Squaring a complex number is a key algorithm needed for rendering the Mandelbrot set, among other things. I noticed Kalles Fraktaler uses a method with 3 real number squares and no multiplications, so I compared a few algorithms:

- naive
- standard method commonly used as a definition
- \[ (x + i y)^2 = (x^2 - y^2) + i (2 x y) \]
- 1 multiplication, 2 squares, 1 additions
- Karatsuba
- modified using difference of squares factorisation
- \[ (x + i y)^2 = (x + y) (x - y) + i (2 x y) \]
- 2 multiplications, 2 additions
- Kalles
- as implemented by Kalles Fraktaler
- \[ (x + i y)^2 = (x^2 - y^2) + i ((x + y)^2 - (x^2 + y^2)) \]
- 3 squares, 4 additions (the subexpressions can be shared)

Because the relative cost of these three operations (multiplication, squaring, addition - multiplication by 2 is constant time) varies depending on the precision of the real variables, I decided to benchmark the different algorithms. Here are the results from one machine, the values in the table are the number of iterations of \( z \to z^2 + c \) that could be performed with a 10 second timeout:

precision | naive | Karatsuba | Kalles |
---|---|---|---|

100 | 28868287 | 40797727 | 24932491 |

316 | 20344558 | 27222494 | 19755743 |

1000 | 9549975 | 11122112 | 9755611 |

3162 | 2475382 | 2726978 | 2810427 |

10000 | 446160 | 502960 | 510899 |

31623 | 78343 | 88965 | 91640 |

100000 | 15013 | 17294 | 17419 |

316228 | 2955 | 3513 | 3403 |

1000000 | 591 | 760 | 673 |

The best algorithm at each precision is highlighted in green. As you can see, Karatsuba is best for low precisions and very high precisions, but between around 3000 and 100000 bits Kalles method is faster. I'm not sure if accuracy will suffer for any of these methods, at least the MPC library based on MPFR uses the Karatsuba algorithm in its mpc_sqr() function so it should be reasonably reliable.

You can download the source: complex-bench.c which contains compilation instructions at the top.