We parallelized the code for rendering and state evolution above. We did not, however, yet parallelize the FFT code. The reason for this is that we initially thought that there was a loop-carried dependency that could not be resolved:

The issue is the <code>T *= phiT;</code> line. This means that every time the loop counter <code>l</code> increases, <code>T</code> gets multiplied by <code>phiT</code>. This statement seems painfully obvious, but it prevents us from parallelizing the code since the iterations can't be done in an arbitrary order. What it does mean, however, is that we can remove that line and replace any usage of <code>T</code> with <code>pow(phiT, l)</code>. We can then parallelize it, since the iterations are now order-invariant. When we do that, the FPS somehow does not change. In fact, if we remove the parallel for construct, the FPS drops to a meager 1 frame per second. This is awful, and likely because the <code>pow()</code> operation is very computationally expensive. Are we stuck? Of course not. We can apply math. There is a property of complex numbers which allows us to turn exponentiation into multiplication. If we write the complex number <code>phiT</code> as <code>phiT = cos(arg) + i * sin(arg)</code>, and we can since it has norm 1, we have <code>phiT ** l = cos(l * arg) + i * sin(l * arg)</code>. This gave a tremendous speedup since the trigonometric functions are apparently less costly than exponentiation. The code and new vTune analyses are below.