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 Python (3.4.3) (judge verdict: Runtime Error)

N,K = map(int, input().split())
X_list =[]
if N<K:
    pro = 0.0
    for n in range(N):
        if K % (2*(n+1))==0:
            X_list.append(K / (2*(n+1))-1)
        else:
            X_list.append(K // (2*(n+1))+1)
    X_max = max(X_list)
    for _ in range(len(X_list)):
        X_list[_] =X_max-X_list[_]
        X_list[_] =2**X_list[_]

    pro += sum(X_list)/(N*2**X_max)

    print(pro)

else:
    pro = float((N-K+1)/N)

    for n in range(K-1):
        if K % (2*(n+1))==0:
            X_list.append( K / (2*(n+1))-1)
        else:
            X_list.append( K // (2*(n+1))+1)

    X_max = max(X_list)
    for _ in range(len(X_list)):
        X_list[_] = X_max - X_list[_]
        X_list[_] = 2 ** X_list[_]
    pro += sum(X_list)/(N*2**X_max)

    print(pro)