Average Error: 31.1 → 17.2
Time: 54.2s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.266849055505758 \cdot 10^{+89}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(-re\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 7.762022248986236 \cdot 10^{+136}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot \left(\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\left(\log re \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -7.266849055505758 \cdot 10^{+89}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(-re\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\

\mathbf{elif}\;re \le 7.762022248986236 \cdot 10^{+136}:\\
\;\;\;\;\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot \left(\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\left(\log re \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)\\

\end{array}
double f(double re, double im) {
        double r2014865 = re;
        double r2014866 = r2014865 * r2014865;
        double r2014867 = im;
        double r2014868 = r2014867 * r2014867;
        double r2014869 = r2014866 + r2014868;
        double r2014870 = sqrt(r2014869);
        double r2014871 = log(r2014870);
        double r2014872 = 10.0;
        double r2014873 = log(r2014872);
        double r2014874 = r2014871 / r2014873;
        return r2014874;
}
↓
double f(double re, double im) {
        double r2014875 = re;
        double r2014876 = -7.266849055505758e+89;
        bool r2014877 = r2014875 <= r2014876;
        double r2014878 = 1.0;
        double r2014879 = 10.0;
        double r2014880 = log(r2014879);
        double r2014881 = sqrt(r2014880);
        double r2014882 = r2014878 / r2014881;
        double r2014883 = -r2014875;
        double r2014884 = log(r2014883);
        double r2014885 = r2014884 * r2014882;
        double r2014886 = r2014882 * r2014885;
        double r2014887 = 7.762022248986236e+136;
        bool r2014888 = r2014875 <= r2014887;
        double r2014889 = sqrt(r2014882);
        double r2014890 = r2014889 * r2014889;
        double r2014891 = im;
        double r2014892 = r2014891 * r2014891;
        double r2014893 = r2014875 * r2014875;
        double r2014894 = r2014892 + r2014893;
        double r2014895 = sqrt(r2014894);
        double r2014896 = log(r2014895);
        double r2014897 = r2014896 * r2014882;
        double r2014898 = r2014890 * r2014897;
        double r2014899 = log(r2014875);
        double r2014900 = r2014899 * r2014882;
        double r2014901 = r2014900 * r2014889;
        double r2014902 = r2014889 * r2014901;
        double r2014903 = r2014888 ? r2014898 : r2014902;
        double r2014904 = r2014877 ? r2014886 : r2014903;
        return r2014904;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if re < -7.266849055505758e+89

    1. Initial program 47.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt47.9

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

    4. Applied pow147.9

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-pow47.9

      \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-frac47.9

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}}\]
    7. Using strategy rm

    8. Applied div-inv47.8

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)}\]

    9. Applied associate-*r*47.8

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \frac{1}{\sqrt{\log 10}}}\]

    10. Taylor expanded around -inf 9.3

      \[\leadsto \left(\frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{\left(-1 \cdot re\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\]

    11. Simplified9.3

      \[\leadsto \left(\frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{\left(-re\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\]

    if -7.266849055505758e+89 < re < 7.762022248986236e+136

    1. Initial program 21.4

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt21.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

    4. Applied pow121.4

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-pow21.4

      \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-frac21.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}}\]
    7. Using strategy rm

    8. Applied div-inv21.2

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)}\]

    9. Applied associate-*r*21.2

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \frac{1}{\sqrt{\log 10}}}\]
    10. Using strategy rm

    11. Applied add-sqr-sqrt21.2

      \[\leadsto \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \color{blue}{\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)}\]

    12. Applied associate-*r*21.3

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}}\]
    13. Using strategy rm

    14. Applied associate-*l*21.2

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)}\]

    if 7.762022248986236e+136 < re

    1. Initial program 57.5

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt57.5

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

    4. Applied pow157.5

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-pow57.5

      \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-frac57.5

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}}\]
    7. Using strategy rm

    8. Applied div-inv57.4

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)}\]

    9. Applied associate-*r*57.4

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \frac{1}{\sqrt{\log 10}}}\]
    10. Using strategy rm

    11. Applied add-sqr-sqrt57.4

      \[\leadsto \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \color{blue}{\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)}\]

    12. Applied associate-*r*57.5

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}}\]

    13. Taylor expanded around inf 7.5

      \[\leadsto \left(\left(\frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{re}\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.266849055505758 \cdot 10^{+89}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(-re\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 7.762022248986236 \cdot 10^{+136}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot \left(\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\left(\log re \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))