Average Error: 31.9 → 18.1
Time: 22.7s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.716637846399996097489746475942839452177 \cdot 10^{89}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{elif}\;re \le -2.205324998556483236155843263778022896956 \cdot 10^{-213}:\\ \;\;\;\;\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 1.954753201427298447121007035735344988274 \cdot 10^{-192}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 1.668135135135140487797926891515850101405 \cdot 10^{72}:\\ \;\;\;\;\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]
\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 -4.716637846399996097489746475942839452177 \cdot 10^{89}:\\
\;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\

\mathbf{elif}\;re \le -2.205324998556483236155843263778022896956 \cdot 10^{-213}:\\
\;\;\;\;\frac{\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\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\

\mathbf{elif}\;re \le 1.954753201427298447121007035735344988274 \cdot 10^{-192}:\\
\;\;\;\;\frac{\log im}{\log base}\\

\mathbf{elif}\;re \le 1.668135135135140487797926891515850101405 \cdot 10^{72}:\\
\;\;\;\;\frac{\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\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 r38142 = re;
        double r38143 = r38142 * r38142;
        double r38144 = im;
        double r38145 = r38144 * r38144;
        double r38146 = r38143 + r38145;
        double r38147 = sqrt(r38146);
        double r38148 = log(r38147);
        double r38149 = base;
        double r38150 = log(r38149);
        double r38151 = r38148 * r38150;
        double r38152 = atan2(r38144, r38142);
        double r38153 = 0.0;
        double r38154 = r38152 * r38153;
        double r38155 = r38151 + r38154;
        double r38156 = r38150 * r38150;
        double r38157 = r38153 * r38153;
        double r38158 = r38156 + r38157;
        double r38159 = r38155 / r38158;
        return r38159;
}
↓
double f(double re, double im, double base) {
        double r38160 = re;
        double r38161 = -4.716637846399996e+89;
        bool r38162 = r38160 <= r38161;
        double r38163 = -1.0;
        double r38164 = r38163 / r38160;
        double r38165 = log(r38164);
        double r38166 = -r38165;
        double r38167 = base;
        double r38168 = log(r38167);
        double r38169 = r38166 / r38168;
        double r38170 = -2.2053249985564832e-213;
        bool r38171 = r38160 <= r38170;
        double r38172 = r38160 * r38160;
        double r38173 = im;
        double r38174 = r38173 * r38173;
        double r38175 = r38172 + r38174;
        double r38176 = sqrt(r38175);
        double r38177 = log(r38176);
        double r38178 = r38177 * r38168;
        double r38179 = atan2(r38173, r38160);
        double r38180 = 0.0;
        double r38181 = r38179 * r38180;
        double r38182 = r38178 + r38181;
        double r38183 = 2.0;
        double r38184 = pow(r38168, r38183);
        double r38185 = r38180 * r38180;
        double r38186 = r38184 + r38185;
        double r38187 = sqrt(r38186);
        double r38188 = r38182 / r38187;
        double r38189 = r38168 * r38168;
        double r38190 = r38189 + r38185;
        double r38191 = sqrt(r38190);
        double r38192 = r38188 / r38191;
        double r38193 = 1.9547532014272984e-192;
        bool r38194 = r38160 <= r38193;
        double r38195 = log(r38173);
        double r38196 = r38195 / r38168;
        double r38197 = 1.6681351351351405e+72;
        bool r38198 = r38160 <= r38197;
        double r38199 = log(r38160);
        double r38200 = -r38199;
        double r38201 = -r38168;
        double r38202 = r38200 / r38201;
        double r38203 = r38198 ? r38192 : r38202;
        double r38204 = r38194 ? r38196 : r38203;
        double r38205 = r38171 ? r38192 : r38204;
        double r38206 = r38162 ? r38169 : r38205;
        return r38206;
}

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 < -4.716637846399996e+89

    1. Initial program 50.5

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

    3. Applied add-sqr-sqrt50.5

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

    4. Applied *-un-lft-identity50.5

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

    5. Applied times-frac50.5

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

    6. Simplified50.5

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

    7. Simplified50.5

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

    8. Taylor expanded around -inf 64.0

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

    9. Simplified10.1

      \[\leadsto \color{blue}{\frac{-\log \left(\frac{-1}{re}\right)}{0 + \log base}}\]

    if -4.716637846399996e+89 < re < -2.2053249985564832e-213 or 1.9547532014272984e-192 < re < 1.6681351351351405e+72

    1. Initial program 18.3

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

    3. Applied add-sqr-sqrt18.3

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

    4. Applied associate-/r*18.2

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

    5. Simplified18.2

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

    if -2.2053249985564832e-213 < re < 1.9547532014272984e-192

    1. Initial program 31.1

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

    2. Taylor expanded around 0 34.4

      \[\leadsto \color{blue}{\frac{\log im}{\log base}}\]

    if 1.6681351351351405e+72 < re

    1. Initial program 47.4

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

    2. Taylor expanded around inf 10.4

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

    3. Simplified10.4

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

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.716637846399996097489746475942839452177 \cdot 10^{89}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{elif}\;re \le -2.205324998556483236155843263778022896956 \cdot 10^{-213}:\\ \;\;\;\;\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 1.954753201427298447121007035735344988274 \cdot 10^{-192}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 1.668135135135140487797926891515850101405 \cdot 10^{72}:\\ \;\;\;\;\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))))