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: Wrong Answer)

require 'bigdecimal'                                                                
                                                                                    
def how_many_times(i, k)                                                            
  result = 0                                                                        
  i2 = i                                                                            
  while i2 <= k do                                                                  
    result += 1                                                                     
    i2 *= 2                                                                         
  end                                                                               
  result                                                                            
end                                                                                 
                                                                                    
FAKE = '100_000_000_000_000'                                                        
                                                                                    
A = gets.split.map(&:to_i)                                                          
N = A[0]                                                                            
K = A[1]                                                                            
                                                                                    
total_pred = BigDecimal('0.0')                                                      
if N < K                                                                            
  (1..N).each do |i|                                                                
    h = how_many_times(i, K)                                                        
    total_pred += BigDecimal("#{0.5**h}") * BigDecimal(FAKE) / BigDecimal("#{N}")    
  end                                                                               
else                                                                                
  (1...K).each do |i|                                                               
    h = how_many_times(i, K)                                                        
    add = BigDecimal("#{0.5**h}") * BigDecimal(FAKE) / BigDecimal("#{N}")           
    total_pred += add                                                               
  end                                                                               
                                                                                    
  add = BigDecimal(FAKE) * BigDecimal("#{(N - K + 1)}") / BigDecimal("#{N}")        
  total_pred += add                                                                 
end                                                                                 
                                                                                    
puts (total_pred / BigDecimal(FAKE)).to_s('F')