - 1 - n.(!i); decr i done;
+ n.(0) <- - n.(0) - 1;
+ if n.(0) < -1 then (n.(0) <- n.(0) + base; Array.append [| -1 |] n) else
+ if n.(0) = - base then (n.(0) <- 0; Array.append [| -1 |] n)
+ else normalize n
+ end else (n.(0) <- - n.(0); n)
+
+let push_carry r j =
+ let j = ref j in
+ while !j > 0 & r.(!j) < 0 do
+ r.(!j) <- r.(!j) + base; decr j; r.(!j) <- r.(!j) - 1
+ done;
+ while !j > 0 & r.(!j) >= base do
+ r.(!j) <- r.(!j) - base; decr j; r.(!j) <- r.(!j) + 1
+ done;
+ if r.(0) >= base then (r.(0) <- r.(0) - base; Array.append [| 1 |] r)
+ else if r.(0) < -base then (r.(0) <- r.(0) + 2*base; Array.append [| -2 |] r)
+ else if r.(0) = -base then (r.(0) <- 0; Array.append [| -1 |] r)
+ else normalize r
+
+let add_to r a j =
+ if a = zero then r else begin
+ for i = Array.length r - 1 downto j+1 do
+ r.(i) <- r.(i) + a.(i-j);
+ if r.(i) >= base then (r.(i) <- r.(i) - base; r.(i-1) <- r.(i-1) + 1)
+ done;
+ r.(j) <- r.(j) + a.(0);
+ push_carry r j
+ end
+
+let add n m =
+ let d = Array.length n - Array.length m in
+ if d > 0 then add_to (Array.copy n) m d else add_to (Array.copy m) n (-d)
+
+let sub_to r a j =
+ if a = zero then r else begin
+ for i = Array.length r - 1 downto j+1 do
+ r.(i) <- r.(i) - a.(i-j);
+ if r.(i) < 0 then (r.(i) <- r.(i) + base; r.(i-1) <- r.(i-1) - 1)
+ done;
+ r.(j) <- r.(j) - a.(0);
+ push_carry r j
+ end
+
+let sub n m =
+ let d = Array.length n - Array.length m in
+ if d >= 0 then sub_to (Array.copy n) m d
+ else let r = neg m in add_to r n (Array.length r - Array.length n)
+
+let rec mult m n =
+ if m = zero or n = zero then zero else
+ let l = Array.length m + Array.length n in
+ let r = Array.create l 0 in
+ for i = Array.length m - 1 downto 0 do
+ for j = Array.length n - 1 downto 0 do
+ let p = m.(i) * n.(j) + r.(i+j+1) in
+ let (q,s) =
+ if p < 0
+ then (p + 1) / base - 1, (p + 1) mod base + base - 1
+ else p / base, p mod base in
+ r.(i+j+1) <- s;
+ if q <> 0 then r.(i+j) <- r.(i+j) + q;
+ done
+ done;
+ normalize r
+
+let rec less_than_same_size m n i j =
+ i < Array.length m &&
+ (m.(i) < n.(j) or (m.(i) = n.(j) && less_than_same_size m n (i+1) (j+1)))
+
+let less_than m n =
+ if is_strictly_neg m then
+ is_pos_or_zero n or Array.length m > Array.length n
+ or (Array.length m = Array.length n && less_than_same_size m n 0 0)
+ else
+ is_strictly_pos n && (Array.length m < Array.length n or
+ (Array.length m = Array.length n && less_than_same_size m n 0 0))
+
+let equal m n = (m = n)
+
+let less_or_equal_than m n = equal m n or less_than m n
+
+let less_than_shift_pos k m n =
+ (Array.length m - k < Array.length n)
+ or (Array.length m - k = Array.length n && less_than_same_size m n k 0)
+
+let rec can_divide k m d i =
+ (i = Array.length d) or
+ (m.(k+i) > d.(i)) or
+ (m.(k+i) = d.(i) && can_divide k m d (i+1))
+
+(* computes m - d * q * base^(|m|-k) in-place on positive numbers *)
+let sub_mult m d q k =
+ if q <> 0 then
+ for i = Array.length d - 1 downto 0 do
+ let v = d.(i) * q in
+ m.(k+i) <- m.(k+i) - v mod base;
+ if m.(k+i) < 0 then (m.(k+i) <- m.(k+i) + base; m.(k+i-1) <- m.(k+i-1) -1);
+ if v >= base then m.(k+i-1) <- m.(k+i-1) - v / base;
+ done
+
+let euclid m d =
+ let isnegm, m =
