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> Integers are much faster to process than floating point This was common and accepted knowledge circa 1990-2005. Is that still the case? In 2025, I'm not so

> Integers are much faster to process than floating point This was common and accepted knowledge circa 1990-2005. Is that still the case? In 2025, I'm not so sure, and if speed is indeed what you're after, what you typically want is to pump calculations into both the floating point units and the integer units in parallel using SIMD instructions, something that should be easy on an embarrassingly parallel problem like Mandelbrot. And regarding precision ... I was under the impression that proving with 100% certainty that a point (x,y) actually belongs to the Mandelbrot set is basically impossible unless you prove some sort of theorem about that specific point. The numerical method is always something along the lines of "as long as the iteration hasn't diverged after N iterations we consider the point to be inside". And both N and the definition of "diverged" are usually completely arbitrary ... so precision, meh, unless you're going to try draw something with a zoom of 1e+20 on the frontier.

pixelpoet 1y ago
It's possible to render exact Mandelbrot set images using interval arithmetic, see https://mathr.co.uk/web/m-trustworthy.html https://mathr.co.uk/web/m-trustworthy.html About performance: floating point is definitely much faster than integer arithmetic these days, for most comparable operations. One more point about precision: integer arithmetic is exact, and so if you can work with a reasonably small bound (e.g. [-2,2]^2 for Mandelbrot) then integer arithmetic actually becomes more accurate than FP for the same number of bits.
- dspillett 1y ago
  > floating point is definitely much faster than integer arithmetic these days, for most comparable operations. Is that true of code running on CPUs? My GPU can run rings around the CPU at floating point, and probably itself at integer arithmetic, but I find it hard to believe any CPU is faster at a floating point version of an algorithm than an integer version of the same (assuming both are well, or equally badly, optimised).
- adrian_b 1y ago
  With most available software packages floating point arithmetic remains much faster than integer arithmetic, but because those software packages do not use the more recent CPU features, since thanks to Intel's stupidity there still are many CPUs which do not support those features. Already in 2018, with Cannon Lake, Intel has introduced instructions (IFMA) which can reuse for integer computations the same vector multipliers and adders that are provided for floating-point computations. However, later Intel has removed AVX-512 from its consumer CPUs. The latest models of Intel CPUs have introduced a slower AVX-encoded variant of IFMA, but this is too little and too late, in comparison with what AMD offers. Using IFMA and the other AVX-512 integer vector instructions, where supported, it is possible to obtain very similar throughputs, regardless if the computation is done with integer or floating-point numbers. Hopefully, with the growing installed base of Zen 4 and Zen 5 based computers there will be the incentive to update many software packages to take advantage of their capabilities.

pixelpoet 1y ago

It's possible to render exact Mandelbrot set images using interval arithmetic, see https://mathr.co.uk/web/m-trustworthy.html https://mathr.co.uk/web/m-trustworthy.html About performance: floating point is definitely much faster than integer arithmetic these days, for most comparable operations. One more point about precision: integer arithmetic is exact, and so if you can work with a reasonably small bound (e.g. [-2,2]^2 for Mandelbrot) then integer arithmetic actually becomes more accurate than FP for the same number of bits.

dspillett 1y ago
> floating point is definitely much faster than integer arithmetic these days, for most comparable operations. Is that true of code running on CPUs? My GPU can run rings around the CPU at floating point, and probably itself at integer arithmetic, but I find it hard to believe any CPU is faster at a floating point version of an algorithm than an integer version of the same (assuming both are well, or equally badly, optimised).
adrian_b 1y ago
With most available software packages floating point arithmetic remains much faster than integer arithmetic, but because those software packages do not use the more recent CPU features, since thanks to Intel's stupidity there still are many CPUs which do not support those features. Already in 2018, with Cannon Lake, Intel has introduced instructions (IFMA) which can reuse for integer computations the same vector multipliers and adders that are provided for floating-point computations. However, later Intel has removed AVX-512 from its consumer CPUs. The latest models of Intel CPUs have introduced a slower AVX-encoded variant of IFMA, but this is too little and too late, in comparison with what AMD offers. Using IFMA and the other AVX-512 integer vector instructions, where supported, it is possible to obtain very similar throughputs, regardless if the computation is done with integer or floating-point numbers. Hopefully, with the growing installed base of Zen 4 and Zen 5 based computers there will be the incentive to update many software packages to take advantage of their capabilities.