\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

↓

\[\log \left(\sqrt{re^2 + im^2}^*\right) \cdot \frac{1}{\log base}\]

Error

The X axis uses an exponential scale — Bits error versus `re`

Derivation

Initial program 31.1
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
Simplified0.4
\[\leadsto \color{blue}{\frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log base}}\]
Using strategy rm
Applied div-inv0.4
\[\leadsto \color{blue}{\log \left(\sqrt{re^2 + im^2}^*\right) \cdot \frac{1}{\log base}}\]
Final simplification0.4
\[\leadsto \log \left(\sqrt{re^2 + im^2}^*\right) \cdot \frac{1}{\log base}\]

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0)) (+ (* (log base) (log base)) (* 0 0))))