Average Error: 31.3 → 16.9
Time: 55.3s
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -4.143132872378192 \cdot 10^{+120}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base}} \cdot \frac{\log \left(-im\right) \cdot \log base}{\sqrt{\log base \cdot \log base}}\\ \mathbf{elif}\;im \le 7.520128843874597 \cdot 10^{-237}:\\ \;\;\;\;\frac{\log base \cdot \log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)\right)}{\sqrt{\log base \cdot \log base}} \cdot \frac{1}{\sqrt{\log base \cdot \log base}}\\ \mathbf{elif}\;im \le 6.3791666397320534 \cdot 10^{-173}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \mathbf{elif}\;im \le 3.022768847954413 \cdot 10^{+124}:\\ \;\;\;\;\frac{\log base \cdot \log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)\right)}{\sqrt{\log base \cdot \log base}} \cdot \frac{1}{\sqrt{\log base \cdot \log base}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if im < -4.143132872378192e+120

    1. Initial program 54.7

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

    3. Applied add-sqr-sqrt54.7

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

    4. Applied *-un-lft-identity54.7

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

    5. Applied times-frac54.7

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

    6. Simplified54.7

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

    7. Simplified54.7

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

    8. Taylor expanded around -inf 8.2

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

    9. Simplified8.2

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

    if -4.143132872378192e+120 < im < 7.520128843874597e-237 or 6.3791666397320534e-173 < im < 3.022768847954413e+124

    1. Initial program 19.9

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

    3. Applied add-sqr-sqrt19.9

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

    4. Applied *-un-lft-identity19.9

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

    5. Applied times-frac19.9

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

    6. Simplified19.9

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

    7. Simplified19.9

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

    9. Applied add-cube-cbrt19.9

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

    if 7.520128843874597e-237 < im < 6.3791666397320534e-173

    1. Initial program 31.1

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

    2. Taylor expanded around inf 34.1

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

    3. Simplified34.0

      \[\leadsto \color{blue}{\frac{-\log re}{-\log base}}\]

    if 3.022768847954413e+124 < im

    1. Initial program 54.9

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

    2. Taylor expanded around 0 7.6

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

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -4.143132872378192 \cdot 10^{+120}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base}} \cdot \frac{\log \left(-im\right) \cdot \log base}{\sqrt{\log base \cdot \log base}}\\ \mathbf{elif}\;im \le 7.520128843874597 \cdot 10^{-237}:\\ \;\;\;\;\frac{\log base \cdot \log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)\right)}{\sqrt{\log base \cdot \log base}} \cdot \frac{1}{\sqrt{\log base \cdot \log base}}\\ \mathbf{elif}\;im \le 6.3791666397320534 \cdot 10^{-173}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \mathbf{elif}\;im \le 3.022768847954413 \cdot 10^{+124}:\\ \;\;\;\;\frac{\log base \cdot \log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)\right)}{\sqrt{\log base \cdot \log base}} \cdot \frac{1}{\sqrt{\log base \cdot \log base}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

Runtime

Time bar (total: 55.3s)Debug logProfile

herbie shell --seed 2018215 
(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))))