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;

namespace _126C
{
    class Program
    {
        static void Main(string[] args)
        {

            string[] S = Console.ReadLine().Split(' ');
            int N = int.Parse(S[0]);
            int K = int.Parse(S[1]);

            double Ans = 0;

            if (N >= K - 1)
            {
                Ans = ((double)N - (double)K + 1) / (double)N;

                for (int i = 1; i <= K-1; i++)
                {



                    double Log2_Ki = Math.Log((double)K / i, 2);

                    Log2_Ki = Math.Ceiling(Log2_Ki);

                    int A = (int)Log2_Ki;
                    double AS = 1;

                    for (int j = 0; j < A; j++)
                    {
                        AS = AS * 2;
                    }

                    //Console.WriteLine("{0},{1},{2}", i, A, AS);
                    Ans += (1 / AS) / N;

                    AS = 1;

                }

            }
            else
            {
                for (int i = 1; i <= N; i++)
                {


                    double Log2_Ki = Math.Log((double)K / i, 2);

                    Log2_Ki = Math.Ceiling(Log2_Ki);

                    int A = (int)Log2_Ki;
                    double AS = 1;

                    for (int j = 0; j < A; j++)
                    {
                        AS = AS * 2;
                    }

                    //Console.WriteLine("{0},{1},{2}", i, A, AS);
                    Ans += (1 / AS)/N;

                    AS = 1;



                }
            }

         
            Console.WriteLine(Ans);

        }

    }
}