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.Linq;
using System.Collections.Generic;

namespace AtCoder_Beginner_Contest_126
{
    class Program
    {
        static void Main(string[] args)
        {
            var NK = Console.ReadLine().Split().Select(int.Parse).ToArray();
            var N = NK[0];
            var K = NK[1];
            int max = (int)Math.Ceiling(Math.Log(K, 2));
            var li = new double[max + 1];
            li[0] = 1;
            for (int i = 1; i <= max; ++i)
            { li[i] = li[i - 1] * 0.5; }
            double p = 0;

            var nd = N;
            if (K <= N)
                nd = K-1;

            for (int i = 1; i <= nd; ++i)
            {
                int jyo = (int)Math.Ceiling(Math.Log((double)K / i, 2));
                p += li[jyo];
            }
            double hosei = 0;
            if (K <= N)
                hosei = (double)(N + 1 - K) / N;
            double ans = p / NK[0] + hosei;
            Console.WriteLine(ans);                        
        }
    }
}