Average Error: 31.9 → 18.0
Time: 19.6s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1318044099381772512265940808597708967248000:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 6.261598578150243116619193101283974553908 \cdot 10^{58}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(-{0.0}^{3}\right) \cdot 0.0 + {\left(\log base\right)}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log re \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \end{array}\]
\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 -1318044099381772512265940808597708967248000:\\
\;\;\;\;\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 6.261598578150243116619193101283974553908 \cdot 10^{58}:\\
\;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(-{0.0}^{3}\right) \cdot 0.0 + {\left(\log base\right)}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\

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

\end{array}
double f(double re, double im, double base) {
        double r54269 = re;
        double r54270 = r54269 * r54269;
        double r54271 = im;
        double r54272 = r54271 * r54271;
        double r54273 = r54270 + r54272;
        double r54274 = sqrt(r54273);
        double r54275 = log(r54274);
        double r54276 = base;
        double r54277 = log(r54276);
        double r54278 = r54275 * r54277;
        double r54279 = atan2(r54271, r54269);
        double r54280 = 0.0;
        double r54281 = r54279 * r54280;
        double r54282 = r54278 + r54281;
        double r54283 = r54277 * r54277;
        double r54284 = r54280 * r54280;
        double r54285 = r54283 + r54284;
        double r54286 = r54282 / r54285;
        return r54286;
}
↓
double f(double re, double im, double base) {
        double r54287 = re;
        double r54288 = -1.3180440993817725e+42;
        bool r54289 = r54287 <= r54288;
        double r54290 = -1.0;
        double r54291 = r54290 * r54287;
        double r54292 = log(r54291);
        double r54293 = base;
        double r54294 = log(r54293);
        double r54295 = r54292 * r54294;
        double r54296 = im;
        double r54297 = atan2(r54296, r54287);
        double r54298 = 0.0;
        double r54299 = r54297 * r54298;
        double r54300 = r54295 + r54299;
        double r54301 = r54294 * r54294;
        double r54302 = r54298 * r54298;
        double r54303 = r54301 + r54302;
        double r54304 = r54300 / r54303;
        double r54305 = 6.261598578150243e+58;
        bool r54306 = r54287 <= r54305;
        double r54307 = r54287 * r54287;
        double r54308 = r54296 * r54296;
        double r54309 = r54307 + r54308;
        double r54310 = sqrt(r54309);
        double r54311 = log(r54310);
        double r54312 = r54311 * r54294;
        double r54313 = r54312 + r54299;
        double r54314 = 3.0;
        double r54315 = pow(r54298, r54314);
        double r54316 = -r54315;
        double r54317 = r54316 * r54298;
        double r54318 = 4.0;
        double r54319 = pow(r54294, r54318);
        double r54320 = r54317 + r54319;
        double r54321 = r54313 / r54320;
        double r54322 = r54301 - r54302;
        double r54323 = r54321 * r54322;
        double r54324 = log(r54287);
        double r54325 = r54324 * r54294;
        double r54326 = r54325 + r54299;
        double r54327 = sqrt(r54303);
        double r54328 = r54326 / r54327;
        double r54329 = r54328 / r54327;
        double r54330 = r54306 ? r54323 : r54329;
        double r54331 = r54289 ? r54304 : r54330;
        return r54331;
}

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 < -1.3180440993817725e+42

    1. Initial program 44.9

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

    2. Taylor expanded around -inf 11.4

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

    if -1.3180440993817725e+42 < re < 6.261598578150243e+58

    1. Initial program 22.6

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

    3. Applied flip-+22.6

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

    4. Applied associate-/r/22.7

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

    5. Simplified22.7

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

    if 6.261598578150243e+58 < re

    1. Initial program 45.3

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

    3. Applied add-sqr-sqrt45.3

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

    4. Applied associate-/r*45.2

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

    5. Taylor expanded around inf 11.6

      \[\leadsto \frac{\frac{\log \color{blue}{re} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1318044099381772512265940808597708967248000:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 6.261598578150243116619193101283974553908 \cdot 10^{58}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(-{0.0}^{3}\right) \cdot 0.0 + {\left(\log base\right)}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log re \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(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))))