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: Runtime Error)

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

namespace ABC126
{
    class C
    {
        static void Main(string[] args)
        {
            uint[] nk = Console.ReadLine().Split().Select(uint.Parse).ToArray();
            ulong ch = 0, mo = nk[0];
            uint[] bo = new uint[17];bo[0] = nk[1];
            int i = 0;
            int flag = -1;
            if (nk[1] <= nk[0]) { flag = 0; }
            double count = nk[1];
            while (count>=1)
            {
                i++;
                count /= 2;
                bo[i] = (uint)Math.Floor(count) + 1;
                Console.Write(bo[i]+" "+flag+" "+i+" ");
                if (bo[i] <= nk[0]&&flag==-1) { flag = i; }
            }
            ch += (nk[0]+1 - bo[flag]) * (ulong)Math.Pow(2, i - flag);
            mo *= (ulong)Math.Pow(2, i);
            for (int j = 1; j <= i; j++)
            {
                if (j<=flag) { continue; }
                Console.WriteLine(ch);
                ch += (bo[j - 1] - bo[j]) * (ulong)Math.Pow(2, i - j);
            }
            Console.WriteLine((double)ch/mo);
        }
    }
}