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 PHP7 (7.0.15) (judge verdict: Accepted)

#!/usr/bin/env php
<?php

function genInputs() {
    while(($input = rtrim(fgets(STDIN))) !== false) {
        foreach (explode(' ', $input) as $v) {
            yield $v;
        }
    }
}
function getNext($gen) {
    $value = $gen->current();
    $gen->next();
    return $value;
}
$gen = genInputs();

$N = (int)getNext($gen);
$K = (int)getNext($gen);

$q = 0;
while (pow(2, $q) < $K) { $q++; }
$per = pow(2, $q);
$sum = 1;

$p = 0;
for ($i = $N; $i > 1; $i--) {
    while ($i * pow(2, $p) < $K) { $p++; }
    $sum += pow(2, $q - $p);
}

echo ($sum / $per / $N).PHP_EOL;