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 Rust (1.15.1) (judge verdict: Wrong Answer)

use std::io::stdin;

macro_rules! read {
    ($t: ty) => {{
        let mut line = String::new();
        stdin().read_line(&mut line);
        line.trim().parse::<$t>().unwrap()
    }};
    ($($t: ty), *) => {{
        let mut line = String::new();
        stdin().read_line(&mut line);
        let mut iter = line.split_whitespace();
        (
            $(iter.next().unwrap().parse::<$t>().unwrap(), )*
        )
    }}
}
macro_rules! reads {
    ($t:ty) => {{
        let mut v = vec![];
        let mut line = String::new();
        stdin().read_line(&mut line);
        let mut iter = line.split_whitespace();
        while let Some(n) = iter.next() {
            v.push(n.parse::<$t>().unwrap());
        }
        v
    }};
    ($e:expr, $t:ty) => {{
        let mut v = vec![];
        for _ in 0..$e {
            let mut line = String::new();
            stdin().read_line(&mut line);
            v.push(line.trim().parse::<$t>().unwrap());
        }
        v
    }};
}

fn main() {
    let (n, k) = read!(usize, usize);

    let mut a: f64 = 0.0;
    for i in 0..n {
        let mut d = i + 1;
        let s = 1.0 / n as f64;
        let mut c = 0;
        while d < k {
            c = c + 1;
            d = d * 2;
        }
        let b = s * (1.0 / 2.0f64.powi(c));
        a = a + b;
    }
    println!("a = {}", a);
}