Alice and Bob have joined a competitive $\pi e$ game! Initially, there are $X$ slices of good $\pi e$, and $Y$ slices of bad $\pi e$. With Alice going first, they will take turns picking one of the slices of $\pi e$ uniformly at random and eating it. If a player eats a good slice of $\pi e$, they gain one point, and if they eat a bad slice of $\pi e$, they lose one point. After each turn, they will bake a good/bad slice of $\pi e$ to replace the type of $\pi e$ that was just eaten. Then, they will bake $Z$ additional slices of $\pi e$, each with equal probability of being either good or bad.

Alice and Bob will spend the rest of eternity competing in this $\pi e$ game! Determine the limit of the expected value of Alice's points minus Bob's points after $n$ turns, as $n$ approaches infinity. To ensure the integrity of your solution, there will be $T$ testcases.

Constraints

$1\le T\le 314159$
$X+Y\ge 1$

Subtask 1 [27%]

$0\le X,Y\le 314$
$1\le Z\le 2$

Subtask 𝓮 [31%]

$0\le X,Y\le 314$
$1\le Z\le 271$

Subtask π [42%]

$0\le X,Y\le 314159$
$1\le Z\le 271828$

Input Specification

The first line contains one integer, $T$, the number of testcases.

The next $T$ lines each contain three space-separated integers, $X$, $Y$, and $Z$, describing each testcase. $X$ is the initial number of good slices of $\pi e$, $Y$ is the initial number of bad slices of $\pi e$, and $Z$ is the number of randomly good/bad $\pi e$ slices which are added after each turn. It is guarenteed that there is at least one slice of $\pi e$ initially.

Output Specificiation

Output $T$ lines, the $i^{th}$ line containing one real number, the answer to the $i^{th}$ testcase. Your answers will be considered correct if their absolute error is less than ${10}^{-6}$.

Sample Input

3
1 0 2
2 4 3
5 2 4

Sample Output

0.7853981633974483
-0.20456854629336979
0.2687567584009584

Explanation for Sample

For the first testcase, it can be shown that the limit of the expected value approaches $\frac{\pi }{4}=0.7853981633974483\dots$

For the second testcase, it can be shown that the limit of the expected value approaches $\frac{2}{3}\ln (2)-\frac{2}{3}=-0.20456854629336979\dots$

For the third testcase, it can be shown that the limit of the expected value approaches $\frac{3\sqrt{2}}{4}\ln (1+\sqrt{2})-\frac{3\sqrt{2}}{8}\pi +1=0.2687567584009584\dots$