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Derivation

1. Split input into 4 regimes

2. if re < -1.0097875590329335e+31

   1. Initial program 41.6

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

   2. Initial simplification41.6

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

   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. Simplified11.3

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

   if -1.0097875590329335e+31 < re < -2.8261408316959366e-210 or -2.637588705690773e-285 < re < 5.376920275859527e+53

   1. Initial program 20.7

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

   2. Initial simplification20.7

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

   4. Applied times-frac20.7

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

   5. Simplified20.7

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

   if -2.8261408316959366e-210 < re < -2.637588705690773e-285

   1. Initial program 31.0

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

   2. Initial simplification31.0

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

   3. Taylor expanded around 0 34.3

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

   if 5.376920275859527e+53 < re

   1. Initial program 44.9

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

   2. Initial simplification44.9

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

   4. Applied times-frac44.9

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

   5. Simplified44.9

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

   6. Taylor expanded around inf 11.7

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

4. Final simplification17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.0097875590329335 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{\log base} \cdot \log \left(\frac{-1}{re}\right)\\ \mathbf{elif}\;re \le -2.8261408316959366 \cdot 10^{-210}:\\ \;\;\;\;\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}\\ \mathbf{elif}\;re \le -2.637588705690773 \cdot 10^{-285}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 5.376920275859527 \cdot 10^{+53}:\\ \;\;\;\;\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

Runtime

Time bar (total: 22.4s)Debug logProfile

herbie shell --seed 2018290 
(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))))