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.*;
import java.io.*;

public class Main {
  public static void main(String ... args) throws Exception {
    int n = 0, k = 0;
    String s;
    try (BufferedReader isr = new BufferedReader(new InputStreamReader(System.in))) {
      String buf = isr.readLine();
      System.err.println(buf);
      //
      String[] input = buf.split(" ");
      n = Integer.parseInt(input[0]);
      k = Integer.parseInt(input[1]);
      /*
      buf = isr.readLine();
      System.err.println(buf);
      // */
      s = buf;
    }

    System.err.println(Math.log1p((double)k/n - 1));
    System.err.println(Math.pow(0.5, (int)Math.ceil(1.2)) / 3);
    double res = 0;
    for (int i=1; i<=n; i++) {
      double d = Math.log1p((double)k/n - 1);
      res += Math.pow(0.5, (int)Math.ceil(d)) / n;
    }
    
    System.out.println(res);
  }

}