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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 pow10.4

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

5. Applied log-pow0.4

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

6. Applied associate-/l*0.4

   \[\leadsto \color{blue}{\frac{1}{\frac{\log base}{\log \left(\sqrt{re^2 + im^2}^*\right)}}}\]
7. Using strategy rm

8. Applied div-inv0.5

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

9. Applied associate-/r*0.5

   \[\leadsto \color{blue}{\frac{\frac{1}{\log base}}{\frac{1}{\log \left(\sqrt{re^2 + im^2}^*\right)}}}\]
10. Using strategy rm

11. Applied div-inv0.5

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

12. Applied *-un-lft-identity0.5

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

13. Applied times-frac0.5

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

14. Simplified0.5

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

15. Simplified0.4

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

16. Final simplification0.4

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

Reproduce

herbie shell --seed 2018365 +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))))

Details

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