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 C (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>

int main()
{
	int n, k, time = 0, tmp = 1;
	double ans = 0, subans = 0;

	scanf("%d %d", &n, &k);

	while (tmp < k)
	{
		tmp *= 2;

		time++;
	}

	subans = (double)pow(0.5, time) / n;
	ans += subans;

	tmp = 1;

	if (n < k)
	{
		for (int i = 2; i <= n; i++)
		{
			if (i - 1 == tmp)
			{
				subans *= 2.0;
				tmp *= 2;
			}

			ans += subans;
		}
	}
	else if (n != 1)
	{
		for (int i = 2; i <= k - 1; i++)
		{
			if (i - 1 == tmp && tmp * 2 <= k)
			{
				subans *= 2.0;
				tmp *= 2;
			}

			ans += subans;
		}

		if (k != 1)
			ans += (double)(n - k + 1) / n;
		else
			ans += (double)(n - k) / n;
	}

	printf("%.11lf\n", ans);

	return 0;
}