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;

public class Prog
{
    public static void Main()
    {
        //var input = Console.ReadLine().Split();
        //int[] X = Array.ConvertAll(input, int.Parse);
        var input = Console.ReadLine().Split();
        int N = int.Parse(input[0]);
        int K = int.Parse(input[1]);
        double rate = 0.0;
        if(N >= K)
        {
            for(int i = 1; i < K; i++)
            {
                int j = 0;
                int x = i;
                while(x < K)
                {
                    x *= 2;
                    j++;
                }
                rate += Math.Pow(1.0/2.0,j) / (double)N;
            }
            rate += (double)(N - K + 1) / (double)N;
        }
        else
        {
            for(int i = 1; i <= N; i++)
            {
                int j = 0;
                int x = i;
                while (x < K)
                {
                    x *= 2;
                    j++;
                }
                rate += Math.Pow(1.0 / 2.0, j) / (double)N;
            }
        }
        Console.WriteLine(rate);
    }
}

//Array.ConvertAll(int.Parse)