Problem:
Score : 300 points

Problem Statement

Takahashi, who lives on the number line, is now at coordinate X. He will make exactly K moves of distance D in the positive or negative direction.

More specifically, in one move, he can go from coordinate x to x + D or x - D.

He wants to make K moves so that the absolute value of the coordinate of the destination will be the smallest possible.

Find the minimum possible absolute value of the coordinate of the destination.

Constraints

-10^{15} \leq X \leq 10^{15}

1 \leq K \leq 10^{15}

1 \leq D \leq 10^{15}

All values in input are integers.

Input

Input is given from Standard Input in the following format:

X K D

Output

Print the minimum possible absolute value of the coordinate of the destination.

Sample Input 1

6 2 4

Sample Output 1

2

Takahashi is now at coordinate 6. It is optimal to make the following moves:

Move from coordinate 6 to (6 - 4 =) 2.

Move from coordinate 2 to (2 - 4 =) -2.

Here, the absolute value of the coordinate of the destination is 2, and we cannot make it smaller.

Sample Input 2

7 4 3

Sample Output 2

1

Takahashi is now at coordinate 7. It is optimal to make, for example, the following moves:

Move from coordinate 7 to 4.

Move from coordinate 4 to 7.

Move from coordinate 7 to 4.

Move from coordinate 4 to 1.

Here, the absolute value of the coordinate of the destination is 1, and we cannot make it smaller.

Sample Input 3

10 1 2

Sample Output 3

8

Sample Input 4

1000000000000000 1000000000000000 1000000000000000

Sample Output 4

1000000000000000

The answer can be enormous.

Solution in OCaml (4.10.0) (judge verdict: Wrong Answer)

open Batteries
open Printf

let ($) g f x = g (f x)

let swap = Tuple2.swap

let s2i = int_of_string
let i2s = string_of_int
let s2f = float_of_string
let f2s = string_of_float
let c2i = int_of_char
let i2c = char_of_int
let f2i = int_of_float
let i2f = float_of_int 

let c2s = String.of_char
let s2cl = String.to_list
let cl2s = String.of_list

let puts s = print_endline s
let putw ?yn:(yn=("Yes", "No")) b = puts (if b then fst yn else snd yn)
let puti i = puts @@ i2s i



module StaleList = struct
  include List

  let nmem x l = not @@ mem x l

  let len = length
  let size = length
  let nempty l = not @@ is_empty l

  let hdop l = if nempty l then Some (hd l) else None

  (* let taker = drop *)
  let takewh = take_while

  (* let dropr = take *)
  let dropwh = drop_while

  let split_by = span

  let chunk = ntake

  let foldl = fold_left
  let foldli = fold_lefti
  let foldr f z l = fold_right f l z
  let foldri f z l = fold_righti f l z

  let partition_to_map f l =
    foldr (fun x m -> Map.modify_opt (f x) (function
      | None -> Some [x] 
      | Some [] -> Some [x] 
      | Some vl -> Some (x::vl)) m) Map.empty l;;

  let scanl f z ls =
    let rec loop acl acc = function
      | [] -> rev acl
      | h::t ->
        let r = f acc h in
        loop (r::acl) r t
    in loop [] z ls

  let scanr f z ls =
    let rec loop acl acc = function
      | [] -> acl 
      | h::t ->
        let r = f h acc in
        loop (r::acl) r t
    in loop [] z (rev ls)

  (* let scanli = scan_lefti *)
  (* let scanri = scan_righti *)
  
  let flatmap = concat_map

  let zipf = map2
  let zipfi = map2i

end

module L = StaleList

module StaleArray = struct
  include Array 

  let nmem x l = not @@ mem x l

  let len = length
  let size = length
  let is_empty r = len r = 0
  let nempty r = len r > 0

  let take n r = left r n
  let taker n r = right r n

  let takeb x r =
    let l = len r in
    let rec loop i =
      if i >= l then r
      else if r.(i) = x then take i r
      else loop (i + 1)
    in loop 0
     
  (* let takewh = take_while *)

  let drop n r = tail r n
  let dropr n r = take ((len r) - n) r
  (* let dropwh = drop_while *)

  let index_of x r =
    let l = len r in
    let rec loop i =
      if i >= l then None
      else if r.(i) = x then Some i
      else loop (i + 1)
    in loop 0

  (* let split_by = span *)

  (* let chunk = ntake *)

  let foldl = fold_left
  let foldli = fold_lefti
  let foldr f z l = fold_right f l z
  let foldri f z l = fold_righti f l z

  (* let partition_to_map f l = *)
  (*   foldr (fun x m -> Map.modify_opt (f x) (function *)
  (*     | None -> Some [x]  *)
  (*     | Some [] -> Some [x]  *)
  (*     | Some vl -> Some (x::vl)) m) Map.empty l;; *)

