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.Linq;

namespace abc126
{
    class Program
    {
        static void Main(string[] args)
        {
            var tmp = Console.ReadLine().Split(' ').Select(_ => int.Parse((string) _)).ToArray();
            // var tmp = new int[] {10000, 5};
            // var tmp = new int[] {3, 10};
            var n = tmp[0];
            var k = tmp[1];

            var kratio = 1;
            var krate = 1.0;
            var total = 0.0;
            var inversen = 1.0 / n;
            var firstmaxNext = n;

            for (var ik = 0; ik < k; ik++)
            {
                var firstmax = firstmaxNext;
                var firstmin = Math.Min((k + kratio - 1) / kratio, firstmax);

                if (firstmin < 1)
                {
                    break;
                }

                if (firstmin <= n)
                {
                    firstmax = Math.Min(firstmax, n);
                    var rate = krate * (firstmax - firstmin + 1) * inversen;
                    total += rate;
                }

                firstmaxNext = firstmin - 1;
                if (firstmaxNext < 1)
                {
                    break;
                }
                krate *= 0.5;
                kratio *= 2;
            }

            Console.WriteLine(total);
        }
    }
}