Error

Try it out

Derivation

1. Split input into 2 regimes

2. if re < -1.2142195196570871e+89

   1. Initial program 48.2

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

   2. Simplified48.2

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

   3. Taylor expanded around -inf 62.8

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

   4. Simplified9.3

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

   6. Applied add-cube-cbrt9.3

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

   7. Applied log-prod9.4

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

   8. Applied distribute-lft-in9.4

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

   10. Applied pow1/39.4

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

   if -1.2142195196570871e+89 < re

   1. Initial program 21.2

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

   2. Simplified21.2

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

   4. Applied times-frac21.1

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

   5. Simplified21.1

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base} \cdot \color{blue}{1}\]
3. Recombined 2 regimes into one program.

4. Final simplification16.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2142195196570871 \cdot 10^{+89}:\\ \;\;\;\;\log \left(\sqrt[3]{\frac{-1}{re}} \cdot \sqrt[3]{\frac{-1}{re}}\right) \cdot \frac{-1}{\log base} + \log \left({\left(\frac{-1}{re}\right)}^{\frac{1}{3}}\right) \cdot \frac{-1}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 
(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

Time bar (total: 50.7s)Debug log