+ if is_strictly_neg m then (-1),neg m else 1,Array.copy m in
+ let isnegd, d = if is_strictly_neg d then (-1),neg d else 1,d in
+ if d = zero then raise Division_by_zero;
+ let q,r =
+ if less_than m d then (zero,m) else
+ let ql = Array.length m - Array.length d in
+ let q = Array.create (ql+1) 0 in
+ let i = ref 0 in
+ while not (less_than_shift_pos !i m d) do
+ if m.(!i)=0 then incr i else
+ if can_divide !i m d 0 then begin
+ let v =
+ if Array.length d > 1 && d.(0) <> m.(!i) then
+ (m.(!i) * base + m.(!i+1)) / (d.(0) * base + d.(1) + 1)
+ else
+ m.(!i) / d.(0) in
+ q.(!i) <- q.(!i) + v;
+ sub_mult m d v !i
+ end else begin
+ let v = (m.(!i) * base + m.(!i+1)) / (d.(0) + 1) in
+ q.(!i) <- q.(!i) + v / base;
+ sub_mult m d (v / base) !i;
+ q.(!i+1) <- q.(!i+1) + v mod base;
+ if q.(!i+1) >= base then
+ (q.(!i+1) <- q.(!i+1)-base; q.(!i) <- q.(!i)+1);
+ sub_mult m d (v mod base) (!i+1)
+ end
+ done;
+ (normalize q, normalize m) in
+ (if isnegd * isnegm = -1 then neg q else q),
+ (if isnegm = -1 then neg r else r)
+
+(* Parsing/printing ordinary 10-based numbers *)
+
+let of_string s =
+ let isneg = String.length s > 1 & s.[0] = '-' in
+ let n = if isneg then 1 else 0 in
+ let d = ref n in
+ while !d < String.length s && s.[!d] = '0' do incr d done;
+ if !d = String.length s then zero else
+ let r = (String.length s - !d) mod size in
+ let h = String.sub s (!d) r in
+ if !d = String.length s - 1 && isneg && h="1" then neg_one else
+ let e = if h<>"" then 1 else 0 in
+ let l = (String.length s - !d) / size in
+ let a = Array.create (l + e + n) 0 in
+ if isneg then begin
+ a.(0) <- (-1);
+ let carry = ref 0 in
+ for i=l downto 1 do
+ let v = int_of_string (String.sub s ((i-1)*size + !d +r) size)+ !carry in
+ if v <> 0 then (a.(i+e)<- base - v; carry := 1) else carry := 0
+ done;
+ if e=1 then a.(1) <- base - !carry - int_of_string h;
+ end
+ else begin
+ if e=1 then a.(0) <- int_of_string h;
+ for i=1 to l do
+ a.(i+e-1) <- int_of_string (String.sub s ((i-1)*size + !d + r) size)
+ done
+ end;
+ a
+
+let to_string_pos sgn n =
+ if Array.length n = 0 then "0" else
+ sgn ^
+ String.concat ""
+ (string_of_int n.(0) :: List.map format_size (List.tl (Array.to_list n)))
+
+let to_string n =
+ if is_strictly_neg n then to_string_pos "-" (neg n)
+ else to_string_pos "" n
+
+(******************************************************************)
+(* Optimized operations on (unbounded) integer numbers *)
+(* integers smaller than base are represented as machine integers *)
+(******************************************************************)
+
+type bigint = Obj.t
+
+let ints_of_int n =
+ if n >= base then [| n / base; n mod base |]
+ else if n <= - base then [| n / base - 1; n mod base + base |]
+ else if n = 0 then [| |] else [| n |]
+
+let big_of_int n =
+ if n >= base then Obj.repr [| n / base; n mod base |]
+ else if n <= - base then Obj.repr [| n / base - 1; n mod base + base |]
+ else Obj.repr n
+
+let big_of_ints n =
+ let n = normalize n in
+ if n = zero then Obj.repr 0 else
+ if Array.length n = 1 then Obj.repr n.(0) else
+ Obj.repr n
+
+let coerce_to_int = (Obj.magic : Obj.t -> int)
+let coerce_to_ints = (Obj.magic : Obj.t -> int array)
