Migrated from Lutece 799 Game with Randomness

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Description

Alice and Bob like playing games. On the first day, they randomly get two different integers from ${1, 2, \cdots, N}$, which means each pair of different integers have the same probility to be chosen. We call the smaller one $x$, and the larger one $y$. $x$ and $y$ keep unchanged all the time. On each of the following days, Alice will choose an integer $d_x$ from ${0, 1, \cdots, x}$ and Bob will choose an integer $d_y$ from ${0, 1, \cdots, y}$ simultaneously. After then, Alice will get a score $S_x$ of that day, which is either $x+d_x$ or $x-d_x$ with equal probability. And similarly, Bob will get a score $S_y$, which is either $y+d_y$ or $y-d_y$ with equal probability. If $S_x$ is no less than $S_y$, Alice will be the winner of that day. Otherwise Bob is.

The strategy of a player on a specific day is the way of choosing $d_x$ or $d_y$ on that day. On the second day, Alice's strategy is to randomly choose $d_x$ from ${0, 1, \cdots, x}$. And Bob's strategy is to randomly choose $d_y$ from ${0, 1, \cdots, y}$. We also know both players' strategies on the $K_{th}$ ($K > 2$) day.

If $K$ is odd, Bob's strategy keeps the same as the previous day, which means different $d_y$s have the same probabilities with the previous day to be chosen by Bob. However, Alice's strategy changes. She knows Bob's strategy on this day and she will choose a $d_x$ from ${0, 1, \cdots, x}$ to maximize her winning probability. If there are several $d_x$es leading to her maximum winning probability, her strategy is to randomly choose one.

If $K$ is even, Alice will do what Bob do and Bob will do what Alice do when $K$ is odd, which means Alice's strategy keeps the same as the previous day and Bob's strategy is to choose a $d_y$ from ${0, 1, \cdots, y}$ to maximize his winning probability. Similarly, if there are several $d_y$s leading to Bob's maximum winning probability, his strategy is to randomly choose one.

You are given an int $N$ and an int $K$. Answer Alice's winning probability on the $K_{th}$ day.

Input

The input contains $2$ integers $N, K$ ($2\leq N, K\leq 10^6$), as mentioned above.

Output

Output Alice's winning probability on the $K_{th}$ day, rounded to $4$ digits after decimal point.

Samples

2 2

0.3750

2 3

0.4167

99999 99

0.2500

Note

Alice knows Bob's strategy on a specific day means that Alice knows the probabilities of different $d_y$s to be chosen by Bob, but she may not know which $d_y$ Bob will choose on that day, and vice versa.

When you calculate Alice's winning probability on the $K_{th}$ day, $x$ and $y$ haven't been chosen yet.

Resources

the 12th UESTC Programming Contest Preliminary