Error

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Derivation

1. Split input into 2 regimes

2. if re < -6.766300819866214e+134

   1. Initial program 57.0

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

   2. Initial simplification57.0

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

   4. Applied times-frac57.0

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

   5. Simplified57.0

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

   7. Applied div-inv57.0

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

   8. Taylor expanded around -inf 7.9

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

   9. Simplified7.9

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

   if -6.766300819866214e+134 < re

   1. Initial program 20.1

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

   2. Initial simplification20.1

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

   4. Applied times-frac20.1

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

   5. Simplified20.1

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

   7. Applied pow1/220.1

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

   8. Applied log-pow20.1

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

   9. Applied associate-/l*20.1

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

   11. Applied +-commutative20.1

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

4. Final simplification16.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.766300819866214 \cdot 10^{+134}:\\ \;\;\;\;\log \left(-re\right) \cdot \frac{1}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{\log base}{\log \left(im \cdot im + re \cdot re\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 
(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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