Integrated circuits with specialized processing blocks are provided. A specialized processing block may include one real addition stage and one real multiplier stage. The multiplier stage may simultaneously feed its output to the addition stage and directly to an adjacent specialized processing block. The addition stage may also produce sum and difference outputs in parallel. A group of four such specialized processing blocks may be connected in a chain to implement a radix-2 fast Fourier transform (FFT) butterfly. Multiple radix-2 butterflies may be stacked to form yet higher order radix butterflies. If desired, the specialized processing block may also be used to implement a complex multiply operation. Three or four specialized processing blocks may be chained together and along with one or more adders outside the specialized processing blocks, real and imaginary portions of a complex product can be generated.