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Derivation

1. Split input into 3 regimes

2. if re < -7.954449727655894e+140

   1. Initial program 58.4

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

   2. Initial simplification58.4

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

   3. Taylor expanded around -inf 62.8

      \[\leadsto \color{blue}{-1 \cdot \frac{\log \left(\frac{-1}{re}\right)}{\log -1 - \log \left(\frac{-1}{base}\right)}}\]

   4. Simplified8.2

      \[\leadsto \color{blue}{\frac{-1}{\log base} \cdot \log \left(\frac{-1}{re}\right)}\]

   if -7.954449727655894e+140 < re < 4.551727447137176e+52

   1. Initial program 21.6

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

   2. Initial simplification21.6

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

   4. Applied add-cbrt-cube21.7

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

   5. Applied add-cbrt-cube21.8

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

   6. Applied add-cbrt-cube21.9

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

   7. Applied cbrt-unprod21.8

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

   8. Applied cbrt-undiv21.7

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

   9. Simplified21.7

      \[\leadsto \sqrt[3]{\color{blue}{\frac{{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}^{3}}{\frac{\log base \cdot \log base}{\frac{1}{\log base}}}}}\]
   10. Using strategy rm

   11. Applied div-inv21.7

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

   12. Applied cube-mult21.7

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

   13. Applied times-frac21.7

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

   14. Applied cbrt-prod21.8

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

   if 4.551727447137176e+52 < re

   1. Initial program 43.1

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

   2. Initial simplification43.1

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

   3. Taylor expanded around inf 11.2

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}}\]

   4. Simplified11.2

      \[\leadsto \color{blue}{\frac{-\log re}{-\log base}}\]
3. Recombined 3 regimes into one program.

4. Final simplification17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.954449727655894 \cdot 10^{+140}:\\ \;\;\;\;\frac{-1}{\log base} \cdot \log \left(\frac{-1}{re}\right)\\ \mathbf{elif}\;re \le 4.551727447137176 \cdot 10^{+52}:\\ \;\;\;\;\sqrt[3]{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\frac{1}{\frac{1}{\log base}}}} \cdot \sqrt[3]{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base \cdot \log base}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

Runtime

Time bar (total: 52.4s)Debug logProfile

herbie shell --seed 2018273 
(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))))