cf Vasily the Bear and Fly_one beautiful day vasily the bear painted 2m circl-优快云博客

本文链接：https://blog.youkuaiyun.com/bossup/article/details/9883989

本博客介绍了一个有趣的数学问题：熊Vasily通过画圆来引导一只苍蝇在坐标平面上飞行，并计算苍蝇在m²天内平均飞行的距离。通过输入熊画的圆的数量m和圆的半径R，读者可以学习如何解决这个问题并计算出答案。

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One beautiful day Vasily the bear painted 2m circles of the same radius R on a coordinate plane. Circles with numbers from 1 to m had centers at points (2R - R, 0), (4R - R, 0), ..., (2Rm - R, 0), respectively. Circles with numbers from m + 1 to 2m had centers at points(2R - R, 2R), (4R - R, 2R), ..., (2Rm - R, 2R), respectively.

Naturally, the bear painted the circles for a simple experiment with a fly. The experiment continued for m² days. Each day of the experiment got its own unique number from 0 to m² - 1, inclusive.

On the day number i the following things happened:

The fly arrived at the coordinate plane at the center of the circle with number $\text{[math]}$ ( $\text{[math]}$ is the result of dividing number xby number y, rounded down to an integer).
The fly went along the coordinate plane to the center of the circle number $\text{[math]}$ ( $\text{[math]}$ is the remainder after dividing number x by number y). The bear noticed that the fly went from the center of circle v to the center of circle ualong the shortest path with all points lying on the border or inside at least one of the 2m circles. After the fly reached the center of circle u, it flew away in an unknown direction.

Help Vasily, count the average distance the fly went along the coordinate plane during each of these m² days.

Input

The first line contains two integers m, R (1 ≤ m ≤ 10⁵, 1 ≤ R ≤ 10).

Output

In a single line print a single real number — the answer to the problem. The answer will be considered correct if its absolute or relative error doesn't exceed 10^- 6.

Sample test(s)

input
1 1

output
2.0000000000

input
2 2

output
5.4142135624

Note

$\text{[math]}$

Figure to the second sample

#include <iostream>
#include<stdio.h>
#include<math.h>
using namespace std;
double f[2];
int main()
{
    int m,r,i,t;
    double addv;

    while(~scanf("%d%d",&m,&r))
    {
        addv=sqrt(2.0)*r;
        t=0;
        f[t]=2*r;
        for(i=2;i<=m;i++)
            {
                f[t^1]=f[t]+2*r+2*(2*r+(2*i-3)*addv+(__int64)(i-1)*(i-2)*r);
                t^=1;
            }
        printf("%.10lf\n",f[t]/((__int64)m*m));
    }
    return 0;
}