Average Error: 31.1 → 17.2
Time: 54.6s
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 r2014870 = re;
        double r2014871 = r2014870 * r2014870;
        double r2014872 = im;
        double r2014873 = r2014872 * r2014872;
        double r2014874 = r2014871 + r2014873;
        double r2014875 = sqrt(r2014874);
        double r2014876 = log(r2014875);
        double r2014877 = 10.0;
        double r2014878 = log(r2014877);
        double r2014879 = r2014876 / r2014878;
        return r2014879;
}
↓
double f(double re, double im) {
        double r2014880 = re;
        double r2014881 = -7.266849055505758e+89;
        bool r2014882 = r2014880 <= r2014881;
        double r2014883 = 1.0;
        double r2014884 = 10.0;
        double r2014885 = log(r2014884);
        double r2014886 = sqrt(r2014885);
        double r2014887 = r2014883 / r2014886;
        double r2014888 = -r2014880;
        double r2014889 = log(r2014888);
        double r2014890 = r2014889 * r2014887;
        double r2014891 = r2014887 * r2014890;
        double r2014892 = 7.762022248986236e+136;
        bool r2014893 = r2014880 <= r2014892;
        double r2014894 = sqrt(r2014887);
        double r2014895 = r2014894 * r2014894;
        double r2014896 = im;
        double r2014897 = r2014896 * r2014896;
        double r2014898 = r2014880 * r2014880;
        double r2014899 = r2014897 + r2014898;
        double r2014900 = sqrt(r2014899);
        double r2014901 = log(r2014900);
        double r2014902 = r2014901 * r2014887;
        double r2014903 = r2014895 * r2014902;
        double r2014904 = log(r2014880);
        double r2014905 = r2014904 * r2014887;
        double r2014906 = r2014905 * r2014894;
        double r2014907 = r2014894 * r2014906;
        double r2014908 = r2014893 ? r2014903 : r2014907;
        double r2014909 = r2014882 ? r2014891 : r2014908;
        return r2014909;
}

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)))