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.Linq;

class Program
{
    static void Main()
    {
        Console.WriteLine(Solve());
        return;
    }

    public static string Solve()
    {
        int[] input = ReadInts();
        int N = input[0], K = input[1];
        double sum = 0;
        for (int i = 1; i <= N; i++)
        {
            double p = 1f / N;
            if (i>=K)
            {
                sum += p;
            }
            else
            {
                p *= Math.Pow(1f/2f,Math.Ceiling(Math.Log((double)K / i,2)));
                sum += p;
            }
        }
        return sum.ToString();
    }

    private static string Read() { return Console.ReadLine(); }
    private static int ReadInt() { return int.Parse(Read()); }
    private static long ReadLong() { return long.Parse(Read()); }
    private static double ReadDouble() { return double.Parse(Read()); }
    private static int[] ReadInts() { return Array.ConvertAll(Read().Split(), int.Parse); }
    private static long[] ReadLongs() { return Array.ConvertAll(Read().Split(), long.Parse); }
    private static double[] ReadDoubles() { return Array.ConvertAll(Read().Split(), double.Parse); }
}