  (* let scanl = scan_left *)
  (* let scanli = scan_lefti *)
  (* let scanr = scan_right *)
  (* let scanri = scan_righti *)
  
  let concat2 = append
  
  let flatmap (f : 'a -> 'b array) a =
    map f a |> enum |> L.of_enum |> concat

  let zipf = map2
  (* let zipfi = map2i *)

end

module A = StaleArray

module StaleString = struct
  include String

  let len = length

  let replace_chars = map
  let map = replace

  let maptol f s = L.map f (s2cl s)
  let maptoli f s = L.map f (s2cl s)

  let zipf f a b = L.zipf f (s2cl a) (s2cl b)
  let zipfi f a b = L.zipfi f (s2cl a) (s2cl b)

  let mkstr f sep ls =
    L.foldl (fun acc x -> acc ^ sep ^ (f x)) (f @@ L.hd ls) (List.tl ls)

end

module S = StaleString

let spsp s = S.split_on_char ' ' s


module StaleSortedSet = struct
  include Set

  module Make (Ord : Set.OrderedType) = struct
    include Set.Make(Ord)

    let nmem x st = not @@ mem x st

    let foldl f s z = fold (fun x a -> f a x) z s

    let size = cardinal
  end

  let nmem x st = not @@ mem x st

  let foldl f s z = fold (fun x a -> f a x) z s

  let size = cardinal
end

module Ss = StaleSortedSet

module StaleSortedMap  = struct
  include Map

  module Make (Ord : Map.OrderedType) = struct
    include Map.Make(Ord)

    let nmem x st = not @@ mem x st

    let get = find
    let get_or x d m = find_default d x m

    let size = cardinal
  end

  let nmem x st = not @@ mem x st

  let get = find
  let get_or x d m = find_default d x m

  let size = cardinal
end

module Ms = StaleSortedMap


module StaleRange = struct

  type t = Range of int * int

  let make lb ub = Range (lb, ub)

  let to_list r =
    let Range (lb, ub) = r in
    L.make (ub - lb) ((+) lb) 

  let to_array r =
    let Range (lb, ub) = r in
      A.make (ub - lb) ((+) lb)

  let iter r f = 
    let Range (lb, ub) = r in
    for i = lb to ub do f i done

  let iter2 r1 r2 f =
    let Range (lb1, ub1) = r1 and Range (lb2, ub2) = r2 in
      for i = lb1 to ub2 do
        for j = lb2 to ub2 do
          f i j
        done
      done
end

module R = StaleRange

let (---) lb ub = R.make lb ub
let (--^) lb ub = R.make lb (ub - 1)

module StaleGenericAlgorithms = struct

  let set_power ls =
    let rec loop ls pls =
      match ls with
      | [] -> pls
      | hd::tl -> loop tl (L.flatmap (fun p -> [p; hd::p]) pls)
    in loop ls [[]]

  let rec bsearch_max_int f lb ub =
    if lb >= ub then (if f lb then lb else (lb - 1))
    else (
      let m = (lb + ub) / 2 in
      if f m then bsearch_max_int f (m + 1) ub
      else bsearch_max_int f lb m
    )

  let partial_sum n ls =
    let rec loop s k = function
      | [] -> s
      | h::t ->
        if k >= n then s
        else loop (s + h) (k + 1) t
    in
    loop 0 0 ls

end

module Gal = StaleGenericAlgorithms


module StaleIntArithmetic = struct

  let rec power x e =
    if e = 0 then x
    else 
      let rec loop r x2 e' =
        if e' = 0 then r
        else if e' land 1 = 0 then loop r (x2 * x2) (e' lsr 1)
        else loop (r * x2) (x2 * x2) (e' lsr 1)
      in loop 1 x e
    

  let rec powers_le_1000 x =
    if x > 32 then [x]
    else
      L.init 9 (fun e -> power x (e + 1))
      |> L.filter (fun y -> y < 1000)

  let primes_le_1000 = [2;3;5;7;11;13;17;19;23;29;31;37;41;
    43;47;53;59;61;67;71;73;79;83;89;97;101;
    103;107;109;113;127;131;137;139;149;151;157;163;167;
    173;179;181;191;193;197;199;211;223;227;229;233;239;
    241;251;257;263;269;271;277;281;283;293;307;311;313;
    317;331;337;347;349;353;359;367;373;379;383;389;397;
    401;409;419;421;431;433;439;443;449;457;461;463;467;
    479;487;491;499;503;509;521;523;541;547;557;563;569;
    571;577;587;593;599;601;607;613;617;619;631;641;643;
    647;653;659;661;673;677;683;691;701;709;719;727;733;
    739;743;751;757;761;769;773;787;797;809;811;821;823;
    827;829;839;853;857;859;863;877;881;883;887;907;911;
    919;929;937;941;947;953;967;971;977;983;991;997]

