Telescope

Dr. Extreme experimentally built an extremely precise telescope to investigate extremely curious phenomena at an extremely distant place. To make the telescope precise enough to investigate phenomena at such an extremely distant place, even a very small distortion is not allowed. However, he forgot about the influence of the internal gas affected by low-frequency vibration of magnetic flux passing through the telescope. The cylinder of the telescope is not affected by the low-frequency vibration, but the internal gas is.

The cross section of the telescope is a perfect circle. If he forms a coil by winding an extremely thin wire along the (inner) circumference, he can measure (the average vertical component of) the temporal variation of magnetic flux, and that measurement would be useful to estimate the influence. But the points on the circumference at which the wire can be fixed are limited, and the number of special clips to fix the wire is also limited. To obtain the highest sensitivity, he wants to form a coil of a polygonal shape with the largest area by stringing the wire among carefully selected points on the circumference.

Your job is to write a program that reports the maximum area of all possible m-polygons (polygons with exactly m vertices), each of whose vertices is one of the n points given on a circumference with a radius of 1. An example of the case n = 4 and m = 3 is illustrated below.

In the figure above, equations such as “p1 = 0.0” indicate the locations of the n given points, and decimals such as “1.000000” on m-polygons indicate the areas of m-polygons.

Parameter pi denotes the location of the i-th given point on the circumference (1 ≤ i ≤ n). The location p of a point on the circumference is in the range 0 ≤ p < 1, corresponding to the range of rotation angles from 0 to 2π radians. That is, the rotation angle from the point at p to the point at 0 equals 2πp radians. (π is the circular constant 3.14159265358979323846…)

You may rely on the fact that the area of an isosceles triangle ABC (AB = AC = 1) with an interior angle BAC of α radians (0 < α < π) is (1/2)·sin α, and the area of a polygon inside a circle with a radius of 1 is less than π.

The input consists of multiple subproblems followed by a line containing two zeros that indicates the end of the input. Each subproblem is given in the following format.

n m
p1 p2 ··· pn

n is the number of points on the circumference (3 ≤ n ≤ 40). m is the number of vertices to form m-polygons (3 ≤ m ≤ n). The locations of the n points, p1, p2, ..., pn, are given as decimals and are separated by either a space character or a newline. In addition, you may assume that 0 ≤ p1 < p2 < ··· < pn < 1.

For each subproblem, output the maximum area, each on a separate line. Each value in the output may not have an error greater than 0.000001, and its fractional part should be represented by 6 decimal digits after the decimal point.