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Derivation

1. Split input into 4 regimes

2. if re < -4.0813384213579333e+98

   1. Initial program 49.9

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

   2. Initial simplification49.9

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

   4. Applied div-inv49.9

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

   5. Taylor expanded around -inf 9.1

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

   6. Simplified9.1

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

   if -4.0813384213579333e+98 < re < -8.489122960153244e-97 or 4.005350697321153e-192 < re < 7.776510812884469e+102

   1. Initial program 15.9

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

   2. Initial simplification15.9

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

   4. Applied times-frac15.8

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

   5. Simplified15.8

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

   if -8.489122960153244e-97 < re < 4.005350697321153e-192

   1. Initial program 27.1

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

   2. Initial simplification27.1

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

   4. Applied times-frac27.1

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

   5. Simplified27.1

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

   7. Applied pow1/227.1

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

   8. Applied log-pow27.1

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

   9. Applied associate-/l*27.1

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

   10. Taylor expanded around 0 35.4

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{1}{2} \cdot \frac{\log base}{\log im}}} \cdot 1\]

   if 7.776510812884469e+102 < re

   1. Initial program 50.0

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

   2. Initial simplification50.0

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

   4. Applied times-frac50.0

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

   5. Simplified50.0

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

   7. Applied pow1/250.0

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

   8. Applied log-pow50.0

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

   9. Applied associate-/l*50.0

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

   10. Taylor expanded around inf 9.8

       \[\leadsto \frac{\frac{1}{2}}{\frac{\log base}{\color{blue}{-2 \cdot \log \left(\frac{1}{re}\right)}}} \cdot 1\]

   11. Simplified9.8

       \[\leadsto \frac{\frac{1}{2}}{\frac{\log base}{\color{blue}{\log re \cdot 2}}} \cdot 1\]
3. Recombined 4 regimes into one program.

4. Final simplification19.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.0813384213579333 \cdot 10^{+98}:\\ \;\;\;\;\left(\log base \cdot \log \left(-re\right)\right) \cdot \frac{1}{\log base \cdot \log base}\\ \mathbf{elif}\;re \le -8.489122960153244 \cdot 10^{-97}:\\ \;\;\;\;\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}\\ \mathbf{elif}\;re \le 4.005350697321153 \cdot 10^{-192}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{1}{2} \cdot \frac{\log base}{\log im}}\\ \mathbf{elif}\;re \le 7.776510812884469 \cdot 10^{+102}:\\ \;\;\;\;\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{\log base}{\log re \cdot 2}}\\ \end{array}\]

Runtime

Time bar (total: 41.8s)Debug logProfile

herbie shell --seed 2018263 
(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))))