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)

import decimal
n,k=map(int,input().split())
num=[]
num2=[]
num4=0
num5=n
if n>=k:
    num4=decimal.Decimal((n-k+1)/n)
    n=k-1
num6=-1*(k//(-1*n))
for i in range(18):
    num.append(2**i)
for i in range(17):
    if num6<=num[i+1]:
        num6=i
        break
for i in range(17):
    if num[i]>=n:
        num2.append(n-num[i-1])
        break
    else:
        if i==0:
            num2.append(1)
        else:
            num2.append(num[i]-num[i-1])
ans=0
num3=len(num2)
for i in range(len(num2)):
    ans+=num2[i]*(2**(i))
print((decimal.Decimal(decimal.Decimal(ans/num5)/(2**(num3+num6))))+num4)