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 Common Lisp (SBCL 1.1.14) (judge verdict: Wrong Answer)

(defun dice (n k)
  (let ((log2 (log 2)))
  (labels ((dice0 (n0 rv)
                  (if (or (= n0 k) (> n0 n)) rv
                    (let ((nx (ceiling (/ (log (/ k n0)) log2))))
                      ;(print `(:dice0 ,n0 ,nx))
                      (dice0 (+ n0 1) (+ rv (/ 1 (expt 2 nx))))))))
    (let ((p0 (/ (dice0 1 0) n)))
      (if (< n k) p0
        (+ p0
           (/ (+ (- n k) 1) n)))))))

#|
(print (dice 3 10))
(print (dice 100000 5))
(format t "~,10f" (dice 3 10))
(format t "~,10f" (dice 100000 5))
|#

(format t "~,10f" (dice (read) (read)))