Computing the DFT generally requires complex multiplications. Consider the product X + JY = (A + JB) (C + JD) = (AC – BD) + J (BC + AD). In this form, a complex multiplication requires four real multiplications and two real additions. Verify that a complex multiplication can be performed with three real multiplications and five additions using the algorithm 

X = (A – B) D + (C – D) A,

Y = (A – B) D + (C + D) B.