Average Error: 31.9 → 17.9
Time: 7.0s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.49575566169960319 \cdot 10^{123}:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 4.3665212724914114 \cdot 10^{58}:\\ \;\;\;\;\frac{\frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]
\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 -2.49575566169960319 \cdot 10^{123}:\\
\;\;\;\;\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 4.3665212724914114 \cdot 10^{58}:\\
\;\;\;\;\frac{\frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\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}}\\

\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 r40275 = re;
        double r40276 = r40275 * r40275;
        double r40277 = im;
        double r40278 = r40277 * r40277;
        double r40279 = r40276 + r40278;
        double r40280 = sqrt(r40279);
        double r40281 = log(r40280);
        double r40282 = base;
        double r40283 = log(r40282);
        double r40284 = r40281 * r40283;
        double r40285 = atan2(r40277, r40275);
        double r40286 = 0.0;
        double r40287 = r40285 * r40286;
        double r40288 = r40284 + r40287;
        double r40289 = r40283 * r40283;
        double r40290 = r40286 * r40286;
        double r40291 = r40289 + r40290;
        double r40292 = r40288 / r40291;
        return r40292;
}
↓
double f(double re, double im, double base) {
        double r40293 = re;
        double r40294 = -2.4957556616996032e+123;
        bool r40295 = r40293 <= r40294;
        double r40296 = -1.0;
        double r40297 = r40296 * r40293;
        double r40298 = log(r40297);
        double r40299 = base;
        double r40300 = log(r40299);
        double r40301 = r40298 * r40300;
        double r40302 = im;
        double r40303 = atan2(r40302, r40293);
        double r40304 = 0.0;
        double r40305 = r40303 * r40304;
        double r40306 = r40301 + r40305;
        double r40307 = r40300 * r40300;
        double r40308 = r40304 * r40304;
        double r40309 = r40307 + r40308;
        double r40310 = r40306 / r40309;
        double r40311 = 4.3665212724914114e+58;
        bool r40312 = r40293 <= r40311;
        double r40313 = r40293 * r40293;
        double r40314 = r40302 * r40302;
        double r40315 = r40313 + r40314;
        double r40316 = sqrt(r40315);
        double r40317 = cbrt(r40316);
        double r40318 = r40317 * r40317;
        double r40319 = r40318 * r40317;
        double r40320 = log(r40319);
        double r40321 = r40320 * r40300;
        double r40322 = r40321 + r40305;
        double r40323 = sqrt(r40309);
        double r40324 = r40322 / r40323;
        double r40325 = r40324 / r40323;
        double r40326 = 1.0;
        double r40327 = r40326 / r40293;
        double r40328 = log(r40327);
        double r40329 = r40326 / r40299;
        double r40330 = log(r40329);
        double r40331 = r40328 / r40330;
        double r40332 = r40312 ? r40325 : r40331;
        double r40333 = r40295 ? r40310 : r40332;
        return r40333;
}

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 < -2.4957556616996032e+123

    1. Initial program 54.4

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

    2. Taylor expanded around -inf 8.0

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

    if -2.4957556616996032e+123 < re < 4.3665212724914114e+58

    1. Initial program 22.2

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

    3. Applied add-sqr-sqrt22.2

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

    4. Applied associate-/r*22.2

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

    6. Applied add-cube-cbrt22.2

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

    if 4.3665212724914114e+58 < re

    1. Initial program 46.4

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

    2. Taylor expanded around inf 11.2

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.49575566169960319 \cdot 10^{123}:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 4.3665212724914114 \cdot 10^{58}:\\ \;\;\;\;\frac{\frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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))))