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)

//import java.math.BigDecimal
import java.util.*

fun main(args: Array<String>) {
    val scanner = Scanner(System.`in`)
    val n = scanner.nextInt()
    val k = scanner.nextInt()

    val counts = (1..n).map { reqFace(it, k) }
    val maxCount = counts.max()!!
    val sum = counts.map { Math.pow(2.0, (maxCount - it).toDouble()) }.sum()
    print(sum / (n * (Math.pow(2.0, maxCount.toDouble()))))
//    val sum = counts.map { BigDecimal(2).pow(maxCount - it) }.sumBy { it.toInt() }
//    val ans = BigDecimal(sum).divide(BigDecimal(n).multiply(BigDecimal(2).pow(maxCount)))
//    print(ans.toPlainString())
}

fun reqFace(base: Int, req: Int): Int {
    if (base >= req) return 0

    var count = 1
    var score = base
    while (score * 2 < req) {
        score *= 2
        count++
    }
    return count
}