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 <cstddef>
#include <complex>
#include <iostream>
#include <iomanip>
#include <string>
#include <sstream>
#include <vector>
#include <array>
#include <list>
#include <unordered_map>
#include <unordered_set>
#include <algorithm>
#include <numeric>
#include <limits>
#include <set>
#include <map>
#include <queue>
#include <stack>
#include <bitset>
#include <valarray>
#include <utility>
#include <tuple>
#include <cmath>
#include <memory>
#include <random>
#include <fstream>
#include <locale>
#include <bitset>

using ll = long long;
using ull = unsigned long long;

//vector
template <class T>
using vec = std::vector<T>;

template <class T>
using vvec = vec<vec<T>>;

constexpr std::size_t operator""_sz(ull n) { return std::size_t(n); }

ll gcd(ll a, ll b)
{
    if (a > b)
        std::swap(a, b);

    while (true)
    {
        auto m = b % a;
        if (m == 0)
            return a;
        b = a;
        a = m;
    }
}

int win(ll x, ll k)
{
    if (x >= k)
        return 0;
    int i = 1;
    while (x * (1ll << i) < k)
        ++i;

    return i;
}

int main()
{
    using namespace std;

    ll n, k;
    cin >> n >> k;

    long double p{};

    for (int i = 1; i <= n; ++i)
    {
        if (i < k)
        {
            auto x = win(i, k);
            p += powl(0.5l, x) / n;
        }
        else
        {
            p += 1.0l / n;
        }
    
    }

    cout << fixed << setprecision(15) << p << endl;

    return 0;
}