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: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

class Program
{
    void Run()
    {
        var sc = new Scanner();
        int n = sc.NextInt();
        int k = sc.NextInt();
        long[] den = new long[n];
        for (int i = 1; i <= n; i++)
        {
            int s = i;
            int c = 0;
            int d = 1;
            while (s <= k - 1)
            {
                s *= 2;
                d *= 2;
            }
            den[i - 1] = d;
        }
        long gcd = den[0];
        for (int i = 1; i < n; i++) gcd = Gcd(gcd, den[i]);
        long lcm = den[0];
        for (int i = 1; i < n; i++) lcm = lcm * den[i] / gcd;
        long sum = 0;
        for (int i = 0; i < n; i++)
            sum += lcm / den[i];
        Console.WriteLine(((double)sum / lcm / n).ToString("F10"));
    }

    static void Main(string[] args)
    {
        new Program().Run();
    }

    static long Gcd(long a, long b)
    {
        if (a < b) return Gcd(b, a);
        if (b == 0) return a;
        return Gcd(b, a % b);
    }
}

class Scanner
{
    int index = -1;
    string[] buf;

    public Scanner()
    {
    }

    public int NextInt()
    {
        CheckNext();
        return int.Parse(buf[index++]);
    }

    public long NextLong()
    {
        CheckNext();
        return long.Parse(buf[index++]);
    }

    public double NextDouble()
    {
        CheckNext();
        return double.Parse(buf[index++]);
    }

    public string NextString()
    {
        CheckNext();
        return buf[index++];
    }

    private void CheckNext()
    {
        if (index >= 0 && index < buf.Length) return;

        buf = Console.ReadLine().Split(' ');
        index = 0;
    }
}