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 (Clang 3.8.0) (judge verdict: Wrong Answer)

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
typedef long long int lli;

int main() {
  int i, j;
  long long int N;
  long long int K;
  long long int t_k;

  long long int p_k;
  long long int c;

  double sum;

  
  // scanf("%d%d", &N, &K);
  scanf("%lld%lld", &N, &K);

  // printf("end");
  // return 0;

  t_k = K;
  lli p_N = N;
  lli diff;
  // printf("%lld\n", t_k);
  c = 0;
  do{
    if(t_k > N){
      sum += 0;
    }
    else {
      if (p_N){
        diff = p_N - t_k + 1;
        p_N = 0;
      }
      else{
        diff = p_k - t_k;
      }
      double prob = pow(0.5, c);
      // printf("%f", prob);
      // sum = sum * (1-prob) + diff * prob;
      // sum += (p_k - t_k + 1) * prob;
      sum += diff * prob;
    }

    // afterward
    c++;
    p_k = t_k;
    if(t_k % 2 == 1){
      t_k = t_k/2 + 1;
    }
    else{
      t_k = t_k / 2;
    }
    // printf("%lld\n", t_k);
  } while (p_k > 1);

  printf("%f", sum / N);

  return 0;
}