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.Collections.Generic;
using System.IO;
using System.Linq;

namespace Contest
{
    struct Edge
    {
        public int U, V;
        public int Dist;

        public Edge(int u, int v, int w)
        {
            U = u;
            V = v;
            Dist = w;
        }
    }
    class Program
    {

        public void Solve()
        {
            var sc = new Scanner();
            int N = sc.NextInt();
            int K = sc.NextInt();

            int b = 0;
            int c = 0;
            double ans = 0;
            for (int i = 1; i <= N; i++)
            {
                int cnt = 0;
                while (true)
                {
                    if ((i << cnt) < K) cnt++;
                    else break;
                }

                double p = Math.Pow(0.5, cnt) / N;
                //Console.WriteLine($"{i} {p}");
                ans += p;
            }
            Console.WriteLine(ans);
        }

        static void Main() => new Program().Solve();
    }
}

class Scanner
{
    public Scanner()
    {
        _stream = new StreamReader(Console.OpenStandardInput());
        _pos = 0;
        _line = new string[0];
        _separator = ' ';
    }
    private char _separator;
    private StreamReader _stream;
    private int _pos;
    private string[] _line;
    #region get a element
    public string Next()
    {
        if (_pos >= _line.Length)
        {
            _line = _stream.ReadLine().Split(_separator);
            _pos = 0;
        }
        return _line[_pos++];
    }
    public int NextInt()
    {
        return int.Parse(Next());
    }
    public long NextLong()
    {
        return long.Parse(Next());
    }
    public double NextDouble()
    {
        return double.Parse(Next());
    }
    #endregion
    #region convert array
    private int[] ToIntArray(string[] array)
    {
        var result = new int[array.Length];
        for (int i = 0; i < array.Length; i++)
        {
            result[i] = int.Parse(array[i]);
        }

        return result;
    }
    private long[] ToLongArray(string[] array)
    {
        var result = new long[array.Length];
        for (int i = 0; i < array.Length; i++)
        {
            result[i] = long.Parse(array[i]);
        }

        return result;
    }
    private double[] ToDoubleArray(string[] array)
    {
        var result = new double[array.Length];
        for (int i = 0; i < array.Length; i++)
        {
            result[i] = double.Parse(array[i]);
        }

        return result;
    }
    #endregion
    #region get array
    public string[] Array()
    {
        if (_pos >= _line.Length)
            _line = _stream.ReadLine().Split(_separator);
        _pos = _line.Length;
        return _line;
    }
    public int[] IntArray()
    {
        return ToIntArray(Array());
    }
    public long[] LongArray()
    {
        return ToLongArray(Array());
    }
    public double[] DoubleArray()
    {
        return ToDoubleArray(Array());
    }
    #endregion
}