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.Text.RegularExpressions;

namespace C_DiceAndCoin
{
    class Program
    {
        static void Main(string[] args)
        {
            double[] numlist = {1,  2,  4,  8, 16 , 32 , 64,
                                128,256,512,1024,2048,4096,
                                8192,16384,32768,65536,131072};

            string[] tmplist = Regex.Split(Console.ReadLine(), @"\s+");

            int num;
            double Ans = 0;

            for (double tmpnum = 1; tmpnum <= Convert.ToDouble(tmplist[0]); ++tmpnum)
            { 
                num = Convert.ToInt32(Math.Ceiling(Math.Log10(Convert.ToDouble(tmplist[1])/tmpnum) / Math.Log10(2)));
                if (num < 0)
                    num = 0;
                Ans += 1.0/numlist[num];
            }
            Console.WriteLine(Ans/Convert.ToDouble(tmplist[0]));
        }
    }
}