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)

#include <bits/stdc++.h>
using namespace std;
#define rep(i, n) for (int i = 0; i < (n); i++)
#define repi(i, s, n) for (int i = (s); i < (n); i++)
#ifdef LOCAL
#define INPUT_FILE                              \
  ifstream in("atcoder-problems/abc126_c.txt"); \
  cin.rdbuf(in.rdbuf());
#else
#define INPUT_FILE
#endif
#define CIN_OPTIMIZE \
  cin.tie(0);        \
  ios::sync_with_stdio(false);
typedef pair<int, int> P;
typedef long long ll;
const int INF = 100100100;
const ll LINF = 1e18 + 100;
const int MOD = 1e9 + 7;

int loga(int x, int a) {
  if (x == 0 || x == 1) return 0;
  int cnt = 0;
  while (x != 0) {
    x /= a;
    cnt++;
  }
  return cnt;
}

int main() {
  INPUT_FILE CIN_OPTIMIZE;

  int N, K;
  cin >> N >> K;

  double ans;
  int n;

  if (N < K) {
    n = N;
    ans = 0;
    int factor = loga(K, 2);
    ll sum = 0;
    repi(i, 1, n + 1) {
      int f = loga(K / i, 2);
      sum += pow(2, factor - f);
    }
    ans += (double)sum / (pow(2, factor) * N);

  } else {
    n = K - 1;
    int factor = loga(K, 2);
    ll sum = 0;
    repi(i, 1, n + 1) {
      int f = loga(K / i, 2);
      sum += pow(2, factor - f);
    }
    ans = (double(sum / (pow(2, factor)) + N - K) / N);
  }

  printf("%.12lf", ans);
}