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Derivation

1. Split input into 4 regimes

2. if re < -7.266161600389518e+142

   1. Initial program 59.3

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

   2. Initial simplification59.3

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

   4. Applied clear-num59.3

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

   5. Taylor expanded around -inf 62.8

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

   6. Simplified7.1

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

   if -7.266161600389518e+142 < re < 2.0267169770552964e-281 or 1.0336891779074329e-137 < re < 8.623049544666381e+99

   1. Initial program 19.1

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

   2. Initial simplification19.1

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

   4. Applied clear-num19.2

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

   if 2.0267169770552964e-281 < re < 1.0336891779074329e-137

   1. Initial program 29.4

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

   2. Initial simplification29.4

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

   4. Applied associate-/r*29.4

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

   5. Taylor expanded around 0 33.9

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

   if 8.623049544666381e+99 < re

   1. Initial program 49.7

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

   2. Initial simplification49.7

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

   4. Applied clear-num49.7

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

   5. Taylor expanded around inf 8.4

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

   6. Simplified8.4

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

4. Final simplification17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.266161600389518 \cdot 10^{+142}:\\ \;\;\;\;\frac{1}{\frac{-\log base}{\log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \le 2.0267169770552964 \cdot 10^{-281}:\\ \;\;\;\;\frac{1}{\frac{\log base \cdot \log base}{\log base \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)}}\\ \mathbf{elif}\;re \le 1.0336891779074329 \cdot 10^{-137}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 8.623049544666381 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\frac{\log base \cdot \log base}{\log base \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-\log base}{-\log re}}\\ \end{array}\]

Runtime

Time bar (total: 43.9s)Debug logProfile

herbie shell --seed 2018295 
(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))))