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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.util.*;

public class Main {
	public static void main(String[] args) {
		int i, j;
		Scanner sc = new Scanner(System.in);
		int n = Integer.parseInt(sc.next());
		int k = Integer.parseInt(sc.next());
		sc.close();
		sc = null;
		double nbunno1 = 1.0 / n;
		double kaku = 0.0;
		int kai;
		double[] nibunnno1 = new double[18];
		double ni = 2.0;
		for(i = 0; i <= 17; i++) {
			ni /= 2.0;
			nibunnno1[i] = ni;
		}
//		int aa = 0;
		int max;
		if(n >= k) {
			max = k -1;
		} else {
			max = n;
		}
		for(i = 1; i <= max; i++) {
			kai = (int) ((Math.log(k / i)/Math.log(2))+ 1.0);
//			System.out.println(kai); /*  */
//			aa +=  Math.pow(2, 17-kai); 
			kaku += nbunno1 * nibunnno1[kai]; 
		}
//		System.out.println(kaku);
//		System.out.println((double)aa/n/131072);
		if(n >= k) {
			kaku += nbunno1 * (n - k + 1);
		}
//		System.out.println(nbunno1 * (n - k));
//		System.out.println(((double)aa/131072+(double)(n-k))/n);
		System.out.println(kaku);
	}
}