Average Error: 31.0 → 16.6
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;-im \le -1.57800062106739 \cdot 10^{+138}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{if}\;-im \le -3.515003298199428 \cdot 10^{-285}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base}} \cdot \frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \log base}{\sqrt{\log base \cdot \log base}}\\ \mathbf{if}\;-im \le 6.334346309973972 \cdot 10^{-282}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{if}\;-im \le 1.5092936986322766 \cdot 10^{+115}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base}} \cdot \frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \log base}{\sqrt{\log base \cdot \log base}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Split input into 4 regimes

  2. if (- im) < -1.57800062106739e+138

    1. Initial program 57.2

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

    2. Taylor expanded around 0 6.9

      \[\leadsto \color{blue}{\frac{\log im}{\log base}}\]

    if -1.57800062106739e+138 < (- im) < -3.515003298199428e-285 or 6.334346309973972e-282 < (- im) < 1.5092936986322766e+115

    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. Applied simplify19.9

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

    7. Applied simplify19.9

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

    if -3.515003298199428e-285 < (- im) < 6.334346309973972e-282

    1. Initial program 34.3

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

    2. Taylor expanded around -inf 28.7

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

    3. Applied simplify28.7

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

    if 1.5092936986322766e+115 < (- im)

    1. Initial program 53.7

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

    3. Applied flip3-+53.7

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\color{blue}{\frac{{\left(\log base \cdot \log base\right)}^{3} + {\left(0 \cdot 0\right)}^{3}}{\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) + \left(\left(0 \cdot 0\right) \cdot \left(0 \cdot 0\right) - \left(\log base \cdot \log base\right) \cdot \left(0 \cdot 0\right)\right)}}}\]

    4. Applied associate-/r/53.7

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{{\left(\log base \cdot \log base\right)}^{3} + {\left(0 \cdot 0\right)}^{3}} \cdot \left(\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) + \left(\left(0 \cdot 0\right) \cdot \left(0 \cdot 0\right) - \left(\log base \cdot \log base\right) \cdot \left(0 \cdot 0\right)\right)\right)}\]

    5. Applied simplify53.7

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{{\left(\log base\right)}^{3} \cdot \left(\log base \cdot \log base\right)}} \cdot \left(\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) + \left(\left(0 \cdot 0\right) \cdot \left(0 \cdot 0\right) - \left(\log base \cdot \log base\right) \cdot \left(0 \cdot 0\right)\right)\right)\]

    6. Taylor expanded around -inf 8.7

      \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot im\right)}}{{\left(\log base\right)}^{3} \cdot \left(\log base \cdot \log base\right)} \cdot \left(\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) + \left(\left(0 \cdot 0\right) \cdot \left(0 \cdot 0\right) - \left(\log base \cdot \log base\right) \cdot \left(0 \cdot 0\right)\right)\right)\]

    7. Applied simplify8.5

      \[\leadsto \color{blue}{\frac{\log \left(-im\right)}{\frac{\log base}{1}}}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify16.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;-im \le -1.57800062106739 \cdot 10^{+138}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{if}\;-im \le -3.515003298199428 \cdot 10^{-285}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base}} \cdot \frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \log base}{\sqrt{\log base \cdot \log base}}\\ \mathbf{if}\;-im \le 6.334346309973972 \cdot 10^{-282}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{if}\;-im \le 1.5092936986322766 \cdot 10^{+115}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base}} \cdot \frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \log base}{\sqrt{\log base \cdot \log base}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \end{array}}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' 
(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))))