Average Error: 31.3 → 17.1
Time: 19.3s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.850726757232304656097215039461175225007 \cdot 10^{116}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 9282772713307497733004856393728:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log re\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.850726757232304656097215039461175225007 \cdot 10^{116}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\

\mathbf{elif}\;re \le 9282772713307497733004856393728:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log re\\

\end{array}
double f(double re, double im) {
        double r39731 = re;
        double r39732 = r39731 * r39731;
        double r39733 = im;
        double r39734 = r39733 * r39733;
        double r39735 = r39732 + r39734;
        double r39736 = sqrt(r39735);
        double r39737 = log(r39736);
        double r39738 = 10.0;
        double r39739 = log(r39738);
        double r39740 = r39737 / r39739;
        return r39740;
}
↓
double f(double re, double im) {
        double r39741 = re;
        double r39742 = -9.850726757232305e+116;
        bool r39743 = r39741 <= r39742;
        double r39744 = 1.0;
        double r39745 = 10.0;
        double r39746 = log(r39745);
        double r39747 = sqrt(r39746);
        double r39748 = r39744 / r39747;
        double r39749 = -r39741;
        double r39750 = pow(r39749, r39748);
        double r39751 = log(r39750);
        double r39752 = r39748 * r39751;
        double r39753 = 9.282772713307498e+30;
        bool r39754 = r39741 <= r39753;
        double r39755 = im;
        double r39756 = r39755 * r39755;
        double r39757 = r39741 * r39741;
        double r39758 = r39756 + r39757;
        double r39759 = sqrt(r39758);
        double r39760 = log(r39759);
        double r39761 = r39748 / r39747;
        double r39762 = r39760 * r39761;
        double r39763 = log(r39741);
        double r39764 = r39761 * r39763;
        double r39765 = r39754 ? r39762 : r39764;
        double r39766 = r39743 ? r39752 : r39765;
        return r39766;
}

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 < -9.850726757232305e+116

    1. Initial program 55.6

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

    3. Applied add-sqr-sqrt55.6

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

    4. Applied pow155.6

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

    5. Applied log-pow55.6

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

    6. Applied times-frac55.6

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

    8. Applied add-log-exp55.6

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

    9. Simplified55.5

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

    10. Taylor expanded around -inf 8.2

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

    11. Simplified8.2

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

    if -9.850726757232305e+116 < re < 9.282772713307498e+30

    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-frac21.3

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

    8. Applied add-log-exp21.3

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

    9. Simplified21.2

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

    11. Applied log-pow21.2

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

    12. Applied associate-*r*21.2

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

    13. Simplified21.2

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

    if 9.282772713307498e+30 < re

    1. Initial program 42.1

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

    3. Applied add-sqr-sqrt42.1

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

    4. Applied pow142.1

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

    5. Applied log-pow42.1

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

    6. Applied times-frac42.1

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

    8. Applied add-log-exp42.1

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

    9. Simplified42.0

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

    11. Applied log-pow42.0

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

    12. Applied associate-*r*42.0

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

    13. Simplified42.0

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

    14. Taylor expanded around inf 12.0

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

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.850726757232304656097215039461175225007 \cdot 10^{116}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 9282772713307497733004856393728:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))