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 Ruby (2.3.3) (judge verdict: Accepted)

N, K = gets.chomp.split.map(&:to_i)
# S = gets.chomp

# Ex.
# N = 3, K = 10
# Start: 1 or 2 or 3
# If start is 1, 
  # 1, 2, 4, 8, 16, so we have to win 4 times. This is (1/2)^4
# If start is 2,
  # 2, 4, 8, 16, so we have to win 3 times. This is (1/2)^3
# If start is 3,
  # 3, 6, 12, so we have to win 2 times. This is (1/2)^2
# So, ((1/2)^4 + (1/2)^3 + (1/2)^2)/3

pairs = []

r = 0
1.upto(N).each do |n|  # n is start
  i = 0
  while (n < K) do
    n *= 2
    i += 1
  end
  r += 0.5 ** i
end

r /= N

p r