Suppose two persons independently evaluate the same exam. The first reports $$A$$ mistakes, the second reports $$B$$ mistakes, and $$C$$ mistakes are reported by both. How can you estimate how many errors remain undiscovered?

Let $$M$$ be the total number of mistakes in the exam. Then the number of mistakes that are undiscovered by the two persons is $$M - (A + B - C)$$. Let $$p$$ and $$q$$ be the probabilities that the first and second person, respectively, notice any given mistake. Then $$A \approx pM$$ and $$B \approx qM$$. And because they work independently, the chance that they both find a given mistake is $$C \approx pqM$$.

But now \[M = \frac{pM \times qM}{pqM} \approx \frac{AB}{C}\] and the number of mistakes that remain unnoticed is just \[M - (A + B - C) \approx \frac{AB}{C} - (A + B - C) = \frac{(A - C)(B - C)}{C}\,.\] This means that as long as the persons work independently, you can estimate the number of errors they've overlooked without even knowing how skillful they are. If they find a large number of mistakes in common but relatively few independently, then the exam is probably evaluated in a relatively correct way. But if they generate large independent lists of errors with few in common, there are probably many mistakes remaining to be found (which matches our intuition).

Input

Three positive integers, each on a separate line, that indicate respectively the number of mistakes found by the first person, the number of mistakes found by the second person and the number of mistakes that are found by both.

Output

The sentence

There are d.dd undiscovered errors.

where the fragment in italic has to be filled up with an estimate of the number of errors that remained undiscovered by both persons. This value must be represented as a floating point number with two decimal digits. Rounding must be used to determine the digits of the floating point number.

Example

Input:

17
21
15

Output:

There are 0.80 undiscovered errors.

Resources

• Pólya G (1976). Probabilities in Proofreading. American Mathematical Monthly 83(1), 42. ¹