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Derivation

1. Initial program 0.8

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

3. Applied log1p-expm1-u0.7

   \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}} - 1)^*)}\]
4. Using strategy rm

5. Applied add-sqr-sqrt0.7

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

6. Applied *-un-lft-identity0.7

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

7. Applied times-frac0.7

   \[\leadsto \log_* (1 + (e^{\color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}} - 1)^*)\]
8. Using strategy rm

9. Applied div-inv0.7

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

10. Applied associate-*r*0.7

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

11. Final simplification0.7

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

Runtime

Time bar (total: 11.8s)Debug logProfile

herbie shell --seed 2018340 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))