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 Fortran (gfortran v4.8.4) (judge verdict: Accepted)

program main
	implicit none
	integer :: N, K, counter, i, m, N_const
	real(8) :: Right, Left, ans
	read(*, *) N, K
	ans = 0.0d0
	counter = 0
	N_const = N
	if(N >= K) then
		ans = ans + dble(N-K+1)  / dble(N_const)
		N   = K
	end if
	do m = 1, 20
		Right = dble(K)*(0.50d0)**(m-1)
		Left  = dble(K)*(0.50d0)**m
		counter = 0
		do i = N, 1, -1
			if(Left <= i .AND. i < Right)then
				counter = counter + 1
			end if
		end do
		ans = ans + (dble(counter)/dble(N_const))*(0.50d0)**m
		N = N - counter
!		write(*, *) Left, Right, ans, N
		if(Left <= 1) then
			exit
		end if
	end do
	write(*, *) ans
end program main