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Derivation

1. Split input into 3 regimes

2. if re < -5.264515794033503e+146

   1. Initial program 60.1

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

   2. Initial simplification60.1

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

   4. Applied times-frac60.1

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

   5. Simplified60.1

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

   6. Taylor expanded around -inf 6.7

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

   7. Simplified6.7

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

   if -5.264515794033503e+146 < re < 1.5126207120265142e+120

   1. Initial program 20.6

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

   2. Initial simplification20.6

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

   4. Applied times-frac20.5

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

   5. Simplified20.5

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

   7. Applied add-cube-cbrt20.5

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

   8. Applied sqrt-prod20.5

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

   9. Simplified20.5

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

   11. Applied pow1/320.5

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

   if 1.5126207120265142e+120 < re

   1. Initial program 53.2

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

   2. Initial simplification53.2

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

   4. Applied times-frac53.1

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

   5. Simplified53.1

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

   6. Taylor expanded around inf 7.8

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

4. Final simplification16.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.264515794033503 \cdot 10^{+146}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;re \le 1.5126207120265142 \cdot 10^{+120}:\\ \;\;\;\;\frac{\log \left(\sqrt{{\left(im \cdot im + re \cdot re\right)}^{\frac{1}{3}}} \cdot \left|\sqrt[3]{im \cdot im + re \cdot re}\right|\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

Runtime

Time bar (total: 30.8s)Debug logProfile

herbie shell --seed 2018296 
(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))))