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: Accepted)

fn main() {
    let std_in = std::io::stdin();
    let mut br = std::io::BufReader::new(std_in.lock());

    let tmp = rvn::<i32>(&mut br);
    let (N, K) = (tmp[0], tmp[1]);
    let mut sm = 0.0;
    for i in 1..N+1 {
        let mut bb = 1;
        while i * bb < K {
            bb *= 2;
        }
        sm += 1.0 / bb as f64;
    }
    println!("{}", sm / N as f64);
}


//////////////////////////////////////////////////////////////////////////////////
use std::io::{BufReader, BufRead, StdinLock};

#[allow(dead_code)]
fn rs(br: &mut BufReader<StdinLock>) -> Vec<char> {
    let mut buf = String::new();
    br.read_line(&mut buf).ok();
    buf.trim().chars().collect()
}

#[allow(dead_code)]
fn rn<T: std::str::FromStr>(br: &mut BufReader<StdinLock>) -> T {
    let mut buf = String::new();
    br.read_line(&mut buf).ok();
    buf.trim().parse().ok().unwrap()
}

#[allow(dead_code)]
fn rvs(br: &mut BufReader<StdinLock>) -> Vec<Vec<char>> {
    let mut buf = String::new();
    br.read_line(&mut buf).ok();
    buf.trim().split_whitespace().map(|e| e.chars().collect()).collect()
}

#[allow(dead_code)]
fn rvn<T: std::str::FromStr>(br: &mut BufReader<StdinLock>) -> Vec<T> {
    let mut buf = String::new();
    br.read_line(&mut buf).ok();
    buf.trim().split_whitespace().map(|e| e.parse().ok().unwrap()).collect()
}