/lib/bigint.ml
OCaml | 504 lines | 297 code | 70 blank | 137 comment | 137 complexity | 654e9b170946fab2fab7bef1ed4b4a48 MD5 | raw file
Possible License(s): LGPL-2.1
- (************************************************************************)
- (* v * The Coq Proof Assistant / The Coq Development Team *)
- (* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
- (* \VV/ **************************************************************)
- (* // * This file is distributed under the terms of the *)
- (* * GNU Lesser General Public License Version 2.1 *)
- (************************************************************************)
- (***************************************************)
- (* Basic operations on (unbounded) integer numbers *)
- (***************************************************)
- (* An integer is canonically represented as an array of k-digits blocs,
- i.e. in base 10^k.
- 0 is represented by the empty array and -1 by the singleton [|-1|].
- The first bloc is in the range ]0;base[ for positive numbers.
- The first bloc is in the range [-base;-1[ for numbers < -1.
- All other blocs are numbers in the range [0;base[.
- Negative numbers are represented using 2's complementation :
- one unit is "borrowed" from the top block for complementing
- the other blocs. For instance, with 4-digits blocs,
- [|-5;6789|] denotes -43211
- since -5.10^4+6789=-((4.10^4)+(10000-6789)) = -43211
- 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 on 32-bits machines, 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). On 64-bits machines, k=9.
- *)
- (* 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 if size = 9 then Printf.sprintf "%09d"
- 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
- (******************************************************************)
- (* First, we represent all numbers by int arrays.
- Later, we will optimize the particular case of small integers *)
- (******************************************************************)
- module ArrayInt = struct
- (* Basic numbers *)
- let zero = [||]
- let neg_one = [|-1|]
- (* An array is canonical when
- - it is empty
- - it is [|-1|]
- - its first bloc is in [-base;-1[U]0;base[
- and the other blocs are in [0;base[. *)
- let canonical n =
- let ok x = (0 <= x && x < base) in
- let rec ok_tail k = (k = 0) || (ok n.(k) && ok_tail (k-1)) in
- let ok_init x = (-base <= x && x < base && x <> -1 && x <> 0)
- in
- (n = [||]) || (n = [|-1|]) ||
- (ok_init n.(0) && ok_tail (Array.length n - 1))
- (* [normalize_pos] : removing initial blocks of 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)
- (* [normalize_neg] : avoid (-1) as first bloc.
- input: an array with -1 as first bloc and other blocs in [0;base[
- output: a canonical array *)
- 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')
- (* [normalize] : avoid 0 and (-1) as first bloc.
- input: an array with first bloc in [-base;base[ and others in [0;base[
- output: a canonical array *)
- let rec normalize n =
- if Array.length n = 0 then n
- else if n.(0) = -1 then normalize_neg n
- else if n.(0) = 0 then normalize_pos n
- else n
- (* Opposite (expects and returns canonical arrays) *)
- 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.(0) <- - n.(0);
- (* n.(0) cannot be 0 since m is canonical *)
- if n.(0) = -1 then normalize_neg n
- else if n.(0) = base then (n.(0) <- 0; Array.append [| 1 |] n)
- else n
- end else begin
- (* here n.(!i) <> 0, hence 0 < base - n.(!i) < base for n canonical *)
- n.(!i) <- base - n.(!i); decr i;
- while !i > 0 do n.(!i) <- base - 1 - n.(!i); decr i done;
- (* since -base <= n.(0) <= base-1, hence -base <= -n.(0)-1 <= base-1 *)
- n.(0) <- - n.(0) - 1;
- (* since m is canonical, m.(0)<>0 hence n.(0)<>-1,
- and m=-1 is already handled above, so here m.(0)<>-1 hence n.(0)<>0 *)
- n
- end
- 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;
- (* here r.(0) could be in [-2*base;2*base-1] *)
- 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 normalize r (* in case r.(0) is 0 or -1 *)
- 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
- (* Comparisons *)
- 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 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))
- (* For this equality test it is critical that n and m are canonical *)
- let equal m n = (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))
- (* For two big nums m and d and a small number q,
- computes m - d * q * base^(|m|-|d|-k) in-place (in m).
- Both m d and q are positive. *)
- 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 begin
- m.(k+i-1) <- m.(k+i-1) - v / base;
- let j = ref (i-1) in
- while m.(k + !j) < 0 do (* result is positive, hence !j remains >= 0 *)
- m.(k + !j) <- m.(k + !j) + base; decr j; m.(k + !j) <- m.(k + !j) -1
- done
- end
- done
- (** Euclid division m/d = (q,r)
- This is the "Floor" variant, as with ocaml's /
- (but not as ocaml's Big_int.quomod_big_int).
