Error

Try it out

Derivation

1. Split input into 3 regimes

2. if im < -4.135402362840663e+167 or -1.8502929772009478e-155 < im < 1.66338613518387e-124

   1. Initial program 38.6

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

   2. Initial simplification38.6

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

   3. Taylor expanded around -inf 20.5

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

   4. Simplified20.5

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

   6. Applied times-frac20.4

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

   7. Simplified20.4

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

   if -4.135402362840663e+167 < im < -1.8502929772009478e-155 or 1.66338613518387e-124 < im < 6.266361841679746e+28

   1. Initial program 17.3

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

   2. Initial simplification17.3

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

   4. Applied times-frac17.2

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

   5. Simplified17.2

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

   if 6.266361841679746e+28 < im

   1. Initial program 40.9

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

   2. Initial simplification40.9

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

   3. Taylor expanded around 0 12.1

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

4. Final simplification17.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -4.135402362840663 \cdot 10^{+167}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;im \le -1.8502929772009478 \cdot 10^{-155}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\\ \mathbf{elif}\;im \le 1.66338613518387 \cdot 10^{-124}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;im \le 6.266361841679746 \cdot 10^{+28}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 
(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: 18.3s)Debug log