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 <iomanip>
#include <cmath>
#include <functional>
#include <algorithm>
#include <numeric>
#include <vector>
#include <list>
#include <queue>
#include <stack>
#include <deque>
#include <map>
#include <set>
#include <unordered_map>
#include <unordered_set>
#include <boost/range/irange.hpp>
#include <boost/container/static_vector.hpp>
#include <boost/utility/string_ref.hpp>
#include <boost/multiprecision/cpp_int.hpp>
#include <boost/math/common_factor_rt.hpp>

class input
{
    std::istream& cin;

public:
    input() : cin(std::cin)
    {
        std::cin.tie(nullptr);
        std::ios::sync_with_stdio(false);
    }

    template <class T>
    inline input& operator>>(T& var)
    {
        cin >> var;
        return *this;
    }
    template <class T>
    inline input& operator,(T& var)
    {
        cin >> var;
        return *this;
    }

    template <class T>
    inline T read()
    {
        T var;
        cin >> var;
        return var;
    }

    template <class InputIterator>
    inline void read(InputIterator head, std::size_t n)
    {
        for (std::size_t i = 0; i < n; ++i) {
            cin >> *head;
            ++head;
        }
    }

    template <class T>
    inline std::vector<T> read(std::size_t n)
    {
        std::vector<T> res;
        res.reserve(n);

        for (std::size_t i = 0; i < n; ++i) {
            T tmp;
            cin >> tmp;
            res.emplace_back(tmp);
        }
        return res;
    }

    template <class T, class F>
    inline void online(std::size_t n, F&& f)
    {
        T var;
        for (std::size_t i = 0; i < n; ++i) {
            cin >> var;
            f(var);
        }
    }
};
template <class T, std::size_t n>
std::istream& operator>>(std::istream& in, std::array<T, n>& var)
{
    for (std::size_t i = 0; i < n; ++i) {
        in >> var[i];
    }
    return in;
}

class output
{
    std::ostream& cout;

public:
    template <bool nl>
    class proxy
    {
        std::ostream& cout;

    public:
        proxy() : cout(std::cout) {}
        ~proxy() { final(); }
        template <class T>
        proxy& operator,(T&& val)
        {
            cout << ' ' << val;
            return *this;
        }
        template <bool f = nl>
        auto final() -> std::enable_if_t<f> { cout << '\n'; }
        template <bool f = nl>
        auto final() -> std::enable_if_t<!f> {}
    };

public:
    output() : cout(std::cout)
    {
        cout << std::fixed << std::setprecision(15);
    }

    template <class T>
    proxy<true> operator,(T&& val)
    {
        cout << val;
        return {};
    }
    proxy<false> noreturn() { return {}; }

    template <class T>
    void print(const T& container)
    {
        bool first = true;
        for (auto&& x : container) {
            if (!first) {
                cout << ' ';
            }
            cout << x;
            first = false;
        }
    }
};

void prog()
{
    input in;
    output out;

    int n, k;
    in >> n, k;

    double p = 0;
    for (int i : boost::irange(1, n + 1)) {
        double t = 1;
        while (true) {
            if (k <= i) {
                p += t / n;
                break;
            }
            i *= 2;
            t /= 2;
        }
    }
    out, p;
}

int main()
{
    prog();
    return 0;
}