Data dependencies and pipelining

Our analysis of the CPU capabilities so far has shown that the processor can execute multiple operations at once as long as the operands are already in the registers: we can evaluate a fairly complex expression that depends on just two values in exactly as much time as it takes to add these values. The depends on just two values qualifier is, unfortunately, a very serious restriction. We now consider a more realistic code example, and we don't have to make many changes to our code:

for (size_t i = 0; i < N; ++i) {
     a1 += (p1[i] + p2[i])*(p1[i] - p2[i]);
}

Recall that the old code had the same loop with a simpler body: a1 += (p1[i] + p2[i]);. Also, p1[i] is just an alias for the vector element v1[i], same for p2 and v2. Why is this code more complex? We have already seen that the processor can do addition, subtraction, and multiplication in a single cycle, and the expression still depends on just two values...