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.util.*;

public class Main {

    private static Map<Integer, Double> map = new HashMap<>();

    static {
        map.put(1, 2.0);
        map.put(2, 4.0);
        map.put(3, 8.0);
        map.put(4, 16.0);
        map.put(5, 32.0);
        map.put(6, 64.0);
        map.put(7, 128.0);
        map.put(8, 256.0);
        map.put(9, 512.0);
        map.put(10, 1024.0);
        map.put(11, 2048.0);
        map.put(12, 4096.0);
        map.put(13, 8192.0);
        map.put(14, 16384.0);
        map.put(15, 32768.0);
        map.put(16, 65536.0);
        map.put(17, 131072.0);
    }

    public static void main(String[] args) {
//        long startTime = System.currentTimeMillis();
        Scanner scanner = new Scanner(System.in);
        int N = scanner.nextInt();
        int K = scanner.nextInt();
        scanner.close();

        double ans = 0.0;

        for (int a = 1; a <= N; a++) {
            int n = 0;
            int t = a;
            while (t < K) {
                n++;
                t *= 2;
            }
            if (n != 0) {
                ans += 1.0 / N * Math.pow(0.50, (double) n);
            } else {
                ans += 1.0 / N;
            }
        }

//        for (int a = 1; a <= N; a++) {
//            double c1 = (double) (a + K) / a;
//            System.out.println("c1:"+c1);
//            int mapIdx = 0;
//            for (int i = 1; i <= 16; i++) {
//                if (map.get(i) < c1) {
//                    continue;
//                } else {
//                    mapIdx = i;
//                    break;
//                }
//            }
//            System.out.println("mapIdx:"+mapIdx);
//            if (mapIdx != 0) {
//                ans += 1.0 / N * Math.pow(0.5, (double) mapIdx);
//            } else {
//                ans += 1.0 / N;
//            }
//        }

        System.out.println(ans);

//        System.out.println(System.currentTimeMillis() - startTime);
    }


}