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 ABC126
{
	class Program
	{
		static void Main(String[] args)
		{
			// 入力
			int[] input = Array.ConvertAll(Console.ReadLine().Split(' '), int.Parse);
			int N = input[0];
			int K = input[1];
			// 計算
			double probability = 0.0;
			int pow = 1;
			int x = N+1;//x~N(Nを含める)までの確率計算をprobabilityに加算した、という情報。最初はNを計算できていないのでN+1で初期化
			while (x > 1)
			{
				// i*pow>=Kとなるようなiを求める(i)
				int i = K / pow;
				if (K%pow!=0){
					// 割り切れない場合は、K=i*pow+alpha(pow>alpha>0)よりi*pow<K && (i+1)*pow>=Kなので、iを1増やす
					i++;
				}
				// iがxより小さくなったら、何個の確率計算がこのループでできるかを調べる
				int count = x - i;
				if (count>0)
				{
					// xを更新
					x = i;
					// probabilityに、調べた個数分の確率を追加する（1つ当たり 1/N/pow である）
					double add = 1.0 * count / N / pow;
					probability += add;
				}
				// powの更新
				pow *= 2;
				// x<=1なら、計算終了
			}
			// 出力
			Console.WriteLine(probability);
		}
	}
}