Average Error: 31.7 → 17.4
Time: 9.6s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.980267520528827579962452667551640625594 \cdot 10^{108}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\sqrt{\frac{1}{2}} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}}\\ \mathbf{elif}\;re \le 5.596984149511623767977842891630738092037 \cdot 10^{118}:\\ \;\;\;\;\frac{\left|\sqrt[3]{1}\right|}{\log 10} \cdot \left(\left(\sqrt{\frac{\sqrt[3]{1}}{2}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \sqrt{\frac{1}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\sqrt{\frac{1}{2}} \cdot \left(\log re \cdot 2\right)}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.980267520528827579962452667551640625594 \cdot 10^{108}:\\
\;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\sqrt{\frac{1}{2}} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}}\\

\mathbf{elif}\;re \le 5.596984149511623767977842891630738092037 \cdot 10^{118}:\\
\;\;\;\;\frac{\left|\sqrt[3]{1}\right|}{\log 10} \cdot \left(\left(\sqrt{\frac{\sqrt[3]{1}}{2}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \sqrt{\frac{1}{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\sqrt{\frac{1}{2}} \cdot \left(\log re \cdot 2\right)}}\\

\end{array}
double f(double re, double im) {
        double r57851 = re;
        double r57852 = r57851 * r57851;
        double r57853 = im;
        double r57854 = r57853 * r57853;
        double r57855 = r57852 + r57854;
        double r57856 = sqrt(r57855);
        double r57857 = log(r57856);
        double r57858 = 10.0;
        double r57859 = log(r57858);
        double r57860 = r57857 / r57859;
        return r57860;
}
↓
double f(double re, double im) {
        double r57861 = re;
        double r57862 = -1.9802675205288276e+108;
        bool r57863 = r57861 <= r57862;
        double r57864 = 0.5;
        double r57865 = sqrt(r57864);
        double r57866 = 10.0;
        double r57867 = log(r57866);
        double r57868 = 2.0;
        double r57869 = -1.0;
        double r57870 = r57869 / r57861;
        double r57871 = log(r57870);
        double r57872 = r57868 * r57871;
        double r57873 = -r57872;
        double r57874 = r57865 * r57873;
        double r57875 = r57867 / r57874;
        double r57876 = r57865 / r57875;
        double r57877 = 5.596984149511624e+118;
        bool r57878 = r57861 <= r57877;
        double r57879 = 1.0;
        double r57880 = cbrt(r57879);
        double r57881 = fabs(r57880);
        double r57882 = r57881 / r57867;
        double r57883 = r57880 / r57868;
        double r57884 = sqrt(r57883);
        double r57885 = r57861 * r57861;
        double r57886 = im;
        double r57887 = r57886 * r57886;
        double r57888 = r57885 + r57887;
        double r57889 = log(r57888);
        double r57890 = r57884 * r57889;
        double r57891 = r57890 * r57865;
        double r57892 = r57882 * r57891;
        double r57893 = log(r57861);
        double r57894 = r57893 * r57868;
        double r57895 = r57865 * r57894;
        double r57896 = r57867 / r57895;
        double r57897 = r57865 / r57896;
        double r57898 = r57878 ? r57892 : r57897;
        double r57899 = r57863 ? r57876 : r57898;
        return r57899;
}

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 < -1.9802675205288276e+108

    1. Initial program 52.3

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

    3. Applied pow152.3

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

    4. Applied sqrt-pow152.3

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{2}\right)}\right)}}{\log 10}\]

    5. Applied log-pow52.3

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

    6. Applied associate-/l*52.2

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt52.3

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\]

    9. Applied associate-/l*52.2

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\frac{1}{2}}}}}\]

    10. Simplified52.2

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\color{blue}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\frac{1}{2}}}}}\]
    11. Using strategy rm

    12. Applied div-inv52.2

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\frac{\color{blue}{\log 10 \cdot \frac{1}{\log \left(re \cdot re + im \cdot im\right)}}}{\sqrt{\frac{1}{2}}}}\]

    13. Applied associate-/l*52.2

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\color{blue}{\frac{\log 10}{\frac{\sqrt{\frac{1}{2}}}{\frac{1}{\log \left(re \cdot re + im \cdot im\right)}}}}}\]

    14. Simplified52.2

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\color{blue}{\sqrt{\frac{1}{2}} \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

    15. Taylor expanded around -inf 8.8

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\sqrt{\frac{1}{2}} \cdot \color{blue}{\left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}}}\]

    if -1.9802675205288276e+108 < re < 5.596984149511624e+118

    1. Initial program 21.6

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

    3. Applied pow121.6

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

    4. Applied sqrt-pow121.6

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{2}\right)}\right)}}{\log 10}\]

