cmath vs math.hypot Magnitudes: Localhost Lab

Intro — what this post promises

2-D magnitude and phase with math.hypot / math.atan2 vs abs(complex) / cmath.polar / cmath.phase. This lab reports ops/s on Linux localhost for 200000 synthetic (re, im) pairs.

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Lab honesty (1 Oct 2026 IST): Python 3.13.5. Affiliates: 0. Stdlib only — useful when you treat latency error components, IQ samples, or vector residuals as complex numbers.

Verdict up front: abs(complex) ~34917055 ops/s; math.hypot ~16959583; cmath.polar r ~11928627; manual sqrt(a*a+b*b) ~23608807. Phase: cmath.phase ~23945233 vs atan2 ~19534098.

Arms

Seven rounds, p50. Sample check: hypot 65.11528238439882 matches abs (match=True).

Lab topology

n=200000 pairs · 7 rounds · p50
metric: ops/s = n / p50_s

Script: lab-evidence/134-cmath-vs-math-hypot/results/run_lab.py.

Lead table (p50 ops/s)

abs(complex) led magnitude. cmath.polar paid for computing both r and φ even when only r was used.

Reading it for SRE work

• Already holding complex values → abs(z) (fastest magnitude here).
• Separate float channels without complex objects → math.hypot (overflow-safer than manual sqrt).
• Need angle too → prefer one cmath.polar over abs+phase separately if both used (measure your path).
• Never use manual sqrt(a*a+b*b) near float extremes — hypot exists for a reason.

Why hypot still matters

Even though hypot was slower than abs(complex) on this run (~16959583 vs ~34917055), hypot avoids intermediate overflow when |a| or |b| is huge. Telemetry pipelines with wild units should stay on hypot unless values are already complex.

cmath.polar at ~11928627 ops/s is the wrong tool if you only need magnitude — you pay for phase work you discard.

Phase peers

cmath.phase (~23945233) edged atan2 (~19534098) slightly here. Pick the API that matches your types; do not convert float pairs to complex solely for phase unless the rest of the pipeline is complex-native.

Type-driven choice

If your pipeline already boxes samples as complex, stay there and call abs (~34917055 ops/s here). If sensors give two floats, call math.hypot and skip allocating complex objects just to take a magnitude. Conversion cost is outside this bench but shows up in real collectors — measure end-to-end before rewriting.

For dashboards that need both magnitude and angle once per sample, cmath.polar can still win on clarity even at ~11928627 ops/s, because one call documents intent better than abs+phase glue.

Pitfalls

• Using cmath.polar only for r in a hot loop.
• Manual squared sum overflowing.
• Comparing arms with different pre-boxed types without stating allocation.
• Assuming numpy hypot matches these CPython numbers.

Reproduce

python3 lab-evidence/134-cmath-vs-math-hypot/results/run_lab.py

Evidence: summary.json, summary.txt.

Limits

One Linux box. Pure Python loops over lists. Not NumPy ufuncs.

Takeaway

For magnitudes on complex values, abs(z) ~34917055 ops/s led; for float pairs prefer math.hypot (~16959583) over manual sqrt. Use cmath.polar only when you need both radius and phase.