Average Error: 30.4 → 17.9
Time: 16.2s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -5.252223443509709 \cdot 10^{+144}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{im}\right)}{\log 10} \cdot -1\\ \mathbf{elif}\;im \le -1.4780679115634563 \cdot 10^{-120}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \log \left(im \cdot im + re \cdot re\right)\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \mathbf{elif}\;im \le -1.8298455091974634 \cdot 10^{-246}:\\ \;\;\;\;\left(\left(2 \cdot \log re\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \mathbf{elif}\;im \le 2.264720310277972 \cdot 10^{+61}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \log \left(im \cdot im + re \cdot re\right)\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\left(2 \cdot \log im\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;im \le -5.252223443509709 \cdot 10^{+144}:\\
\;\;\;\;\frac{\log \left(\frac{-1}{im}\right)}{\log 10} \cdot -1\\

\mathbf{elif}\;im \le -1.4780679115634563 \cdot 10^{-120}:\\
\;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \log \left(im \cdot im + re \cdot re\right)\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\

\mathbf{elif}\;im \le -1.8298455091974634 \cdot 10^{-246}:\\
\;\;\;\;\left(\left(2 \cdot \log re\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\

\mathbf{elif}\;im \le 2.264720310277972 \cdot 10^{+61}:\\
\;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \log \left(im \cdot im + re \cdot re\right)\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\left(2 \cdot \log im\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\

\end{array}
double f(double re, double im) {
        double r1276686 = re;
        double r1276687 = r1276686 * r1276686;
        double r1276688 = im;
        double r1276689 = r1276688 * r1276688;
        double r1276690 = r1276687 + r1276689;
        double r1276691 = sqrt(r1276690);
        double r1276692 = log(r1276691);
        double r1276693 = 10.0;
        double r1276694 = log(r1276693);
        double r1276695 = r1276692 / r1276694;
        return r1276695;
}
↓
double f(double re, double im) {
        double r1276696 = im;
        double r1276697 = -5.252223443509709e+144;
        bool r1276698 = r1276696 <= r1276697;
        double r1276699 = -1.0;
        double r1276700 = r1276699 / r1276696;
        double r1276701 = log(r1276700);
        double r1276702 = 10.0;
        double r1276703 = log(r1276702);
        double r1276704 = r1276701 / r1276703;
        double r1276705 = r1276704 * r1276699;
        double r1276706 = -1.4780679115634563e-120;
        bool r1276707 = r1276696 <= r1276706;
        double r1276708 = 0.5;
        double r1276709 = cbrt(r1276708);
        double r1276710 = cbrt(r1276703);
        double r1276711 = r1276709 / r1276710;
        double r1276712 = r1276696 * r1276696;
        double r1276713 = re;
        double r1276714 = r1276713 * r1276713;
        double r1276715 = r1276712 + r1276714;
        double r1276716 = log(r1276715);
        double r1276717 = r1276711 * r1276716;
        double r1276718 = r1276711 * r1276711;
        double r1276719 = r1276717 * r1276718;
        double r1276720 = -1.8298455091974634e-246;
        bool r1276721 = r1276696 <= r1276720;
        double r1276722 = 2.0;
        double r1276723 = log(r1276713);
        double r1276724 = r1276722 * r1276723;
        double r1276725 = r1276724 * r1276711;
        double r1276726 = r1276725 * r1276718;
        double r1276727 = 2.264720310277972e+61;
        bool r1276728 = r1276696 <= r1276727;
        double r1276729 = log(r1276696);
        double r1276730 = r1276722 * r1276729;
        double r1276731 = r1276730 * r1276711;
        double r1276732 = r1276718 * r1276731;
        double r1276733 = r1276728 ? r1276719 : r1276732;
        double r1276734 = r1276721 ? r1276726 : r1276733;
        double r1276735 = r1276707 ? r1276719 : r1276734;
        double r1276736 = r1276698 ? r1276705 : r1276735;
        return r1276736;
}

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 4 regimes

  2. if im < -5.252223443509709e+144

    1. Initial program 59.5

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

    3. Applied pow1/259.5

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

    4. Applied log-pow59.5

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

    5. Applied associate-/l*59.5

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

    7. Applied +-commutative59.5

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

    8. Taylor expanded around -inf 6.9

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

    if -5.252223443509709e+144 < im < -1.4780679115634563e-120 or -1.8298455091974634e-246 < im < 2.264720310277972e+61

    1. Initial program 19.6

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

    3. Applied pow1/219.6

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

    4. Applied log-pow19.6

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

    5. Applied associate-/l*19.7

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

    7. Applied pow119.7

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

    8. Applied log-pow19.7

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

    9. Applied add-cube-cbrt20.2

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

    10. Applied times-frac20.2

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

    11. Applied add-cube-cbrt19.6

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

    12. Applied times-frac19.5

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

    13. Simplified19.5

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

    14. Simplified19.5

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

    if -1.4780679115634563e-120 < im < -1.8298455091974634e-246

    1. Initial program 25.3

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

    3. Applied pow1/225.3

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

    4. Applied log-pow25.3

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

    5. Applied associate-/l*25.4

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

    7. Applied pow125.4

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

    8. Applied log-pow25.4

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

    9. Applied add-cube-cbrt25.9

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

    10. Applied times-frac25.8

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

    11. Applied add-cube-cbrt25.3

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

    12. Applied times-frac25.2

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

    13. Simplified25.2

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

    14. Simplified25.2

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

    15. Taylor expanded around inf 37.5

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

    16. Simplified37.5

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

    if 2.264720310277972e+61 < im

    1. Initial program 44.4

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

    3. Applied pow1/244.4

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

    4. Applied log-pow44.4

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

    5. Applied associate-/l*44.4

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

    7. Applied pow144.4

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

    8. Applied log-pow44.4

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

    9. Applied add-cube-cbrt44.6

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

    10. Applied times-frac44.6

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

    11. Applied add-cube-cbrt44.4

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

    12. Applied times-frac44.4

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

    13. Simplified44.4

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

    14. Simplified44.3

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

    15. Taylor expanded around 0 10.1

      \[\leadsto \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \color{blue}{\left(2 \cdot \log im\right)}\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -5.252223443509709 \cdot 10^{+144}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{im}\right)}{\log 10} \cdot -1\\ \mathbf{elif}\;im \le -1.4780679115634563 \cdot 10^{-120}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \log \left(im \cdot im + re \cdot re\right)\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \mathbf{elif}\;im \le -1.8298455091974634 \cdot 10^{-246}:\\ \;\;\;\;\left(\left(2 \cdot \log re\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \mathbf{elif}\;im \le 2.264720310277972 \cdot 10^{+61}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \log \left(im \cdot im + re \cdot re\right)\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\left(2 \cdot \log im\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))