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 (1.42.0) (judge verdict: Accepted)

#[allow(dead_code)]

pub struct Scanner {
    idx: usize, 
    buf: Vec<String>,
}

impl Scanner {
    pub fn new<T: std::io::Read>(inf: &mut T) -> Scanner {
        Self {
            idx:0,
            buf: {
                let mut s = String::new();
                inf.read_to_string(&mut s).expect("I/O error");
                s.split_whitespace().map(|x| x.to_owned()).collect()
            },
        }
    }

    pub fn read<T: std::str::FromStr> (&mut self) -> T 
    where
        <T as std::str::FromStr>::Err: std::fmt::Debug,
    {
        if self.empty() {
            panic!("reached the end of input")
        }
        let ret = self.buf[self.idx].parse::<T>().expect("parse error");
        self.idx += 1;
        return ret;
    }

    pub fn empty(&self) -> bool {
        return self.idx >= self.buf.len();
    }
}

fn main() {
    let mut sc = Scanner::new(&mut std::io::stdin());
    let n: usize = sc.read();
    let k: usize = sc.read();

    let mut ans = 0.0;

    for mut i in 1..=n {
        if i >= k {
            ans += 1.0 / n as f64;
        }
        else {
            let mut c = 1.0 / n as f64;
            while i < k {
                i *= 2;
                c /= 2.0;
            }
            ans += c;
        }
    }

    println!("{}", ans);
}