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# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Linq;
using System.Collections.Generic;

namespace Contest
{
    class Program
    {
        static void Main(string[] args)
        {
            ABC126.C();
        }

        class ABC126
        {
            public static void A()
            {
                var N = Read.Int();
                var K = Read.Int() - 1;
                var S = Read.Str();
                Console.WriteLine(String.Join("", S.Select((x, i) => (char)(i == K ? x - 'A' + 'a' : x))));
            }
            public static void B()
            {
                var S = Read.Str();
                var v1 = Convert.ToInt32(S.Substring(0, 2));
                var v2 = Convert.ToInt32(S.Substring(2, 2));
                if (1 <= v1 && v1 <= 12)
                {
                    if (1 <= v2 && v2 <= 12) { Console.WriteLine("AMBIGUOUS"); } else { Console.WriteLine("MMYY"); }
                }
                else
                {
                    if (1 <= v2 && v2 <= 12) { Console.WriteLine("YYMM"); } else { Console.WriteLine("NA"); }
                }
            }
            public static void C()
            {
                var N = Read.Int();
                var K = Read.Int();
                double prob = 0;
                for(int n = 1; n <= N; ++n)
                {
                    int po = n;
                    double pr = 1.0 / N;
                    while (po < K)
                    {
                        po *= 2;
                        pr /= 2;
                    }
                    prob += pr;
                }
                Console.WriteLine(prob);
            }
            public static void D()
            {
                var N = Read.Int();
                var ADJ = Read.AdjacencyListWithWeight(N, N - 1);

                long[] dist = Enumerable.Range(0, N + 1).Select(i => (long)-1).ToArray();
                dist[1] = 0;
                Queue<int> queue = new Queue<int>();
                queue.Enqueue(1);
                while (queue.Count > 0)
                {
                    var v = queue.Dequeue();
                    foreach(var kvp in ADJ[v])
                    {
                        var w = kvp.Key;
                        if (dist[w] < 0)
                        {
                            queue.Enqueue(w);
                            dist[w] = dist[v] + kvp.Value;
                        }
                    }
                }
                for(int i = 1; i <= N; ++i)
                {
                    Console.WriteLine(dist[i] % 2 == 0 ? "0" : "1");
                }
            }
            public static void E()
            {
                var N = Read.Int();
                var M = Read.Int();
                var XY = Read.AdjacencyList(N, M);

                bool[] check = new bool[N + 1];
                int count = 0;
                for (int n = 1; n <= N; ++n)
                {
                    if (check[n]) { continue; }
                    count++;
                    Queue<int> queue = new Queue<int>();
                    queue.Enqueue(n);
                    while (queue.Count > 0)
                    {
                        var v = queue.Dequeue();
                        foreach (var w in XY[v])
                        {
                            if (!check[w])
                            {
                                check[w] = true;
                                queue.Enqueue(w);
                            }
                        }
                    }
                }
                Console.WriteLine(count);
            }
            public static void F()
            {
                var M = Read.Int();
                var K = Read.Int();
                if (K >= (1 << M)) { Console.WriteLine(-1); return; }
                if (M == 1 && K == 1) { Console.WriteLine(-1); return; }
                List<int> list = new List<int>();
                list.Add(K);
                for (int i = 0; i < (1 << M); ++i)
                {
                    if (i != K) { list.Add(i); }
                }
                list.Add(K);
                for (int i = (1 << M) - 1; i >= 0; --i)
                {
                    if (i != K) { list.Add(i); }
                }
                Console.WriteLine(String.Join(" ", list));
            }
        }
        class Util
        {
        }

        static class Read
        {
            static Queue<string> queue = new Queue<string>();
            static void Enqueue() { var ss = Console.ReadLine().Split(' '); foreach (var s in ss) { queue.Enqueue(s); } }
            public static int Int() { if (queue.Count == 0) { Enqueue(); } return Convert.ToInt32(queue.Dequeue()); }
            public static long Long() { if (queue.Count == 0) { Enqueue(); } return Convert.ToInt64(queue.Dequeue()); }
            public static string Str() { if (queue.Count == 0) { Enqueue(); } return queue.Dequeue(); }
            public static int[] IntN() { return Console.ReadLine().Split(' ').Select(s => Convert.ToInt32(s)).ToArray(); }
            public static long[] LongN() { return Console.ReadLine().Split(' ').Select(s => Convert.ToInt64(s)).ToArray(); }
            public static string[] StrN() { return Console.ReadLine().Split(' '); }
            public static Tuple<int, int>[] TupleInt(int n) { List<Tuple<int, int>> list = new List<Tuple<int, int>>(); for (int i = 0; i < n; ++i) { var ab = Read.IntN(); list.Add(new Tuple<int, int>(ab[0], ab[1])); } return list.ToArray(); }
            public static Tuple<long, long>[] TupleLong(int n) { List<Tuple<long, long>> list = new List<Tuple<long, long>>(); for (int i = 0; i < n; ++i) { var ab = Read.LongN(); list.Add(new Tuple<long, long>(ab[0], ab[1])); } return list.ToArray(); }
            public static Tuple<string, string>[] TupleStr(int n) { List<Tuple<string, string>> list = new List<Tuple<string, string>>(); for (int i = 0; i < n; ++i) { var ab = Read.StrN(); list.Add(new Tuple<string, string>(ab[0], ab[1])); } return list.ToArray(); }
            public static Tuple<double, double>[] TupleDouble(int n) { List<Tuple<double, double>> list = new List<Tuple<double, double>>(); for (int i = 0; i < n; ++i) { var ab = Read.LongN(); list.Add(new Tuple<double, double>(ab[0], ab[1])); } return list.ToArray(); }
            public static long[,] LongMatrix(int r, int c)
            {
                var mat = new long[r, c];
                for (int i = 0; i < r; ++i)
                {
                    var x = Read.LongN();
                    for (int j = 0; j < c; ++j)
                    {
                        mat[i, j] = x[j];
                    }
                }
                return mat;
            }
            public static Dictionary<int, HashSet<int>> AdjacencyList(int N, int M)
            {
                Dictionary<int, HashSet<int>> dict = new Dictionary<int, HashSet<int>>();
                for (int i = 0; i <= N; ++i)
                {
                    dict.Add(i, new HashSet<int>());
                }
                for (int i = 0; i < M; ++i)
                {
                    var ab = Read.IntN();
                    dict[ab[0]].Add(ab[1]);
                    dict[ab[1]].Add(ab[0]);
                }
                return dict;
            }
            public static Dictionary<int, Dictionary<int, long>> AdjacencyListWithWeight(int N, int M)
            {
                Dictionary<int, Dictionary<int, long>> dict = new Dictionary<int, Dictionary<int, long>>();
                for (int i = 0; i <= N; ++i)
                {
                    dict.Add(i, new Dictionary<int, long>());
                }
                for (int i = 0; i < M; ++i)
                {
                    var ab = Read.IntN();
                    dict[ab[0]].Add(ab[1], ab[2]);
                    dict[ab[1]].Add(ab[0], ab[2]);
                }
                return dict;
            }
        }
    }
}