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: Accepted)

#include <stdio.h>
 
int main(){
	double N,K,ch,ch2;
	int count=0;
	double num=0.0;
	scanf("%lf%lf",&N,&K);
	for(int i=1.0000;i<=N;i++){
		ch2=1.00;
		ch=i;
		count=0;
		if(ch>=K){
			ch2=1.0000000000;
		}else{
			while(ch<K){
				ch=ch*2.00000000;
				//printf("%lf\n",ch);
				count++;
			}
			//printf("count=%d\n",count);
			if(count==1){
				ch2=0.5;
			}else{
			for(int j=1;j<=count;j++){
				ch2=ch2*(0.5000000000);
				//printf("ch2=%lf\n",ch2);
			}
		}
		}
		num+=(1.00000000/N)*ch2;
		//printf("num=%lf\n",num);
	}
	printf("%.16lf\n",num);
	return 0;
}