/coq-8.3pl2/lib/bigint.ml
OCaml | 411 lines | 303 code | 56 blank | 52 comment | 128 complexity | 2e1c8436d59f5730ea391dcfd94633c5 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 1999-2010 *)
- (* \VV/ **************************************************************)
- (* // * This file is distributed under the terms of the *)
- (* * GNU Lesser General Public License Version 2.1 *)
- (************************************************************************)
- (* $Id: bigint.ml 13323 2010-07-24 15:57:30Z herbelin $ *)
- (*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_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 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 (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)
- (* 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 is_neg_or_zero m then
- odd_rest
- else
- let (quo,rem) = div2_with_rest m in
- pow_aux
- ((* [if m mod 2 = 1]*)
- if rem then
- mult n odd_rest
- else
- odd_rest )
- (* quo = [m/2] *)
- (mult n n) quo
- in
- pow_aux one
- (* 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