Coin tossing

A machine makes coins that land heads with probability $p$ and tails with probability $1-p$. Here $p$ is a number between 0 and 1, and the machine has no setting for it. The machine picks $p$ at random while it makes a coin, so $p$ is uniform on the interval $[0, 1]$.

The machine made two coins. Coin 1 has probability $p$ and coin 2 has probability $q$. Both $p$ and $q$ are uniform on $[0, 1]$ and independent of each other, and there is no way to read off their actual values.

Coin 1 was tossed $n_1$ times and came up heads $m_1$ times. Coin 2 was tossed $n_2$ times and came up heads $m_2$ times. Given these observations, write a program that computes the probability that $p < q$.

The first line contains the number of test cases $T$ ($1 \le T \le 100000$).

Each of the next $T$ lines holds one test case: $n_1$, $m_1$, $n_2$, $m_2$, separated by spaces. ($1 \le n_1, n_2 \le 1000$, $0 \le m_1, m_2 \le 50$, $m_1 \le n_1$, $m_2 \le n_2$)

For each test case, print the probability that $p < q$ on its own line. Round at the seventh decimal place and print exactly six digits after the decimal point. For example, print 0.800000 when the probability is $\frac{4}{5}$.