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: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace abc119d
{
    class Program
    {
        static void Main(string[] args)
        {
            var NK = ReadIntList();
            var N = NK[0];
            var K = NK[1];
            var ans = 0.0;

            for (int i = 1; i <= N; i++)
            {
                if (i >= K)
                    ans += 1.0 / N;
                else
                    ans += 1.0 / (N * (Math.Pow(2, Math.Floor(Math.Log((double) K / i, 2))+1)));
            }

            Console.WriteLine(ans);
#if DEBUG
            Console.ReadKey();
#endif
        }

        private static bool IsRange(int a, int from, int to) => (from <= a && a <= to);
        private static int ReadInt() => int.Parse(Console.ReadLine());
        private static List<int> ReadIntList() => Console.ReadLine().Split().Select(int.Parse).ToList();
        private static double ReadDouble() => double.Parse(Console.ReadLine());
        private static List<double> ReadDoubleList() => Console.ReadLine().Split().Select(double.Parse).ToList();
        private static long ReadLong() => long.Parse(Console.ReadLine());
        private static List<long> ReadLongList() => Console.ReadLine().Split().Select(long.Parse).ToList();
        private static List<int> Int2Digits(int a) => a.ToString().ToCharArray().Select(_ => int.Parse(_.ToString())).ToList();
        private static int BinarySearch<Type>(IReadOnlyList<Type> s, Type target) // target 以下の最大要素のインデックスを返す
            where Type : IComparable
        {
            var lower = 0;
            var upper = s.Count - 1;
            while (upper - lower > 1)
            {
                var mid = lower + (upper - lower) / 2;
                if (s[mid].CompareTo(target) <= 0)
                {
                    lower = mid;
                }
                else
                {
                    upper = mid;
                }

            }
            return lower;
        }
    }
}