Average Error: 32.0 → 18.6
Time: 27.0s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -732279.798828648519702255725860595703125:\\ \;\;\;\;\frac{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log \left(-re\right) \cdot \log base}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 2.521155375520642127611045408258516477621 \cdot 10^{-231}:\\ \;\;\;\;\frac{\frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 6.77316547352422363096540025987530146079 \cdot 10^{-176}:\\ \;\;\;\;\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log im \cdot \log base}{{\left(\log base\right)}^{4} - {0.0}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\ \mathbf{elif}\;re \le 645957920562454986752:\\ \;\;\;\;\frac{\frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}
↓
\begin{array}{l}
\mathbf{if}\;re \le -732279.798828648519702255725860595703125:\\
\;\;\;\;\frac{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log \left(-re\right) \cdot \log base}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\

\mathbf{elif}\;re \le 2.521155375520642127611045408258516477621 \cdot 10^{-231}:\\
\;\;\;\;\frac{\frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\

\mathbf{elif}\;re \le 6.77316547352422363096540025987530146079 \cdot 10^{-176}:\\
\;\;\;\;\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log im \cdot \log base}{{\left(\log base\right)}^{4} - {0.0}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\

\mathbf{elif}\;re \le 645957920562454986752:\\
\;\;\;\;\frac{\frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-\log re}{-\log base}\\

\end{array}
double f(double re, double im, double base) {
        double r60346 = re;
        double r60347 = r60346 * r60346;
        double r60348 = im;
        double r60349 = r60348 * r60348;
        double r60350 = r60347 + r60349;
        double r60351 = sqrt(r60350);
        double r60352 = log(r60351);
        double r60353 = base;
        double r60354 = log(r60353);
        double r60355 = r60352 * r60354;
        double r60356 = atan2(r60348, r60346);
        double r60357 = 0.0;
        double r60358 = r60356 * r60357;
        double r60359 = r60355 + r60358;
        double r60360 = r60354 * r60354;
        double r60361 = r60357 * r60357;
        double r60362 = r60360 + r60361;
        double r60363 = r60359 / r60362;
        return r60363;
}
↓
double f(double re, double im, double base) {
        double r60364 = re;
        double r60365 = -732279.7988286485;
        bool r60366 = r60364 <= r60365;
        double r60367 = im;
        double r60368 = atan2(r60367, r60364);
        double r60369 = 0.0;
        double r60370 = r60368 * r60369;
        double r60371 = -r60364;
        double r60372 = log(r60371);
        double r60373 = base;
        double r60374 = log(r60373);
        double r60375 = r60372 * r60374;
        double r60376 = r60370 + r60375;
        double r60377 = 2.0;
        double r60378 = pow(r60374, r60377);
        double r60379 = r60369 * r60369;
        double r60380 = r60378 + r60379;
        double r60381 = sqrt(r60380);
        double r60382 = r60376 / r60381;
        double r60383 = r60374 * r60374;
        double r60384 = r60383 + r60379;
        double r60385 = sqrt(r60384);
        double r60386 = r60382 / r60385;
        double r60387 = 2.521155375520642e-231;
        bool r60388 = r60364 <= r60387;
        double r60389 = r60364 * r60364;
        double r60390 = r60367 * r60367;
        double r60391 = r60389 + r60390;
        double r60392 = sqrt(r60391);
        double r60393 = log(r60392);
        double r60394 = r60374 * r60393;
        double r60395 = r60394 + r60370;
        double r60396 = r60395 / r60381;
        double r60397 = r60396 / r60385;
        double r60398 = 6.773165473524224e-176;
        bool r60399 = r60364 <= r60398;
        double r60400 = log(r60367);
        double r60401 = r60400 * r60374;
        double r60402 = r60370 + r60401;
        double r60403 = 4.0;
        double r60404 = pow(r60374, r60403);
        double r60405 = pow(r60369, r60403);
        double r60406 = r60404 - r60405;
        double r60407 = r60402 / r60406;
        double r60408 = r60383 - r60379;
        double r60409 = r60407 * r60408;
        double r60410 = 6.45957920562455e+20;
        bool r60411 = r60364 <= r60410;
        double r60412 = log(r60364);
        double r60413 = -r60412;
        double r60414 = -r60374;
        double r60415 = r60413 / r60414;
        double r60416 = r60411 ? r60397 : r60415;
        double r60417 = r60399 ? r60409 : r60416;
        double r60418 = r60388 ? r60397 : r60417;
        double r60419 = r60366 ? r60386 : r60418;
        return r60419;
}

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 re < -732279.7988286485

    1. Initial program 42.1

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt42.1

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

    4. Applied associate-/r*42.0

      \[\leadsto \color{blue}{\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

    5. Simplified42.0

      \[\leadsto \frac{\color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    6. Taylor expanded around -inf 14.1

      \[\leadsto \frac{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log \color{blue}{\left(-1 \cdot re\right)} \cdot \log base}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    7. Simplified14.1

      \[\leadsto \frac{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log \color{blue}{\left(-re\right)} \cdot \log base}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    if -732279.7988286485 < re < 2.521155375520642e-231 or 6.773165473524224e-176 < re < 6.45957920562455e+20

    1. Initial program 22.3

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt22.3

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

    4. Applied associate-/r*22.3

      \[\leadsto \color{blue}{\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

    5. Simplified22.3

      \[\leadsto \frac{\color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    if 2.521155375520642e-231 < re < 6.773165473524224e-176

    1. Initial program 31.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
    2. Using strategy rm

    3. Applied flip-+31.9

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\frac{\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) - \left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right)}{\log base \cdot \log base - 0.0 \cdot 0.0}}}\]

    4. Applied associate-/r/31.9

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) - \left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right)} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)}\]

    5. Simplified31.9

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{{\left(\log base\right)}^{4} - {0.0}^{\left(3 + 1\right)}}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\]

    6. Taylor expanded around 0 34.0

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \color{blue}{\log base \cdot \log im}}{{\left(\log base\right)}^{4} - {0.0}^{\left(3 + 1\right)}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\]

    if 6.45957920562455e+20 < re

    1. Initial program 41.4

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

    2. Taylor expanded around inf 12.7

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

    3. Simplified12.7

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

  4. Final simplification18.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -732279.798828648519702255725860595703125:\\ \;\;\;\;\frac{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log \left(-re\right) \cdot \log base}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 2.521155375520642127611045408258516477621 \cdot 10^{-231}:\\ \;\;\;\;\frac{\frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 6.77316547352422363096540025987530146079 \cdot 10^{-176}:\\ \;\;\;\;\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log im \cdot \log base}{{\left(\log base\right)}^{4} - {0.0}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\ \mathbf{elif}\;re \le 645957920562454986752:\\ \;\;\;\;\frac{\frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))