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++ (GCC 5.4.1) (judge verdict: Accepted)

#include<iostream>
#include<cstdio>

using namespace std;

#define rep(i, n) for (int i = 0; i < n; i++)

int main(){
    long int n, k;
    cin >> n >> k;
    long int a[n];
    rep(i, n){
        long int num = i + 1;
        long int bin = 0;
        while(num < k){
            num *= 2;
            bin++;
        }
        a[i] = bin;    
    }
    // rep(i,n) cout << a[i] << " ";
    // cout << endl;
    long int sm = 0;
    long int sq[a[0]+1];
    sq[0] = 1;
    for(int i = 1; i <= a[0]; i++){
        sq[i] = sq[i-1] * 2;
    }
    // rep(i,a[0]+1) cout << sq[i] << " ";
    // cout << endl;
    rep(i, n){
        sm += sq[a[0]-a[i]];
    }
    // cout << sm << endl;
    double ans = (double)sm / (double)(sq[a[0]] * n);
    printf("%.16f\n", ans);
}