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 Hello
{
    private const int MOD_N = 1000000000 + 7;

    public static void Main()
    {
        var list = Console.ReadLine().Split(' ');
        var n = int.Parse(list[0]);
        var k = int.Parse(list[1]);

        var countList = new List<int>();
        var tn = n;

        var _ = Math.Max(tn - k + 1, 0);
        countList.Add(_);
        tn -= _;

        while (k > 1)
        {
            var half = (int)Math.Ceiling(k * 0.5);
            _ = Math.Max(tn - half + 1, 0);
            countList.Add(_);
            tn -= _;
            k = half;
        }

        double ans = 0.0;
        double p = 1.0;

        for (int i = 0, count = countList.Count; i < count; ++i)
        {
            var c = countList[i];
            ans += p * c;
            p *= 0.5;
        }
        ans *= 1.0 / n;

        Console.WriteLine(ans);
    }
}