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Derivation

1. Split input into 5 regimes

2. if im < -1.3909444433223192e+163

   1. Initial program 62.0

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

   2. Simplified62.0

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

   4. Applied times-frac62.0

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

   5. Simplified62.0

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

   6. Taylor expanded around -inf 51.7

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

   7. Simplified51.7

      \[\leadsto \frac{\log \color{blue}{\left(-re\right)}}{\log base} \cdot 1\]
   8. Using strategy rm

   9. Applied *-un-lft-identity51.7

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

   10. Applied add-sqr-sqrt51.6

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

   11. Applied times-frac51.6

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

   12. Simplified51.6

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

   if -1.3909444433223192e+163 < im < -2.897340525820373e-111

   1. Initial program 16.9

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

   2. Simplified16.9

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

   4. Applied times-frac16.8

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

   5. Simplified16.8

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

   7. Applied pow1/216.8

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

   8. Applied log-pow16.8

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

   9. Applied associate-/l*16.8

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

   if -2.897340525820373e-111 < im < 1.505571081103404e-136 or 7.693766194139374e+18 < im < 1.511238324259529e+52

   1. Initial program 26.9

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

   2. Simplified26.9

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

   3. Taylor expanded around -inf 10.4

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

   4. Simplified10.4

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

   if 1.505571081103404e-136 < im < 7.693766194139374e+18

   1. Initial program 16.2

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

   2. Simplified16.2

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

   4. Applied times-frac16.1

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

   5. Simplified16.1

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

   7. Applied div-inv16.1

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

   if 1.511238324259529e+52 < im

   1. Initial program 43.3

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

   2. Simplified43.3

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

   3. Taylor expanded around 0 11.1

      \[\leadsto \color{blue}{\frac{\log im}{\log base}}\]
3. Recombined 5 regimes into one program.

4. Final simplification17.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -1.3909444433223192 \cdot 10^{+163}:\\ \;\;\;\;\frac{\sqrt{\log \left(-re\right)}}{\log base} \cdot \sqrt{\log \left(-re\right)}\\ \mathbf{elif}\;im \le -2.897340525820373 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{\log base}{\log \left(im \cdot im + re \cdot re\right)}}\\ \mathbf{elif}\;im \le 1.505571081103404 \cdot 10^{-136}:\\ \;\;\;\;\frac{\log base \cdot \log \left(-re\right)}{\log base \cdot \log base}\\ \mathbf{elif}\;im \le 7.693766194139374 \cdot 10^{+18}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \frac{1}{\log base}\\ \mathbf{elif}\;im \le 1.511238324259529 \cdot 10^{+52}:\\ \;\;\;\;\frac{\log base \cdot \log \left(-re\right)}{\log base \cdot \log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(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))))

Details

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