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: Wrong Answer)

import java.util.Scanner

import scala.collection.mutable.ArrayBuffer


object Main {


  def read() = {
    val sc = new Scanner(System.in)
    val n, k = sc.nextInt()
    (n, k)
  }

  def mypow(x: Double, t: Int): Double = {
    if(t == 0){
      return 1
    }
    val y = mypow(x, t / 2)
    if (t % 2 == 0) {
      y * y
    }else{
      x * y * y
    }
  }

  def solve(n: Int, k: Int): Double = {
    val eps = 1e-9
    val tlist = (1 to n).map(i => {
      var t = 0
      var point = i
      while (point <= k) {
        point = 2 * point
        t = t + 1
      }
      t
    })
    val hoge = 22
    val tmax = tlist.max
    var mapt = Array.fill(tmax+1 + hoge)(0)
    for(t <- tlist){
      mapt(t+hoge) = mapt(t+hoge) + 1
    }
    for(t <- (1 to tmax+hoge).reverse){
      mapt(t-1) += mapt(t)/2
      mapt(t) %= 2
    }
    val fuga = mapt.zipWithIndex.map{
      case (x, t) => x * Math.pow(0.5, t-hoge)
    }
    mapt.zipWithIndex.map{
      case (x, t) => x * Math.pow(0.5, t-hoge)
    }.reverse.sum/n
  }

  def main(args: Array[String]): Unit = {
    val (n, k) = read()
    println(solve(n, k))
  }
}