Error

Try it out

Derivation

1. Split input into 4 regimes

2. if im < -1.339088896496117e+154

   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. Taylor expanded around -inf 51.6

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

   4. Simplified51.6

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

   6. Applied add-cube-cbrt51.7

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

   7. Applied associate-*l*51.7

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

   if -1.339088896496117e+154 < im < -5.717705561230724e-79

   1. Initial program 13.9

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

   2. Simplified13.9

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

   4. Applied clear-num13.9

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

   if -5.717705561230724e-79 < im < 2.9780377269530606e-08

   1. Initial program 24.0

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

   2. Simplified24.0

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

   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. Simplified11.5

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

   5. Taylor expanded around -inf 62.8

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

   6. Simplified11.5

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

   if 2.9780377269530606e-08 < im

   1. Initial program 38.9

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

   2. Simplified38.9

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

   3. Taylor expanded around 0 14.1

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

4. Final simplification17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -1.339088896496117 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\log \left(-re\right)} \cdot \sqrt[3]{\log \left(-re\right)}\right) \cdot \left(\log base \cdot \sqrt[3]{\log \left(-re\right)}\right)}{\log base \cdot \log base}\\ \mathbf{elif}\;im \le -5.717705561230724 \cdot 10^{-79}:\\ \;\;\;\;\frac{1}{\frac{\log base \cdot \log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}\\ \mathbf{elif}\;im \le 2.9780377269530606 \cdot 10^{-08}:\\ \;\;\;\;\frac{-1}{\log base} \cdot \log \left(\frac{-1}{re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019021 
(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: 54.9s)Debug log