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 Kotlin (1.0.0) (judge verdict: Accepted)

fun main(arr:Array<String>) {
    val (n,k) = readLine()!!.split(" ").map { it.toInt() }
    goal = k
    val results = (1..n).map { calc(it) }
    val max = results.maxBy { it.total}!!.total
    var win = 0L
    var total = 0L
    results.forEach {
        win += (it.win * (max/it.total))
        total += max
    }
    println("%.10f".format(win.toDouble()/total.toDouble()))
}

class Result(val win:Long, val total:Long)
var goal = 0

val dic = mutableMapOf<Int, Result>()
fun calc(num:Int):Result {
    if(num >= goal) {
        return Result(1, 1)
    }
    if(num <= 0) {
        return Result(0,1)
    }
    dic[num]?.let { return it }
    val ret = calc(num*2)
    val ans = Result(ret.win, ret.total * 2)
    dic[num] = ans
    return ans
}