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Derivation

1. Split input into 3 regimes

2. if re < -2.294531372627966e+104

   1. Initial program 51.1

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

   2. Initial simplification51.1

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

   3. Taylor expanded around -inf 62.8

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

   4. Simplified8.8

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

   if -2.294531372627966e+104 < re < 6.19918186560116e+63

   1. Initial program 21.9

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

   2. Initial simplification21.9

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

   4. Applied times-frac21.8

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

   5. Simplified21.8

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

   7. Applied add-cube-cbrt21.9

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

   8. Applied sqrt-prod21.9

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

   9. Applied log-prod21.9

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

   10. Simplified21.9

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

   12. Applied pow1/321.9

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

   14. Applied pow121.9

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

   15. Applied log-pow21.9

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

   16. Applied pow121.9

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

   17. Applied log-pow21.9

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

   18. Applied distribute-lft-out21.9

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

   19. Applied associate-/l*21.9

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

   if 6.19918186560116e+63 < re

   1. Initial program 44.8

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

   2. Initial simplification44.8

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

   4. Applied times-frac44.8

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

   5. Simplified44.8

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

   6. Taylor expanded around inf 10.2

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

   7. Simplified10.2

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

4. Final simplification17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.294531372627966 \cdot 10^{+104}:\\ \;\;\;\;\frac{-1}{\log base} \cdot \log \left(\frac{-1}{re}\right)\\ \mathbf{elif}\;re \le 6.19918186560116 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log \left(\sqrt{{\left(im \cdot im + re \cdot re\right)}^{\frac{1}{3}}}\right) + \log \left(\left|\sqrt[3]{im \cdot im + re \cdot re}\right|\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

Runtime

Time bar (total: 32.4s)Debug logProfile

herbie shell --seed 2018255 
(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))))