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1s, 512 MB

Alice has been given the responsibility of creating the background of the annual play in her school. The background consists of Mountains. One such Background-

*Figure 1: A Cardboard Mountain Background.*

The Mountains' characteristics are —

Mountains are triangles whose base is on the floor.

A stick is placed vertically from the floor to the peak of the mountain to support it.

The lower right vertex of a mountain starts from the stick of the mountain to the immediate left and the lower right vertex on the stick of the mountain to the immediate right.

The left-most mountain's lower left vertex is on the lower left corner of the background.

The right-most mountain's lower right vertex is on the lower right corner of the background.

The width of the stick is negligible.

Note that, a mountain $m_1$ is defined left (or, right) to another mountain $m_2$ if the stick supporting $m_1$ is strictly on the left (or, right respectively) of $m_2$. Furthermore, $m_1$ is the immediate left (or, right) of $m_2$ if among all of the mountains who are on left (or, right) of $m_2$, $m_1$ is the right-most (or, left-most respectively).

The Mountains are traditionally created this way. But Alice being a problem solver has thought of an easier way to create the mountains. Alice thought of doing the following —

Consider the background as a 2D Cartesian plane, where the lower left corner will be the origin and the lower right corner will be $(X, 0)$, where $X$ is the length of the background.

Fix the coordinates of the peaks of $N$ Mountains on the Cartesian plane, then draw the Mountains respecting their characteristics, on a paper for reference.

Cut a single cardboard shape according to that reference.

Let’s say Alice has generated 5 mountain peaks’ co-ordinates. They are $(8, 6)$, $(22, 8)$, $(4, 10)$, $(13, 12)$ and $(18, 14)$. The length of the background is $25$ units. The final shape of the cardboard shape can be denoted by *figure 1*.

Alice is wondering what would be the area of the final cardboard shape for the Mountains on the background after cutting. Given the coordinates of the $N$ mountain tips, and the length of the background, $X$, can you calculate the area of the final cardboard shape?

The first line contains a single integer $T$ $(1≤T≤10^4)$, the number of test cases.

The first line of each test case contains two space-separated integers, $N$$(1 \le N \le 2 \times 10^5)$ and $X$$(2 \le X \le 10^6)$, denoting the number of peaks and the background’s length respectively.

The $i^{th}$ line of the next $N$lines contain two integers $x_i$ and $y_i$ $(0 < x_i < X, \; 1 \le y_i \le 10^6)$, the coordinate of the $i^{th}$ mountain.

The sum of $N$ all over test cases doesn’t exceed $2 \times 10^5$.

For each test case, output the area of the final cardboard shape. Your answer will be considered correct if the relative or absolute difference is less than $10^{-6}$. That means, if your answer is $A$ and jury’s answer is $B$, your answer will be considered correct if $\frac{|A-B|}{max(1,B)}<10^{-6}$.

Input | Output |
---|---|

1 5 25 8 6 18 14 4 10 13 12 22 8 | 174.16433566 |