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 Scala (2.11.7) (judge verdict: Accepted)

object Main {
  import java.util.Scanner

  implicit class RichScanner(sc: Scanner) {
    def readStrings(n: Int): Array[String] = Array.fill(n) { sc.next() }
    def readInts(n: Int): Array[Int] = Array.fill(n) { sc.nextInt() }
    def readLongs(n: Int): Array[Long] = Array.fill(n) { sc.nextLong() }
    def readLineStrings: Array[String] = sc.nextLine().split(' ')
    def readLineInts: Array[Int] = readLineStrings.map(_.toInt)
    def readLineLongs: Array[Long] = readLineStrings.map(_.toLong)
  }

  def main(args: Array[String]): Unit = solve(new Scanner(System.in))

  def sumProbOfWinning(n: Int, k: Int): Double = {
    if (k == 1) return 0

    val nextBoundary = (k + 1) / 2

    val winIfSuccessfulRange = {
      if (n * 2 < k) 0
      else Math.min(n + 1, k) - nextBoundary
    }

    winIfSuccessfulRange / 2.0 + sumProbOfWinning(n, nextBoundary) / 2.0
  }

  def solve(sc: Scanner): Unit = {
    val Array(n, k) = sc.readLineInts

    val sum = if (n >= k) {
      (n - k + 1) + sumProbOfWinning(k - 1, k)
    } else {
      sumProbOfWinning(n, k)
    }

    println(sum / n)
  }
}