Hello, there is a lot of research on this topic, e.g.,
- http://www.matrix67.com/data/InvSqrt.pdf
- https://cs.uwaterloo.ca/~m32rober/rsqrt.pdf.
Modern modifications to the algorithm include:
- https://www.mdpi.com/1099-4300/23/1/86
- https://ieeexplore.ieee.org/abstract/document/8924302
-...