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: Accepted)

from decimal import Decimal
li = input().split()
n = int(li[0])
k = int(li[1])

# n面サイコロ kは得点

# 1/nの確率で1~nが出る
# 勝つためにx回表を出し続ければ良い
# 目 * 2^x >= kとなればよい

answer = 0
for i in range(n):
    dicenum = i + 1
    winline = dicenum
    x = 0
    while winline < k:
        x += 1
        winline = dicenum * (2**x)
    if dicenum == 1:
        first_x = x
        bunbo = n * 2**x #initialize
        bunsi = 1 #initialize
    elif x == 0:
        bunsi += 2**first_x
    else:
        bunsi += 2**(first_x - x)

answer = Decimal(str(bunsi)) / Decimal(str(bunbo))
print("{:.12f}".format(answer))