Litmus Test

Time Limit: 2 Seconds
Memory Limit: 65536 KB

The pH scale measures the concentration of protons (H^{+}) in
a solution and, therefore, its acidity or alkalinity. The pH value of
a solution is a number between 0 and 14; it is less than 7 if the
solution is acidic, greater than 7 if the solution is basic, and 7 if
it is neutral.

The formula for calculating pH is

pH = -log_{10} [H^{+}]
where [H^{+}] is the concentration of protons measured in moles
per litre.

To calculate the pH value of an acid, one has to determine the
concentration of protons in the solution. When an acid is dissolved
in water, an equilibrium is reached and is governed by the
equation

K_{a} = [H^{+}] [acid ions] / [acid]
where K_{a} is the acidity constant (known for each acid),
[acid ions] is the concentration of the acid ions that have dissolved,
and [acid] is the concentration of the undissolved acid. Before the
acid is added, both [H^{+}] and [acid ions] are assumed to be
0.

For example, the acidity constant of methanoic acid is 1.6 x
10^{-4}. Dissolving one mole of acid molecules results
in one mole of H^{+} and one mole of acid ions.
If the initial concentration of the methanoic acid is 0.1 moles/L
and *x* moles of acid are dissolved (per liter), then the final
concentration at equilibrium would be 0.1 - *x* moles/L for
the acid and *x* moles/L for H^{+} and the acid ions.

Input

The input consists of a number of test cases. Each test case contains
4 numbers on a line: two positive floating-point numbers specifying
the acidity constant K_{a} and the original concentration of
the acid (in moles/liter) added to the water, as well as two positive integers
*m* and *n* indicating that each mole of acid molecules
is dissolved into *m* moles of H^{+} ions and
*n* moles of acid ions. The floating-point numbers are
specified in scientific notation as shown below. The input is
terminated with a line containing four zeros.

Output

For each test case, print on a line the pH value of the solution,
rounded to 3 decimal places.

This problem contains multiple test cases!

The first line of a multiple input is an integer N, then a blank line followed by N input blocks. Each input block is in the
format indicated in the problem description. There is a blank line between input blocks.

The output format consists of N output blocks. There is a blank line between output blocks.

Sample Input

1

1.6e-04 1.0e-01 1 1

1.6e-04 1.0e-01 4 1

1.5e-05 5.0e-02 1 2

0 0 0 0

Sample Output

2.407

2.101

3.216

Source:

**Rocky Mountain 2004**
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