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If you need a right mathematical consequence (appropriately rounded to inside floating level precision), a sq. root is the way in which to go.
It’s true that it is dearer than a multiply, however it’s nonetheless very seemingly not the bottleneck in your app’s efficiency. Even transcendental features nonetheless value far fewer CPU cycles than a single cache miss. And you have seemingly acquired cache misses occurring extra typically than you normalize vectors.
To make this concrete with some numbers, let’s summarize typical cycle counts for floating level operations within the magnitude ranges we have a tendency to make use of, based mostly on timings on an Intel Core i7 CPU from this doc, and cache data from right here (not the very same mannequin, however shut):
| Operation | (Approximate) Cycles |
|---|---|
| Addition | 5 |
| Multiplication | 5 |
| Division | 8 |
| Sqrt | 10 |
| L3 Cache Latency | 42 |
So you are able to do a minimum of 4 sq. roots (very seemingly extra, attributable to pipelining) within the time it takes to drag within the subsequent object to replace that wasn’t already scorching in cache.
Trying to get rid of sqrt at this stage is sort of actually a untimely micro-optimization. I’d guess that you just’re paying a lot better inefficiencies in architectural decisions like information structure which are far simpler to vary than the legal guidelines of geometry.
Very previous video games like Quake III used to make use of an approximation of 1/sqrt(x) for normalizing vectors, which you’ll examine in the Wikipedia article “Fast Inverse Square Root”, however it factors out that this technique is not actually your best choice on fashionable {hardware}:
With subsequent {hardware} developments, particularly the x86 SSE instruction
rsqrtss, this technique will not be typically relevant to normal function computing, although it stays an attention-grabbing instance each traditionally and for extra restricted machines, resembling low-cost embedded methods.
If you are normalizing large batches of vectors the place the sq. root value is a considerable fraction of the whole computation time, you will seemingly get higher features by vectorizing the code so that you compute 4 normalizations directly, slightly than attempting to get intelligent with the way you compute the sq. root itself.
Overall, be cautious of programming by rumour. When people say “sq. root is pricey, keep away from it when you may”, they’re primarily speaking about instances the place the sq. root is pointless to the worth you care about: like evaluating the size of a vector towards a threshold or discovering the shortest/longest vector in a set. Those are instances the place you get the identical reply utilizing squared size, so paying for a sq. root there’s cycles burned for no enchancment in accuracy. Where there’s accuracy to be gained (like in getting an accurate unit vector), it is value paying the modest value of a sq. root and division.
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