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Derivation

1. Split input into 5 regimes

2. if re < -7.373796795114872e+42

   1. Initial program 42.9

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

   2. Initial simplification42.9

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

   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. Simplified10.9

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

   if -7.373796795114872e+42 < re < -1.132734653217697e-203

   1. Initial program 19.3

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

   2. Initial simplification19.3

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

   4. Applied add-cbrt-cube19.4

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

   5. Applied add-cbrt-cube19.5

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

   6. Applied add-cbrt-cube19.6

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

   7. Applied cbrt-unprod19.5

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

   8. Applied cbrt-undiv19.4

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

   9. Simplified19.4

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

   if -1.132734653217697e-203 < re < 3.4345341339819355e-140

   1. Initial program 29.5

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

   2. Initial simplification29.5

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

   4. Applied times-frac29.4

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

   5. Simplified29.4

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

   7. Applied div-inv29.4

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

   8. Taylor expanded around 0 34.6

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

   if 3.4345341339819355e-140 < re < 9.295499610334908e+101

   1. Initial program 16.0

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

   2. Initial simplification16.0

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

   4. Applied add-cbrt-cube16.2

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

   5. Applied add-cbrt-cube16.3

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

   6. Applied add-cbrt-cube16.4

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

   7. Applied cbrt-unprod16.2

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

   8. Applied cbrt-undiv16.2

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

   9. Simplified16.2

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

   11. Applied add-cube-cbrt16.3

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

   12. Applied pow1/216.3

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

   13. Applied log-pow16.3

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

   14. Applied unpow-prod-down16.3

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

   15. Applied times-frac16.3

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

   16. Applied cbrt-prod16.4

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

   17. Simplified16.3

       \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{\frac{1}{8}}{\log base}}{\log base}}} \cdot \sqrt[3]{\frac{{\left(\log \left(re \cdot re + im \cdot im\right)\right)}^{3}}{\sqrt[3]{\frac{\log base \cdot \log base}{\frac{1}{\log base}}}}}\]

   if 9.295499610334908e+101 < re

   1. Initial program 49.8

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

   2. Initial simplification49.8

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

   4. Applied times-frac49.7

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

   5. Simplified49.7

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

   6. Taylor expanded around inf 9.1

      \[\leadsto \frac{\log \color{blue}{re}}{\log base} \cdot 1\]
3. Recombined 5 regimes into one program.

4. Final simplification18.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.373796795114872 \cdot 10^{+42}:\\ \;\;\;\;\frac{-1}{\log base} \cdot \log \left(\frac{-1}{re}\right)\\ \mathbf{elif}\;re \le -1.132734653217697 \cdot 10^{-203}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(\log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)}^{3}}{\frac{\log base \cdot \log base}{\frac{1}{\log base}}}}\\ \mathbf{elif}\;re \le 3.4345341339819355 \cdot 10^{-140}:\\ \;\;\;\;\log im \cdot \frac{1}{\log base}\\ \mathbf{elif}\;re \le 9.295499610334908 \cdot 10^{+101}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(\log \left(im \cdot im + re \cdot re\right)\right)}^{3}}{\sqrt[3]{\frac{\log base \cdot \log base}{\frac{1}{\log base}}}}} \cdot \sqrt[3]{\frac{\frac{\frac{1}{8}}{\log base}}{\log base}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

Runtime

Time bar (total: 34.3s)Debug logProfile

herbie shell --seed 2018339 
(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))))