Exercise 4.2.7

Show how to multiply the complex numbers $a + bi$ and $c + di$ using only three multiplications of real numbers. The algorithm should take $a$, $b$, $c$ and $d$ as input and produce the real component $ac - bd$ and the imaginary component $ad + bc$ separately.

Simple enough. The three multiplications are:

$$ A = (a + b)(c + d) = ac + ad + bc + bd \\ B = ac \quad\qquad C = bd $$

The result is:

$$ (B - C) + (A - B - C)i $$