Test the sum formulae at many angles

A formula is an identity when it holds at every angle. The computer can test five hundred pairs of angles in a moment. You will write the formula for $\sin(x + y)$, find its largest error, and count how often the false rule $\sin(x + y) = \sin x + \sin y$ misses.

The idea. A true identity misses only by the computer's rounding, about $10^{-15}$. A false rule misses by a lot.

Your tasks.

1. Make sin_sum(x, y) return $\sin x \cos y + \cos x \sin y$.
2. Set worst to the largest difference between sin_sum(x, y) and math.sin(x + y) over all the pairs.
3. Set broken to the number of pairs where $\sin(x + y)$ and $\sin x + \sin y$ differ by more than $0.01$.