    let primes_and_powers_le_1000 =
      [2;3;4;5;7;8;9;11;13;16;17;19;23;25;27;29;31;32;37;41;
      43;47;49;53;59;61;64;67;71;73;79;81;83;89;97;
      101;103;107;109;113;121;125;127;128;131;137;139;149;151;
      157;163;167;169;173;179;181;191;193;197;199;211;223;227;
      229;233;239;241;243;251;256;257;263;269;271;277;281;283;
      289;293;307;311;313;317;331;337;343;347;349;353;359;361;
      367;373;379;383;389;397;401;409;419;421;431;433;439;443;
      449;457;461;463;467;479;487;491;499;503;509;512;521;523;
      529;541;547;557;563;569;571;577;587;593;599;601;607;613;
      617;619;625;631;641;643;647;653;659;661;673;677;683;691;
      701;709;719;727;729;733;739;743;751;757;761;769;773;787;
      797;809;811;821;823;827;829;839;841;853;857;859;863;877;
      881;883;887;907;911;919;929;937;941;947;953;961;967;971;
      977;983;991;997]

  let factorize_with ps n =
    let rec loop acc ps e n' =
      if n' <= 1 then ([], n') 
      else
        match ps with
        | [] -> (acc, n')
        | p::t ->
          let r = n' mod p in
          if r = 0 then (
            if n' = p then ((p, e + 1) :: acc, 1)
            else loop acc ps (e + 1) (n' / p)
          ) else
            loop (if e = 0 then acc else (p, e) :: acc) t 0 n'
    in
    let (fs, r) = loop [] ps 0 n in
    (L.rev fs, r)

  let factorize_le_million n =
    factorize_with primes_le_1000 n

  let rec to_bits_le n =
    if n = 0 then []
    else (n mod 2) :: to_bits_le (n lsr 1)

  let rec gcd x y =
    match x mod y with
    | 0 -> y
    | z -> gcd y z

  let ext_euclid a b =
    let rec next (r0, s0, t0) (r1, s1, t1) =
      match r1 with
      | 0 -> (r0, s0, t0)
      | r2 ->
        let q = r0 / r1 in
        next (r1, s1, t1) (r0 - q * r1, s0 - q * s1, t0 - q * t1)
    in
    if a < b then
      let (g, y, x) = next (b, 1, 0) (a, 0, 1) in (g, x, y)
    else
      next (a, 1, 0) (b, 0, 1)

  let modinv x m =
    let (g, y, z) = ext_euclid x m in
    match g with
    | 1 -> y mod m
    | _ -> invalid_arg "Not relatively prime"
      

end

module N = StaleIntArithmetic



let rds () = read_line ()

let rdi () = s2i @@ rds ()

let rdhs () = spsp @@ rds ()
let rdhi () = rdhs () |> L.map s2i

let rdhf () = rdhs () |> L.map s2f

let rdf2 () = match rdhf () with
| a::b::_ -> (a, b)
| _ -> (0., 0.)

let rdf3 () = match rdhf () with
| a::b::c::_ -> (a, b, c)
| _ -> (0., 0., 0.)

let rdf4 () = match rdhf () with
| a::b::c::d::_ -> (a, b, c, d)
| _ -> (0., 0., 0., 0.)

let rds2 () = match rdhs () with
  | a::b::_ -> (a, b)
  | _ -> ("", "")

let rdi2 () = match rdhi () with
  | a::b::_ -> (a, b)
  | _ -> (0, 0)

let rdi3 () = match rdhi () with
| a::b::c::_ -> (a, b, c)
| _ -> (0, 0, 0)

let rdi4 () = match rdhi () with
| a::b::c::d::_ -> (a, b, c, d)
| _ -> (0, 0, 0, 0)

let rdi5 () = match rdhi () with
| a::b::c::d::e::_ -> (a, b, c, d, e)
| _ -> (0, 0, 0, 0, 0)

let rdv n rf =
  let rec loop n =
    if n <= 0 then []
    else (
      let s = rf () in
      s::(loop (n - 1))
    )
  in loop n

let rdvi n = rdv n rdi
let rdvi2 n = rdv n rdi2
let rdvi3 n = rdv n rdi3
let rdvi4 n = rdv n rdi4


let print_list aa = 
  printf "[ "; aa |> L.iter (fun a -> printf "%d " a); puts "]"

let print_array aa = 
  printf "[ "; aa |> A.iter (fun a -> printf "%d " a); puts "]"

let print_carray aa = 
  printf "[ "; aa |> A.iter (fun a -> printf "%c " a); puts "]"

let print_matrix mat =
  puts "----";
  mat |> A.iter (fun r -> print_carray r);
  puts "----"

let _ =
  let (x, k, d) = rdi3 () in
  let xa = abs x in
  let k1 = xa / d and m1 = xa mod d and m2 = d - xa mod d in
  let m =
    if k < k1 then 
      abs (if x >= 0 then abs x - k * d else abs x + k * d)
    else if (k - k1) mod 2 = 0 then m1
    else m2
  in
  puti @@ abs m