+
+let ints_of_z n =
+ if Obj.is_int n then ints_of_int (coerce_to_int n)
+ else coerce_to_ints n
+
+let app_pair f (m, n) =
+ (f m, f n)
+
+let add m n =
+ if Obj.is_int m & Obj.is_int n
+ then big_of_int (coerce_to_int m + coerce_to_int n)
+ else big_of_ints (add (ints_of_z m) (ints_of_z n))
+
+let sub m n =
+ if Obj.is_int m & Obj.is_int n
+ then big_of_int (coerce_to_int m - coerce_to_int n)
+ else big_of_ints (sub (ints_of_z m) (ints_of_z n))
+
+let mult m n =
+ if Obj.is_int m & Obj.is_int n
+ then big_of_int (coerce_to_int m * coerce_to_int n)
+ else big_of_ints (mult (ints_of_z m) (ints_of_z n))
+
+let euclid m n =
+ if Obj.is_int m & Obj.is_int n
+ then app_pair big_of_int
+ (coerce_to_int m / coerce_to_int n, coerce_to_int m mod coerce_to_int n)
+ else app_pair big_of_ints (euclid (ints_of_z m) (ints_of_z n))
+
+let less_than m n =
+ if Obj.is_int m & Obj.is_int n
+ then coerce_to_int m < coerce_to_int n
+ else less_than (ints_of_z m) (ints_of_z n)
+
+let neg n =
+ if Obj.is_int n then big_of_int (- (coerce_to_int n))
+ else big_of_ints (neg (ints_of_z n))
+
+let of_string m = big_of_ints (of_string m)
+let to_string m = to_string (ints_of_z m)
+
+let zero = big_of_int 0
+let one = big_of_int 1
+let sub_1 n = sub n one
+let add_1 n = add n one
+let two = big_of_int 2
+let neg_two = big_of_int (-2)
+let mult_2 n = add n n
+let is_zero n = n=zero
+
+let div2_with_rest n =
+ let (q,b) = euclid n two in
+ (q, b = one)
+
+let is_strictly_neg n = is_strictly_neg (ints_of_z n)
+let is_strictly_pos n = is_strictly_pos (ints_of_z n)
+let is_neg_or_zero n = is_neg_or_zero (ints_of_z n)
+let is_pos_or_zero n = is_pos_or_zero (ints_of_z n)
+
+let pr_bigint n = str (to_string n)
+
+(* Testing suite *)
+
+let check () =
+ let numbers = [
+ "1";"2";"99";"100";"101";"9999";"10000";"10001";
+ "999999";"1000000";"1000001";"99999999";"100000000";"100000001";
+ "1234";"5678";"12345678";"987654321";
+ "-1";"-2";"-99";"-100";"-101";"-9999";"-10000";"-10001";
+ "-999999";"-1000000";"-1000001";"-99999999";"-100000000";"-100000001";
+ "-1234";"-5678";"-12345678";"-987654321";"0"
+ ]
+ in
+ let eucl n m =
+ let n' = abs_float n and m' = abs_float m in
+ let q' = floor (n' /. m') in let r' = n' -. m' *. q' in
+ (if n *. m < 0. & q' <> 0. then -. q' else q'),
+ (if n < 0. then -. r' else r') in
+ let round f = floor (abs_float f +. 0.5) *. (if f < 0. then -1. else 1.) in
+ let i = ref 0 in
+ let compare op n n' =
+ incr i;
+ let s = Printf.sprintf "%30s" (to_string n) in
+ let s' = Printf.sprintf "% 30.0f" (round n') in
+ if s <> s' then Printf.printf "%s: %s <> %s\n" op s s' in
+List.iter (fun a -> List.iter (fun b ->
+ let n = of_string a and m = of_string b in
+ let n' = float_of_string a and m' = float_of_string b in
+ let a = add n m and a' = n' +. m' in
+ let s = sub n m and s' = n' -. m' in
+ let p = mult n m and p' = n' *. m' in
+ let q,r = try euclid n m with Division_by_zero -> zero,zero
+ and q',r' = eucl n' m' in
+ compare "+" a a';
+ compare "-" s s';
+ compare "*" p p';
+ compare "/" q q';
+ compare "%" r r') numbers) numbers;
+ Printf.printf "%i tests done\n" !i
+
+
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