algorithms - Efficient Magnitude Comparison for Complex Numbers

Is there a more efficient algorithm (or what is the most efficient known algorithm) to select the larger of two complex numbers given as $I + jQ$ without having to compute the squared magnitude as

$$I^2+Q^2$$

In particular binary arithmetic solutions that do not require multipliers? This would be for a binary arithmetic solution using AND, NAND, OR, NOR, XOR, XNOR, INV, and shifts and adds without simply replacing required multiplication steps with their shift and add equivalents (or what would be equivalent in processing steps). Also assume the number is represented with rectangular fixed point (bounded integers) coordinates (I, Q).

I am aware of magnitude estimators for complex numbers such as $max(I,Q) + min(I,Q)/2$, and more accurate variants at the expense of multiplying coefficients but they all have a finite error.

I have considered using the CORDIC rotator to rotate each to the real axis and then being able to compare real numbers. This solution also has finite error but I can choose the number of iterations in the CORDIC such that the error is less than $e$ for any $e$ that I choose within my available numeric precision. For that reason this solution would be acceptable.

Are there other solutions that would be more efficient than the CORDIC (which requires time via iterations to achieve accuracy)?

---

Determining Best Answer

There was incredible work done by the participants (and we still welcome competing options if anyone has other other ideas). I posted my proposed approach to ranking the algorithms and welcome debate on the ranking approach before I dive in:

Best Approach to Rank Complex Magnitude Comparision Problem