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)

# すぬけ君は 1 〜 N の整数が等確率で出る N 面サイコロと表と裏が等確率で出るコインを持っています。
# すぬけ君は、このサイコロとコインを使って今から次のようなゲームをします。
# まず、サイコロを 1 回振り、出た目を現在の得点とする。
# 得点が 1 以上 K−1 以下である限り、すぬけ君はコインを振り続ける。
# 表が出たら得点は 2 倍になり、裏が出たら得点は 0 になる。
# 得点が 0 になった、もしくは K 以上になった時点でゲームが終了する。
# このとき、得点が K 以上である場合すぬけ君の勝ち、 0 である場合すぬけ君の負けである。
# N と K が与えられるので、このゲームですぬけ君が勝つ確率を求めてください。

input = gets.chomp.split
n = input[0].to_i
k = input[1].to_i

probability_sum = 0

n.times do |i|
  point = i + 1

  double_count = 0
  while point < k
    point *= 2
    double_count += 1
  end
  
  probability = (1.0 / n) * ((1.0 / 2.0) ** double_count)
  probability_sum += probability
end

puts probability_sum