Average Error: 31.8 → 17.9
Time: 8.2s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.6825219586219489 \cdot 10^{108}:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 3.6965442466504777 \cdot 10^{31}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base \cdot \log base\right)}^{3} + {\left(0.0 \cdot 0.0\right)}^{3}}} \cdot \sqrt{\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - \left(\log base \cdot \log base\right) \cdot \left(0.0 \cdot 0.0\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]
\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 -8.6825219586219489 \cdot 10^{108}:\\
\;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\

\mathbf{elif}\;re \le 3.6965442466504777 \cdot 10^{31}:\\
\;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base \cdot \log base\right)}^{3} + {\left(0.0 \cdot 0.0\right)}^{3}}} \cdot \sqrt{\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - \left(\log base \cdot \log base\right) \cdot \left(0.0 \cdot 0.0\right)\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\

\end{array}
double f(double re, double im, double base) {
        double r48272 = re;
        double r48273 = r48272 * r48272;
        double r48274 = im;
        double r48275 = r48274 * r48274;
        double r48276 = r48273 + r48275;
        double r48277 = sqrt(r48276);
        double r48278 = log(r48277);
        double r48279 = base;
        double r48280 = log(r48279);
        double r48281 = r48278 * r48280;
        double r48282 = atan2(r48274, r48272);
        double r48283 = 0.0;
        double r48284 = r48282 * r48283;
        double r48285 = r48281 + r48284;
        double r48286 = r48280 * r48280;
        double r48287 = r48283 * r48283;
        double r48288 = r48286 + r48287;
        double r48289 = r48285 / r48288;
        return r48289;
}
↓
double f(double re, double im, double base) {
        double r48290 = re;
        double r48291 = -8.682521958621949e+108;
        bool r48292 = r48290 <= r48291;
        double r48293 = -1.0;
        double r48294 = r48293 * r48290;
        double r48295 = log(r48294);
        double r48296 = base;
        double r48297 = log(r48296);
        double r48298 = r48295 * r48297;
        double r48299 = im;
        double r48300 = atan2(r48299, r48290);
        double r48301 = 0.0;
        double r48302 = r48300 * r48301;
        double r48303 = r48298 + r48302;
        double r48304 = r48297 * r48297;
        double r48305 = r48301 * r48301;
        double r48306 = r48304 + r48305;
        double r48307 = r48303 / r48306;
        double r48308 = 3.6965442466504777e+31;
        bool r48309 = r48290 <= r48308;
        double r48310 = 1.0;
        double r48311 = sqrt(r48306);
        double r48312 = r48310 / r48311;
        double r48313 = r48290 * r48290;
        double r48314 = r48299 * r48299;
        double r48315 = r48313 + r48314;
        double r48316 = sqrt(r48315);
        double r48317 = log(r48316);
        double r48318 = r48317 * r48297;
        double r48319 = r48318 + r48302;
        double r48320 = 3.0;
        double r48321 = pow(r48304, r48320);
        double r48322 = pow(r48305, r48320);
        double r48323 = r48321 + r48322;
        double r48324 = sqrt(r48323);
        double r48325 = r48319 / r48324;
        double r48326 = r48304 * r48304;
        double r48327 = r48305 * r48305;
        double r48328 = r48304 * r48305;
        double r48329 = r48327 - r48328;
        double r48330 = r48326 + r48329;
        double r48331 = sqrt(r48330);
        double r48332 = r48325 * r48331;
        double r48333 = r48312 * r48332;
        double r48334 = r48310 / r48290;
        double r48335 = log(r48334);
        double r48336 = r48310 / r48296;
        double r48337 = log(r48336);
        double r48338 = r48335 / r48337;
        double r48339 = r48309 ? r48333 : r48338;
        double r48340 = r48292 ? r48307 : r48339;
        return r48340;
}

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 3 regimes

  2. if re < -8.682521958621949e+108

    1. Initial program 53.1

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

    2. Taylor expanded around -inf 8.9

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

    if -8.682521958621949e+108 < re < 3.6965442466504777e+31

    1. Initial program 22.1

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

    3. Applied add-sqr-sqrt22.1

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

    4. Applied *-un-lft-identity22.1

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

    5. Applied times-frac22.1

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

    7. Applied flip3-+22.2

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

    8. Applied sqrt-div22.2

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

    9. Applied associate-/r/22.2

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

    if 3.6965442466504777e+31 < re

    1. Initial program 42.0

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

    2. Taylor expanded around inf 12.9

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.6825219586219489 \cdot 10^{108}:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 3.6965442466504777 \cdot 10^{31}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base \cdot \log base\right)}^{3} + {\left(0.0 \cdot 0.0\right)}^{3}}} \cdot \sqrt{\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - \left(\log base \cdot \log base\right) \cdot \left(0.0 \cdot 0.0\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))