- We have sign r = sign m *)
- 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 len = String.length s in
- let isneg = len > 1 & s.[0] = '-' in
- let d = ref (if isneg then 1 else 0) in
- while !d < len && s.[!d] = '0' do incr d done;
- if !d = len then zero else
- let r = (len - !d) mod size in
- let h = String.sub s (!d) r in
- let e = if h<>"" then 1 else 0 in
- let l = (len - !d) / size in
- let a = Array.create (l + e) 0 in
- 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;
- if isneg then neg a else 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
- end
- (******************************************************************)
- (* Optimized operations on (unbounded) integer numbers *)
- (* integers smaller than base are represented as machine integers *)
- (******************************************************************)
- open ArrayInt
- type bigint = Obj.t
- (* Since base is the largest power of 10 such that base*base <= max_int,
- we have max_int < 100*base*base : any int can be represented
- by at most three blocs *)
- let small n = (-base <= n) && (n < base)
- let mkarray n =
- (* n isn't small, this case is handled separately below *)
- let lo = n mod base
- and hi = n / base in
- let t = if small hi then [|hi;lo|] else [|hi/base;hi mod base;lo|]
- in
- for i = Array.length t -1 downto 1 do
- if t.(i) < 0 then (t.(i) <- t.(i) + base; t.(i-1) <- t.(i-1) -1)
- done;
- t
- let ints_of_int n =
- if n = 0 then [| |]
- else if small n then [| n |]
- else mkarray n
- let of_int n =
- if small n then Obj.repr n else Obj.repr (mkarray n)
- let of_ints n =
- let n = normalize n in (* TODO: using normalize here seems redundant now *)
- 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 to_ints n =
- if Obj.is_int n then ints_of_int (coerce_to_int n)
- else coerce_to_ints n
- let int_of_ints =
- let maxi = mkarray max_int and mini = mkarray min_int in
- fun t ->
- let l = Array.length t in
- if (l > 3) || (l = 3 && (less_than maxi t || less_than t mini))
- then failwith "Bigint.to_int: too large";
- let sum = ref 0 in
- let pow = ref 1 in
- for i = l-1 downto 0 do
- sum := !sum + t.(i) * !pow;
- pow := !pow*base;
- done;
- !sum
- let to_int n =
- if Obj.is_int n then coerce_to_int n
- else int_of_ints (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 of_int (coerce_to_int m + coerce_to_int n)
- else of_ints (add (to_ints m) (to_ints n))
- let sub m n =
- if Obj.is_int m & Obj.is_int n
- then of_int (coerce_to_int m - coerce_to_int n)
- else of_ints (sub (to_ints m) (to_ints n))
- let mult m n =
- if Obj.is_int m & Obj.is_int n
- then of_int (coerce_to_int m * coerce_to_int n)
- else of_ints (mult (to_ints m) (to_ints n))
- let euclid m n =
- if Obj.is_int m & Obj.is_int n
- then app_pair of_int
- (coerce_to_int m / coerce_to_int n, coerce_to_int m mod coerce_to_int n)
- else app_pair of_ints (euclid (to_ints m) (to_ints 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 (to_ints m) (to_ints n)
- let neg n =
- if Obj.is_int n then of_int (- (coerce_to_int n))
- else of_ints (neg (to_ints n))
- let of_string m = of_ints (of_string m)
- let to_string m = to_string (to_ints m)
- let zero = of_int 0
- let one = of_int 1
- let two = of_int 2
- let sub_1 n = sub n one
- let add_1 n = add n one
- let mult_2 n = add n n
- let div2_with_rest n =
- let (q,b) = euclid n two in
- (q, b = one)
- let is_strictly_neg n = is_strictly_neg (to_ints n)
- let is_strictly_pos n = is_strictly_pos (to_ints n)
- let is_neg_or_zero n = is_neg_or_zero (to_ints n)
- let is_pos_or_zero n = is_pos_or_zero (to_ints n)
- let equal m n = (m = n)
- (* spiwack: computes n^m *)
- (* The basic idea of the algorithm is that n^(2m) = (n^2)^m *)
- (* In practice the algorithm performs :
- k*n^0 = k
- k*n^(2m) = k*(n*n)^m
- k*n^(2m+1) = (n*k)*(n*n)^m *)
- let pow =
- let rec pow_aux odd_rest n m = (* odd_rest is the k from above *)
- if m<=0 then
- odd_rest
- else
- let quo = m lsr 1 (* i.e. m/2 *)
- and odd = (m land 1) <> 0 in
- pow_aux
- (if odd then mult n odd_rest else odd_rest)
- (mult n n)
- quo
- in
- pow_aux one
- (** Testing suite w.r.t. OCaml's Big_int *)
- (*
- module B = struct
- open Big_int
- let zero = zero_big_int
- let to_string = string_of_big_int
- let of_string = big_int_of_string
- let add = add_big_int
- let opp = minus_big_int
- let sub = sub_big_int
- let mul = mult_big_int
- let abs = abs_big_int
- let sign = sign_big_int
- let euclid n m =
- let n' = abs n and m' = abs m in
- let q',r' = quomod_big_int n' m' in
- (if sign (mul n m) < 0 && sign q' <> 0 then opp q' else q'),
- (if sign n < 0 then opp r' else r')
- end
- let check () =
- let roots = [ 1; 100; base; 100*base; base*base ] in
- let rands = [ 1234; 5678; 12345678; 987654321 ] in
- let nums = (List.flatten (List.map (fun x -> [x-1;x;x+1]) roots)) @ rands in
- let numbers =
- List.map string_of_int nums @
- List.map (fun n -> string_of_int (-n)) nums
- in
- let i = ref 0 in
- let compare op x y n n' =
- incr i;
- let s = Printf.sprintf "%30s" (to_string n) in
- let s' = Printf.sprintf "%30s" (B.to_string n') in
- if s <> s' then Printf.printf "%s%s%s: %s <> %s\n" x op y s s' in
- let test x y =
- let n = of_string x and m = of_string y in
- let n' = B.of_string x and m' = B.of_string y in
- let a = add n m and a' = B.add n' m' in
- let s = sub n m and s' = B.sub n' m' in
- let p = mult n m and p' = B.mul n' m' in
- let q,r = try euclid n m with Division_by_zero -> zero,zero
- and q',r' = try B.euclid n' m' with Division_by_zero -> B.zero, B.zero
- in
- compare "+" x y a a';
- compare "-" x y s s';
- compare "*" x y p p';
- compare "/" x y q q';
- compare "%" x y r r'
- in
- List.iter (fun a -> List.iter (test a) numbers) numbers;
- Printf.printf "%i tests done\n" !i
- *)