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 JavaScript (node.js v5.12) (judge verdict: Accepted)

function Main(input) {
  var N = parseInt(input.split(" ")[0]);
  var K = parseInt(input.split(" ")[1]);
  
  var output = 0;
  if(N >= K){
     output += (N - K + 1) / N;
     for(var i = 1; i <= K - 1; i++){
       var count = 0;
       var tmp = i;
       while(true){
          if(tmp >= K){
             break;
          }
         tmp = tmp * 2;
         count++;
       }
       output += (1 / N) * (1 / Math.pow(2, count));
     }
  }else{
    for(var i = 1; i <= N; i++){
       var count = 0;
       var tmp = i;
       while(true){
          if(tmp >= K){
             break;
          }
         tmp = tmp * 2;
         count++;
       }
       output += (1 / N) * (1 / Math.pow(2, count));
     }
  }
  console.log(output);
}

Main(require("fs").readFileSync("/dev/stdin", "utf8"));