Autocorrelation Function

Autocorrelation measures how much a signal matches a shifted copy of itself. For a continuous function $f(x)$ defined on the whole real line, the autocorrelation function $R_f(r)$ is

$R_f(r) = \int_{-\infty}^{\infty} f(x)\,f(x + r)\,dx$

where $r$ is a real number.

Every function in this problem is piecewise linear and continuous. It is given by $n$ endpoints $(x_1, y_1)$ through $(x_n, y_n)$ with distinct $x$-coordinates, and consecutive endpoints are joined by a straight segment. The first and the last endpoint have $y$-coordinate 0, and $f$ is 0 outside the interval $[x_1, x_n]$. The figure below shows one such function.

Given $f$ and $r$, compute $R_f(r)$.

The input holds several data sets.

The first line of each data set has an integer $n$ ($3 \le n \le 100$) and a real number $r$ ($-100 \le r \le 100$). Here $n$ is the number of endpoints that form $f$, and $r$ is the parameter of the autocorrelation function. Each of the next $n$ lines has two integers $x_i$ ($-100 \le x_i \le 100$) and $y_i$ ($-50 \le y_i \le 50$), the coordinates of the $i$-th endpoint. The endpoints come in increasing order of $x$-coordinate, and no two endpoints share an $x$-coordinate. The first and the last endpoint always have $y$-coordinate 0.

A line with $n$ and $r$ both equal to 0 ends the input. That line is not a data set, so do not process it.

For each data set, print the value of $R_f(r)$ on its own line. Round the value at the seventh digit after the decimal point and print exactly six digits after the decimal point. If the value is 0, print 0.000000 and not -0.000000.