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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include<iostream>
#include<cstdlib>
#include<cstring>
#include<stack>
#include<queue>
#include<algorithm>
#include<vector>
#include<math.h>
#include<bitset>
#include<iomanip>
using namespace std;

typedef stack<int> IntStack;
typedef queue<int> IntQueue;

const double pi = 3.141592653589793;

int char_to_int(char c)
{
	if (c == '0') return 0;
	if (c == '1') return 1;
	if (c == '2') return 2;
	if (c == '3') return 3;
	if (c == '4') return 4;
	if (c == '5') return 5;
	if (c == '6') return 6;
	if (c == '7') return 7;
	if (c == '8') return 8;
	if (c == '9') return 9;
	else return -1;
}

/*ここまでテンプレ*//*ここまでテンプレ*//*ここまでテンプレ*//*ここまでテンプレ*/

double win[20];

int main()
{	
	double N, K;
	cin >> N >> K;
	double n = N;
	
	int i = 0;
	while (N >= 1)
	{
		while (K <= N)
		{
			win[i]++;
			N--;
		}
		K /= 2;
		i++;
	}

	double ans = 0;
	double d = 1;
	for (int j = 0; j < 20; j++)
	{
		ans += win[j] / d;
		d *= 2;
	}

	cout << fixed << setprecision(12) << ans / n;

	return 0;
}