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++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#define _USE_MATH_DEFINES
#include <math.h>
#include <iostream>
#include <iomanip>
#include <cstdlib>
#include <algorithm>
#include <numeric>
#include <stdio.h>
#include <vector>

using namespace std;
typedef long long ll;

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

int main() {

    ll N,K;
    cin >> N >> K;
    ll lim = N;
    double out = 0;
    if(N>K){
        lim = K-1;
        out += 1-K/(double)N;
    }
    ll A[lim];
    rep(i,lim) A[i] = i+1;
    ll cnt = 0;
    while(A[0]<K){
        cnt++;
        rep(i,lim){
            A[i] *= 2;
            if(A[i]>K){
                out += pow(2,-cnt)/(double)N;
                if(i!=0) A[i] = 0;
            }
        }
    }
    cout << fixed << setprecision(9) << out << endl;
    return 0;
}