Points on a Circle

Time limit2sMemory limit512 MB

Summary

Given n random points on a unit circle and an angle p, compute -log2 of the probability that all n points fit inside some arc of central angle p.

Level: Hard8 of 10
Topics: Probability, Math, Combinatorics, Implementation
Solved: No attempts yet

Problem

You place $n$ points at random on the circumference of a circle of radius 1. Each point is equally likely to land anywhere on the circumference.

Let $P$ be the probability that all $n$ points lie on one arc of that same circle whose central angle is at most $p$ degrees. The arc may begin at any point of the circumference.

Because $P$ can be extremely small, print $-\log_2 P$ instead of $P$ .

Input

The first line contains the number of points $n$ and the angle $p$ . ( $1 \le n \le 100000$ , $1 \le p < 180$ )

Both $n$ and $p$ are positive integers.

Output

Print $-\log_2 P$ with exactly six digits after the decimal point, rounded at the seventh digit.

Examples3

Example 1
Input
```
1 150
```
Expected output
```
0.000000
```
Example 2
Input
```
2 179
```
Expected output
```
0.008037
```
Example 3
Input
```
5 90
```
Expected output
```
5.678072
```