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;

namespace CDiceAndCoin
{
    class Program
    {
        static void Main(string[] args)
        {
            // 入力の取得
            var inputs = Console.ReadLine().Split().Select(int.Parse).ToArray();
            var n = inputs[0];
            var k = inputs[1];

            // 出目 Dice の値が K/2 ~ K - 1 の場合、１回コインの表が出れば勝ちなので 1/N * 1/2
            // 出目 Dice の値が K/4 ~ K/2 - 1 の場合、２回コインの表が出れば勝ちなので 1/N * (1/2)^2
            // ...
            // 出目 Dice が既に K 以上ならばコインを投げる必要はないため 1/N
            long sum = 0;
            var divCount = 0;
            var prevK = k;
            while (1 < prevK)
            {
                sum *= 2;

                var lower = (int)Math.Ceiling((float)prevK / 2);
                if (lower <= n)
                {
                    sum += Math.Min(prevK - 1, n) - lower + 1;
                }

                prevK = lower;
                divCount++;
            }

            var prob = sum / (n * Math.Pow(2, divCount));

            // K 以上の値ではコインを投げる必要がないため 1/N
            if (k <= n)
            {
                // 計算誤差が大きくなってしまうため 1/N ずつ加算
                for (var i = 0; i < n - k + 1; i++)
                {
                    prob += (double)1 / n;
                }
            }

            // 解答の出力
            Console.WriteLine(prob);
        }
    }
}