    5. Applied log-pow21.6

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

    6. Applied associate-/l*21.6

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt21.7

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\]

    9. Applied associate-/l*21.5

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\frac{1}{2}}}}}\]

    10. Simplified21.5

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\color{blue}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\frac{1}{2}}}}}\]
    11. Using strategy rm

    12. Applied div-inv21.6

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\frac{\color{blue}{\log 10 \cdot \frac{1}{\log \left(re \cdot re + im \cdot im\right)}}}{\sqrt{\frac{1}{2}}}}\]

    13. Applied associate-/l*21.5

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\color{blue}{\frac{\log 10}{\frac{\sqrt{\frac{1}{2}}}{\frac{1}{\log \left(re \cdot re + im \cdot im\right)}}}}}\]

    14. Simplified21.5

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\color{blue}{\sqrt{\frac{1}{2}} \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]
    15. Using strategy rm

    16. Applied div-inv21.5

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\color{blue}{\log 10 \cdot \frac{1}{\sqrt{\frac{1}{2}} \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

    17. Applied *-un-lft-identity21.5

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{1 \cdot 2}}}}{\log 10 \cdot \frac{1}{\sqrt{\frac{1}{2}} \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

    18. Applied add-cube-cbrt21.5

      \[\leadsto \frac{\sqrt{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot 2}}}{\log 10 \cdot \frac{1}{\sqrt{\frac{1}{2}} \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

    19. Applied times-frac21.5

      \[\leadsto \frac{\sqrt{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{2}}}}{\log 10 \cdot \frac{1}{\sqrt{\frac{1}{2}} \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

    20. Applied sqrt-prod21.5

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}} \cdot \sqrt{\frac{\sqrt[3]{1}}{2}}}}{\log 10 \cdot \frac{1}{\sqrt{\frac{1}{2}} \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

    21. Applied times-frac21.5

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}}{\log 10} \cdot \frac{\sqrt{\frac{\sqrt[3]{1}}{2}}}{\frac{1}{\sqrt{\frac{1}{2}} \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

    22. Simplified21.5

      \[\leadsto \color{blue}{\frac{\left|\sqrt[3]{1}\right|}{\log 10}} \cdot \frac{\sqrt{\frac{\sqrt[3]{1}}{2}}}{\frac{1}{\sqrt{\frac{1}{2}} \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

    23. Simplified21.5

      \[\leadsto \frac{\left|\sqrt[3]{1}\right|}{\log 10} \cdot \color{blue}{\left(\left(\sqrt{\frac{\sqrt[3]{1}}{2}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \sqrt{\frac{1}{2}}\right)}\]

    if 5.596984149511624e+118 < re

    1. Initial program 55.3

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

    3. Applied pow155.3

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

    4. Applied sqrt-pow155.3

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{2}\right)}\right)}}{\log 10}\]

    5. Applied log-pow55.3

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

    6. Applied associate-/l*55.3

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt55.3

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\]

    9. Applied associate-/l*55.3

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\frac{1}{2}}}}}\]

    10. Simplified55.3

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\color{blue}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\frac{1}{2}}}}}\]
    11. Using strategy rm

    12. Applied div-inv55.3

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\frac{\color{blue}{\log 10 \cdot \frac{1}{\log \left(re \cdot re + im \cdot im\right)}}}{\sqrt{\frac{1}{2}}}}\]

    13. Applied associate-/l*55.3

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\color{blue}{\frac{\log 10}{\frac{\sqrt{\frac{1}{2}}}{\frac{1}{\log \left(re \cdot re + im \cdot im\right)}}}}}\]

    14. Simplified55.3

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\color{blue}{\sqrt{\frac{1}{2}} \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

    15. Taylor expanded around inf 8.3

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\sqrt{\frac{1}{2}} \cdot \color{blue}{\left(-2 \cdot \log \left(\frac{1}{re}\right)\right)}}}\]

    16. Simplified8.3

      \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\sqrt{\frac{1}{2}} \cdot \color{blue}{\left(\log re \cdot 2\right)}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.980267520528827579962452667551640625594 \cdot 10^{108}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\sqrt{\frac{1}{2}} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}}\\ \mathbf{elif}\;re \le 5.596984149511623767977842891630738092037 \cdot 10^{118}:\\ \;\;\;\;\frac{\left|\sqrt[3]{1}\right|}{\log 10} \cdot \left(\left(\sqrt{\frac{\sqrt[3]{1}}{2}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \sqrt{\frac{1}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\sqrt{\frac{1}{2}} \cdot \left(\log re \cdot 2\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))