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: Accepted)

import java.io.*;
import java.util.*;

public class Main {

    public static void main(String[] args) {
        try (CustomReader in = new CustomReader()) {
            new Main().execute(in);
        } catch (IOException e) {
            e.printStackTrace();
        }
    }

    public void execute(CustomReader in) throws IOException {
        int[] nk = in.readLineAsIntArray();
        final int N = nk[0];
        final int K = nk[1];

        double p = 0.0;

        double mul = 1.0;

        int high = N;
        int k = K;
        while (high > 0) {
            if (high >= k) {
                p += (Math.max(high - k + 1, 0) * mul) / N;
                high = k - 1;
            } else {
            }
            if(k%2==0){
                k = k / 2;
            }else{
                k = k / 2 + 1;
            }
            mul *= 0.5;
        }
        System.out.println(p);

    }


    static class CustomReader extends BufferedReader {

        public CustomReader() throws IOException {
            super(new InputStreamReader(System.in));
        }

        public int[] readLineAsIntArray() throws IOException {
            String[] strArray = this.readLine().split(" ");
            int[] intArray = new int[strArray.length];
            for (int i = 0, n = strArray.length; i < n; i++) {
                intArray[i] = Integer.parseInt(strArray[i]);
            }
            return intArray;
        }

        public int[][] readAsIntMatrix(int rows, int columns) throws IOException {
            int[][] matrix = new int[rows][columns];
            for (int i = 0; i < rows; i++) {
                String[] r = this.readLine().split(" ");
                for (int j = 0; j < columns; j++) {
                    matrix[i][j] = Integer.parseInt(r[j]);
                }
            }
            return matrix;
        }

    }
}