Calculate the relativistic velocity
Mathematica, 17 bytes

+##/(1+##/9*^16)&amp;


An unnamed function taking two integers and returning an exact fraction.

Explanation

This uses two nice tricks with the argument sequence ##, which allows me to avoid referencing the individual arguments u and v separately. ## expands to a sequence of all arguments, which is sort of an "unwrapped list". Here is a simple example:

{x, ##, y}&amp;[u, v]


gives

{x, u, v, y}


The same works inside arbitrary functions (since {...} is just shorthand for List[...]):

f[x, ##, y]&amp;[u, v]


gives

f[x, u, v, y]


Now we can also hand ## to operators which will first treat them as a single operand as far as the operator is concerned. Then the operator will be expanded to its full form f[...], and only then is the sequence expanded. In this case +## is Plus[##] which is Plus[u, v], i.e. the numerator we want.

In the denominator on the other hand, ## appears as the left-hand operator of /. The reason this multiplies u and v is rather subtle. / is implemented in terms of Times:

FullForm[a/b]
(* Times[a, Power[b, -1]] *)


So when a is ##, it gets expanded afterwards and we end up with

Times[u, v, Power[9*^16, -1]]


Here, *^ is just Mathematica's operator for scientific notation.