Average Error: 30.6 → 16.9
Time: 1.3m
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -4.725642240169594 \cdot 10^{+143}:\\ \;\;\;\;\frac{1}{\frac{\log base \cdot \log \left(\sqrt[3]{base}\right) + \log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \log base}{\log \left(-re\right) \cdot \log base}}\\ \mathbf{elif}\;im \le -1.0543741711052671 \cdot 10^{-97}:\\ \;\;\;\;\frac{1}{\log base \cdot \log base} \cdot \left(\log base \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)\\ \mathbf{elif}\;im \le 1.50140156992105 \cdot 10^{-142}:\\ \;\;\;\;\frac{1}{-\frac{\log base}{\log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;im \le 6.752891394528769 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\frac{\log base \cdot \log \left(\sqrt[3]{base}\right) + \log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \log base}{\log base \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Split input into 5 regimes

  2. if im < -4.725642240169594e+143

    1. Initial program 59.1

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

    2. Simplified59.1

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

    4. Applied clear-num59.1

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

    6. Applied add-cube-cbrt59.1

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

    7. Applied log-prod59.1

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

    8. Applied distribute-lft-in59.1

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

    9. Taylor expanded around -inf 50.5

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

    10. Simplified50.5

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

    if -4.725642240169594e+143 < im < -1.0543741711052671e-97

    1. Initial program 15.0

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

    2. Simplified15.0

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

    4. Applied div-inv15.0

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

    if -1.0543741711052671e-97 < im < 1.50140156992105e-142

    1. Initial program 27.0

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

    2. Simplified27.0

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

    4. Applied clear-num27.1

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

    5. Taylor expanded around -inf 62.8

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

    6. Simplified8.4

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

    if 1.50140156992105e-142 < im < 6.752891394528769e+99

    1. Initial program 15.3

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

    2. Simplified15.3

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

    4. Applied clear-num15.3

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

    6. Applied add-cube-cbrt15.3

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

    7. Applied log-prod15.4

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

    8. Applied distribute-lft-in15.4

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

    if 6.752891394528769e+99 < im

    1. Initial program 50.2

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

    2. Simplified50.2

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

    3. Taylor expanded around 0 8.8

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

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -4.725642240169594 \cdot 10^{+143}:\\ \;\;\;\;\frac{1}{\frac{\log base \cdot \log \left(\sqrt[3]{base}\right) + \log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \log base}{\log \left(-re\right) \cdot \log base}}\\ \mathbf{elif}\;im \le -1.0543741711052671 \cdot 10^{-97}:\\ \;\;\;\;\frac{1}{\log base \cdot \log base} \cdot \left(\log base \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)\\ \mathbf{elif}\;im \le 1.50140156992105 \cdot 10^{-142}:\\ \;\;\;\;\frac{1}{-\frac{\log base}{\log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;im \le 6.752891394528769 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\frac{\log base \cdot \log \left(\sqrt[3]{base}\right) + \log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \log base}{\log base \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 
(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))))