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: Accepted)

#include <iostream>
#include <string>
#include <algorithm>
#include <vector>
#include <cmath>
#include <iomanip>

using namespace std;

int main(){
  int N, K;
  cin >> N >> K;
  double ans = 0;
  double halfbeki[100001];
  halfbeki[0] = 1;
  long long twohat[19];
  twohat[0] = 1;
  int times[100001];
  double tmp = 1;
  long long tmpp  = 1;
  for(int i=1; i <= 100000; i++){
    halfbeki[i] = (tmp /= 2);
    if(i < 19) twohat[i] = (tmpp *= 2);
  }

  for(int i=1; i <= N; i++){
    for(int j = 0; j < 19; j++){
      if(i * twohat[j] >= K){
        times[i] = j;
        break;
      }
    }
  }

  for(int i=1; i <= N; i++){
    //cout << halfbeki[times[i]] << endl;
    ans += halfbeki[times[i]];
  }
  //cout << ans << endl;
  ans /= N;
  cout << setprecision(20) << ans << endl;


  return 0;
}