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::*;
use std::str::FromStr;

#[allow(unused_macros)]
macro_rules! scan {
  ($e:expr; $t:ty) => {
    $e.get::<$t>()
  };
  ($e:expr; $($t:ty), *) => {
    ($($e.get::<$t>(),)*)
  };
  ($e:expr => $i:expr; $t:ty) => {
    (0..$i).map(|_| $e.get::<$t>()).collect::<Vec<$t>>()
  };
  ($e:expr => $row:expr, $col:expr; $t:ty) => {
    (0..$i).map(|_| scan!($e => $col; $t)).collect::<Vec<$t>>()
  }
}

struct Scanner<R: Read> {
  reader: R
}

#[allow(dead_code)]
impl<R: Read> Scanner<R> {
  fn new(reader: R) -> Scanner<R> {
    Scanner { reader: reader }
  }
  fn safe_get<T: FromStr>(&mut self) -> Option<T> {
    let token = self.reader.by_ref().bytes().map(|c| c.unwrap() as char)
      .skip_while(|c| c.is_whitespace())
      .take_while(|c| !c.is_whitespace())
      .collect::<String>();
    if token.is_empty() { None } else { token.parse().ok() }
  }
  fn get<T: FromStr>(&mut self) -> T {
    if let Some(v) = self.safe_get() { v } else {
      writeln!(stderr(), "Terminated with EOF").unwrap();
      std::process::exit(0);
    }
  }
}

trait Writable {
  fn write(&self, sep: &'static str) -> String;
}

impl<T: ToString> Writable for Vec<T> {
  fn write(&self, sep: &'static str) -> String {
    self.iter().map(|e| e.to_string()).collect::<Vec<String>>().join(sep)
  }
}

fn main() {
  let cin = stdin();
  let cin = cin.lock();
  let mut sc = Scanner::new(cin);
  let (n, k) = scan!(sc; f64, f64);
  let mut ans = 0.0;
  for i in 1..((n as i32) + 1) {
    let i = i as f64;
    let cnt = coin_toss(i, k);
    ans += pow(2.0, -cnt);
  }
  println!("{}", ans * 1.0 / n);
}

fn coin_toss(cur: f64, goal: f64) -> f64 {
  let mut ans = 0.0;
  let mut cur = cur;
  while cur < goal {
    cur *= 2.0;
    ans += 1.0;
  }
  ans
}

fn pow(val: f64, expo: f64) -> f64 {
  if expo == 0.0 {
    1.0
  }
  else {
    let mut ans = 1.0;
    for _ in 0..abs(expo as i32) {
      if expo < 0.0 {
        ans /= val;
      }
      else {
        ans *= val;
      }
    }
    ans
  }
}

fn abs(val: i32) -> i32 {
  if val > 0 { val } else { -val }
}