Fast computation of $x^{1/p}$, where $x\in\mathbb{R}^+$ and $p=2^{n}$, where $n\in\mathbb{N}$ with bit shifts?

Question

There is plenty of literature regarding the legendary Fast inverse square root routine from Quake, but can we do something similar to compute $x^{1/p}$ as given in the title?

Given that $p$ is a power of 2, there should be some clever trick using bit shifts to achieve this.

Answer 1

Is known, that $y^{\large 2^{\Large p}}$ can be calculated by $p$ multiplication operations. Besides, is not a problem to calculate the MSB position of the result $y=x^{\large\frac1{\large 2^p}}.$

This allows to get the result y, bit after bit, over $pM$ multiplications, where $M$ is a quantity of bits of the result.