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 r48220 = re;
        double r48221 = r48220 * r48220;
        double r48222 = im;
        double r48223 = r48222 * r48222;
        double r48224 = r48221 + r48223;
        double r48225 = sqrt(r48224);
        double r48226 = log(r48225);
        double r48227 = base;
        double r48228 = log(r48227);
        double r48229 = r48226 * r48228;
        double r48230 = atan2(r48222, r48220);
        double r48231 = 0.0;
        double r48232 = r48230 * r48231;
        double r48233 = r48229 + r48232;
        double r48234 = r48228 * r48228;
        double r48235 = r48231 * r48231;
        double r48236 = r48234 + r48235;
        double r48237 = r48233 / r48236;
        return r48237;
}
↓
double f(double re, double im, double base) {
        double r48238 = re;
        double r48239 = -8.682521958621949e+108;
        bool r48240 = r48238 <= r48239;
        double r48241 = -1.0;
        double r48242 = r48241 * r48238;
        double r48243 = log(r48242);
        double r48244 = base;
        double r48245 = log(r48244);
        double r48246 = r48243 * r48245;
        double r48247 = im;
        double r48248 = atan2(r48247, r48238);
        double r48249 = 0.0;
        double r48250 = r48248 * r48249;
        double r48251 = r48246 + r48250;
        double r48252 = r48245 * r48245;
        double r48253 = r48249 * r48249;
        double r48254 = r48252 + r48253;
        double r48255 = r48251 / r48254;
        double r48256 = 3.6965442466504777e+31;
        bool r48257 = r48238 <= r48256;
        double r48258 = 1.0;
        double r48259 = sqrt(r48254);
        double r48260 = r48258 / r48259;
        double r48261 = r48238 * r48238;
        double r48262 = r48247 * r48247;
        double r48263 = r48261 + r48262;
        double r48264 = sqrt(r48263);
        double r48265 = log(r48264);
        double r48266 = r48265 * r48245;
        double r48267 = r48266 + r48250;
        double r48268 = 3.0;
        double r48269 = pow(r48252, r48268);
        double r48270 = pow(r48253, r48268);
        double r48271 = r48269 + r48270;
        double r48272 = sqrt(r48271);
        double r48273 = r48267 / r48272;
        double r48274 = r48252 * r48252;
        double r48275 = r48253 * r48253;
        double r48276 = r48252 * r48253;
        double r48277 = r48275 - r48276;
        double r48278 = r48274 + r48277;
        double r48279 = sqrt(r48278);
        double r48280 = r48273 * r48279;
        double r48281 = r48260 * r48280;
        double r48282 = r48258 / r48238;
        double r48283 = log(r48282);
        double r48284 = r48258 / r48244;
        double r48285 = log(r48284);
        double r48286 = r48283 / r48285;
        double r48287 = r48257 ? r48281 : r48286;
        double r48288 = r48240 ? r48255 : r48287;
        return r48288;
}

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))))