Problem:
Score : 300 points

Problem Statement

Snuke has a fair N-sided die that shows the integers from 1 to N with equal probability and a fair coin. He will play the following game with them:

Throw the die. The current score is the result of the die.

As long as the score is between 1 and K-1 (inclusive), keep flipping the coin. The score is doubled each time the coin lands heads up, and the score becomes 0 if the coin lands tails up.

The game ends when the score becomes 0 or becomes K or above. Snuke wins if the score is K or above, and loses if the score is 0.

You are given N and K. Find the probability that Snuke wins the game.

Constraints

1 ≤ N ≤ 10^5

1 ≤ K ≤ 10^5

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K

Output

Print the probability that Snuke wins the game. The output is considered correct when the absolute or relative error is at most 10^{-9}.

Sample Input 1

3 10

Sample Output 1

0.145833333333

If the die shows 1, Snuke needs to get four consecutive heads from four coin flips to obtain a score of 10 or above. The probability of this happening is \frac{1}{3} \times (\frac{1}{2})^4 = \frac{1}{48}.

If the die shows 2, Snuke needs to get three consecutive heads from three coin flips to obtain a score of 10 or above. The probability of this happening is \frac{1}{3} \times (\frac{1}{2})^3 = \frac{1}{24}.

If the die shows 3, Snuke needs to get two consecutive heads from two coin flips to obtain a score of 10 or above. The probability of this happening is \frac{1}{3} \times (\frac{1}{2})^2 = \frac{1}{12}.

Thus, the probability that Snuke wins is \frac{1}{48} + \frac{1}{24} + \frac{1}{12} = \frac{7}{48} \simeq 0.1458333333.

Sample Input 2

100000 5

Sample Output 2

0.999973749998

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.IO;
using System.Text;
class abc126_a {
    static int __cnt = 0;
    static string __mark = "###";
    static void Main(string[] args) {
#if LOCAL_ENVIRONMENT
        StreamReader __sr = new StreamReader("stdin.txt");
        string __str = __sr.ReadToEnd();
        __sr.Close();
        int __index = __str.IndexOf(__mark, 0);
        while (__index != -1) {
            __index = __str.IndexOf(__mark, __index + __mark.Length);
            __cnt++;
        }
        if (__cnt > 0) {
            Console.Write("Eenter test data no(1-{0})? ", __cnt);
            Console.Out.Flush();
            if (!int.TryParse(Console.ReadLine(), out __cnt)) __cnt = 1;
            Console.WriteLine("Select/Default test data: {0}", __cnt);
        }
        var stdin = new System.IO.StreamReader("stdin.txt");
        Console.SetIn(stdin);
#endif
        SimpleScanner sc = new SimpleScanner(__cnt);
        int N = sc.NextInt(), K = sc.NextInt();
        double ans = 0;
        for (var n = 1; n <= N; n++) {
            int b = 1, r = n;
            for (var m = 0; r < K; m++) {
                r <<= 1;
                b <<= 1;
            }
            ans += (1.0 / (double)N * 1.0 / (double)b);
        }
        Console.WriteLine(ans);
    } //Main

    public class SimpleScanner {
        private int cnt;
        private string[] items;
        public SimpleScanner(int cnt = 0) {
            this.cnt = 0;
            var sb = new StringBuilder();
            var s = Console.ReadLine();
            while (s != null) {
                sb.Append(s + " ");
                s = Console.ReadLine();
            }
            this.items = sb.ToString().Split(' ');
#if LOCAL_ENVIRONMENT
            if (cnt > 0) {
                int index = Array.IndexOf(this.items, __mark + cnt.ToString());
                this.cnt = index + 1;
            }
            Console.Error.WriteLine("SimpleScanner.cnt: {0}", this.cnt);
            Console.Error.WriteLine("SimpleScanner.items: {0}" , this.items.ToStringA());
#endif
        }
        public void SetCnt(int n) { this.cnt = n; }
        public int NextInt() {
            return int.Parse(this.items[this.cnt++]);
        }
        public long NextLong() {
            return long.Parse(this.items[this.cnt++]);
        }
        public string NextString() {
            return this.items[this.cnt++];
        }
        public int[] NextIntArray(int n) {
            if (n < 1) throw new IndexOutOfRangeException("引数は1以上であること！");
            var result = new int[n];
            for (var i = 0; i < n; i++) result[i] = this.NextInt();
#if LOCAL_ENVIRONMENT
            Console.Error.WriteLine("{0}", result.ToStringA());
#endif
            return result;
        }
        public long[] NextLongArray(int n) {
            if (n < 1) throw new IndexOutOfRangeException("引数は1以上であること！");
            var result = new long[n];
            for (var i = 0; i < n; i++) result[i] = this.NextLong();
#if LOCAL_ENVIRONMENT
            Console.Error.WriteLine("{0}", result.ToStringA());
#endif
            return result;
        }
        public string[] NextStringArray(int n) {
            if (n < 1) throw new IndexOutOfRangeException("引数は1以上であること！");
            var result = new string[n];
            for (var i = 0; i < n; i++) result[i] = this.NextString();
#if LOCAL_ENVIRONMENT
            Console.Error.WriteLine("{0}", result.ToStringA());
#endif
            return result;
        }
        public int[][] NextInt2Col(int n) {
            const int M = 2;
            if (n < 1) throw new IndexOutOfRangeException("引数は1以上であること！");
            int[][] result = new int[M][] {
                new int[n],
                new int[n],
            };
            for (var i = 0; i < n; i++) {
                result[0][i] = this.NextInt();
                result[1][i] = this.NextInt();
            }
#if LOCAL_ENVIRONMENT
            Console.Error.WriteLine("NextInt2Col: [");
            for (var i = 0; i < M; i++)
                Console.Error.WriteLine("  {0},", result[i].ToStringA());
            Console.Error.WriteLine("]");
#endif
            return result;
        }
    }
}
static class Extensions {
    public static bool IsEven(this int value) {
        return value % 2 == 0;
    }
    public static bool IsOdd(this int value) {
        return value % 2 != 0;
    }
    public static string ToStringA<T>(this T[] array, string separator = ", ", string braces = "[]") {
        string bL = braces.Length > 0 ? braces[0].ToString() : "";
        string bR = braces.Length > 1 ? braces[1].ToString() : "";
        return bL + String.Join(separator, array) + bR;
    }
    public static T Max<T>(this T[] array) where T: IComparable { // this->params
        if(array.Length < 1) throw new IndexOutOfRangeException("要素は1つ以上であること！");
        T max = array[0];
        for(int i = 1; i < array.Length; i++) {
            max = max.CompareTo(array[i]) > 0 ? max : array[i];
        }
        return max;
    }
    public static T Min<T>(this T[] array) where T: IComparable {
        if(array.Length < 1) throw new IndexOutOfRangeException("要素は1つ以上であること！");
        T min = array[0];
        for(int i = 1; i < array.Length; i++) {
            min = min.CompareTo(array[i]) < 0 ? min : array[i];
        }
        return min;
    }
    public static T[] SortDesc<T>(this T[] array) {
        Array.Sort(array);
        Array.Reverse(array);
        return array;
    }
}