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Derivation

1. Initial program 0.8

   \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
2. Using strategy rm

3. Applied add-sqr-sqrt0.8

   \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

4. Applied *-un-lft-identity0.8

   \[\leadsto \frac{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

5. Applied times-frac0.8

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}\]
6. Using strategy rm

7. Applied add-cube-cbrt1.5

   \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\left(\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}\right) \cdot \sqrt[3]{\sqrt{\log 10}}}}\]

8. Applied *-un-lft-identity1.5

   \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \frac{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}{\left(\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}\right) \cdot \sqrt[3]{\sqrt{\log 10}}}\]

9. Applied times-frac1.5

   \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt[3]{\sqrt{\log 10}}}\right)}\]

10. Simplified0.1

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\color{blue}{\frac{1}{\sqrt[3]{\log 10}}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt[3]{\sqrt{\log 10}}}\right)\]

11. Final simplification0.1

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt[3]{\sqrt{\log 10}}} \cdot \frac{1}{\sqrt[3]{\log 10}}\right)\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018214 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))