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Derivation

1. Initial program 31.0

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

2. Initial simplification0.4

   \[\leadsto \frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log base}\]
3. Using strategy rm

4. Applied add-cube-cbrt0.4

   \[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}}{\log base}\]
5. Using strategy rm

6. Applied pow1/30.4

   \[\leadsto \frac{\log \left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \color{blue}{{\left(\sqrt{re^2 + im^2}^*\right)}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{\log base}\]

7. Applied pow1/30.4

   \[\leadsto \frac{\log \left(\left(\color{blue}{{\left(\sqrt{re^2 + im^2}^*\right)}^{\frac{1}{3}}} \cdot {\left(\sqrt{re^2 + im^2}^*\right)}^{\frac{1}{3}}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{\log base}\]

8. Applied pow-prod-up0.4

   \[\leadsto \frac{\log \left(\color{blue}{{\left(\sqrt{re^2 + im^2}^*\right)}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{\log base}\]

9. Simplified0.4

   \[\leadsto \frac{\log \left({\left(\sqrt{re^2 + im^2}^*\right)}^{\color{blue}{\frac{2}{3}}} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{\log base}\]

10. Final simplification0.4

    \[\leadsto \frac{\log \left({\left(\sqrt{re^2 + im^2}^*\right)}^{\frac{2}{3}} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{\log base}\]

Runtime

Time bar (total: 20.8s)Debug logProfile

herbie shell --seed 2018235 +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))))