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 9.2.1) (judge verdict: Accepted)

#include<bits/stdc++.h>
using namespace std;

#define fastio 			ios::sync_with_stdio(0);	cin.tie(0); cout.tie(0);  cout<<fixed;  cout<<setprecision(12);
#define randomINT 		mt19937 rnd(chrono::steady_clock::now().time_since_epoch().count());
#define	newl 			cout<<"\n"
#define DISP(as)		for(auto it : as) cout<<it<<" ";newl;
#define all(x)         	(x).begin(),(x).end()
#define mset(x,val)    	memset(x,val,sizeof(x))
#define newl           	cout<<"\n"
#define pb             	push_back
#define mp             	make_pair
#define f 				first
#define s 				second
#define dline          cerr<<"///REACHED///\n";
#define deb1(x)        cerr<<#x<<" = "<<x<<'\n';
#define deb2(x,y)      cerr<<'['<<#x<<','<<#y<<"] = "<<'['<<x<<','<<y<<']'<<'\n';
#define deb3(x,y,z)    cerr<<'['<<#x<<','<<#y<<','<<#z<<"] = "<<'['<<x<<','<<y<<','<<z<<']'<<'\n';

typedef long long 				ll;
typedef long double 			ld;
typedef vector<ll> 				vll;
typedef pair<ll , ll> 			pll;
typedef pair<ld, ld> 			pld;
typedef unordered_map<ll, ll> 	um;
typedef vector<pll> 			vpll;

const ll 	MAX5 	= 	1e+5 + 7;
const ll 	MAX7 	= 	1e+7 + 7;
const ll 	MAXN 	= 	MAX7;
const ll   	INF   	=  	0x7f7f7f7f7f7f7f7f;
const int  	INFi 	=  	0x7f7f7f7f;
const ll 	MOD		=  	1e+9 + 7;

const ll N = 3*MAX5;
// <------------- Declare Variables Here ------------> //

ll T = 1;
ll n, x, m, y, k, z, l;
string s, t, d, c;

// <------------- Implement Functions Here ----------> //



// <------------- Start of main() -------------------> //

void MAIN() {
	ld n,k;
	cin >> n >> k;
	ld ans = 0;
	for(ll i=1;i<=n;i++) {
		ld score = i;
		ld val = 1;
		ll cou = 0;
		while(score < k) cou++,score*=2;
		val /= (n);
		val /= pow(2,cou);
		ans += val;
	}
	cout<<ans;
}

int main() {
	fastio; randomINT;
	//cin >> T;
	while (T--) {
		MAIN();
	}
	return 0;
}