https://software.intel.com/sites/landingpage/IntrinsicsGuide...
So all the major platforms, CPUs and GPUs, implement the fast reciprocal square root to decent amounts of accuracy, without any need of bit-twiddling anymore.
Though you can still have vectorised support for regular precision for AVX2.
https://software.intel.com/sites/landingpage/IntrinsicsGuide...
That way you can quickly get the precision that you want ;)
You can always compute newton's method afterwards to improve the accuracy. But getting the maximum accuracy from one cycle is probably best.
With such an instruction, you could exponentially increase the precision in 2 cycles by just using the same instruction twice.
You can't do that on Intel's hardware. As you mention, you'd need to roll your own multi-SIMD-instruction rsqrt newton iteration complex loop, and use it after the first SIMD call.
That's really sad. There is hardware to perform Newton iterations on Intel CPUs, that's how that instruction is implemented, but the ISA only exposes this hardware via the "do a 14-bit rsqrt operation", which means that you can't really use it to increase precision if that does not suffice for your app.