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)

use std::io::stdin;
use std::io::Error;
use std::io::Read;

#[derive(Debug)]
struct Input {
    n: u32,
    k: u32,
}

fn handle_input(input_str: String) -> Result<Input, Error> {
    let mut iter = input_str.split_whitespace();
    let n: u32 = iter.next().unwrap().parse().unwrap();
    let k: u32 = iter.next().unwrap().parse().unwrap();
    let input = Input { n: n, k: k };
    Ok(input)
}

fn handle_output(input: Input) -> Result<String, Error> {
    let mut sum = 0_f64;
    for i in 1..input.n + 1 {
        if i >= input.k {
            sum += 1_f64 / input.n as f64;
            continue;
        }
        let cast = (((input.k - 1) / i) as f64).log2() as i32 + 1;
        sum += 0.5_f64.powi(cast) / input.n as f64;
    }
    Ok(format!("{}", sum))
}

fn main() {
    let mut input_str = String::new();
    stdin().read_to_string(&mut input_str).unwrap();
    let input = handle_input(input_str).unwrap();
    let output = handle_output(input).unwrap();
    println!("{}", output);
}