Average Error: 32.2 → 17.6
Time: 12.1s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.040563023697554117134626098062771221449 \cdot 10^{108}:\\ \;\;\;\;\left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right) \cdot \frac{-1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 1.876761322676858408683984671658326382422 \cdot 10^{119}:\\ \;\;\;\;\log \left({\left({\left(im \cdot im + re \cdot re\right)}^{\left(\frac{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}}}{\left|\sqrt[3]{\log 10}\right|}\right)}\right)}^{\left(\frac{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}}}{\sqrt{\sqrt[3]{\log 10}}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log re \cdot 2\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.040563023697554117134626098062771221449 \cdot 10^{108}:\\
\;\;\;\;\left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right) \cdot \frac{-1}{\sqrt{\log 10}}\\

\mathbf{elif}\;re \le 1.876761322676858408683984671658326382422 \cdot 10^{119}:\\
\;\;\;\;\log \left({\left({\left(im \cdot im + re \cdot re\right)}^{\left(\frac{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}}}{\left|\sqrt[3]{\log 10}\right|}\right)}\right)}^{\left(\frac{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}}}{\sqrt{\sqrt[3]{\log 10}}}\right)}\right)\\

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

\end{array}
double f(double re, double im) {
        double r31818 = re;
        double r31819 = r31818 * r31818;
        double r31820 = im;
        double r31821 = r31820 * r31820;
        double r31822 = r31819 + r31821;
        double r31823 = sqrt(r31822);
        double r31824 = log(r31823);
        double r31825 = 10.0;
        double r31826 = log(r31825);
        double r31827 = r31824 / r31826;
        return r31827;
}
↓
double f(double re, double im) {
        double r31828 = re;
        double r31829 = -9.040563023697554e+108;
        bool r31830 = r31828 <= r31829;
        double r31831 = -1.0;
        double r31832 = r31831 / r31828;
        double r31833 = log(r31832);
        double r31834 = 1.0;
        double r31835 = 10.0;
        double r31836 = log(r31835);
        double r31837 = r31834 / r31836;
        double r31838 = sqrt(r31837);
        double r31839 = r31833 * r31838;
        double r31840 = sqrt(r31836);
        double r31841 = r31831 / r31840;
        double r31842 = r31839 * r31841;
        double r31843 = 1.8767613226768584e+119;
        bool r31844 = r31828 <= r31843;
        double r31845 = im;
        double r31846 = r31845 * r31845;
        double r31847 = r31828 * r31828;
        double r31848 = r31846 + r31847;
        double r31849 = 0.5;
        double r31850 = sqrt(r31849);
        double r31851 = sqrt(r31840);
        double r31852 = r31850 / r31851;
        double r31853 = cbrt(r31836);
        double r31854 = fabs(r31853);
        double r31855 = r31852 / r31854;
        double r31856 = pow(r31848, r31855);
        double r31857 = sqrt(r31853);
        double r31858 = r31852 / r31857;
        double r31859 = pow(r31856, r31858);
        double r31860 = log(r31859);
        double r31861 = log(r31828);
        double r31862 = 2.0;
        double r31863 = r31861 * r31862;
        double r31864 = r31849 / r31840;
        double r31865 = r31863 * r31864;
        double r31866 = r31834 / r31840;
        double r31867 = r31865 * r31866;
        double r31868 = r31844 ? r31860 : r31867;
        double r31869 = r31830 ? r31842 : r31868;
        return r31869;
}

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 < -9.040563023697554e+108

    1. Initial program 54.4

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

    3. Applied add-sqr-sqrt54.4

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

    4. Applied pow1/254.4

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

    5. Applied log-pow54.4

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

    6. Applied times-frac54.4

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

    8. Applied div-inv54.3

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

    9. Applied associate-*r*54.3

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

    10. Simplified54.3

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

    11. Taylor expanded around -inf 10.0

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

    12. Simplified10.0

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

    if -9.040563023697554e+108 < re < 1.8767613226768584e+119

    1. Initial program 22.0

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

    3. Applied add-sqr-sqrt22.0

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

    4. Applied pow1/222.0

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

    5. Applied log-pow22.0

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

    6. Applied times-frac22.0

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

    8. Applied div-inv21.9

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

    9. Applied associate-*r*21.9

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

    10. Simplified21.9

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

    12. Applied add-log-exp21.9

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

    13. Simplified21.7

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

    15. Applied add-cube-cbrt22.6

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

    16. Applied sqrt-prod22.6

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

    17. Applied add-sqr-sqrt22.6

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

    18. Applied sqrt-prod22.8

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

    19. Applied add-sqr-sqrt22.6

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

    20. Applied times-frac21.7

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

    21. Applied times-frac21.7

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

    22. Applied pow-unpow21.6

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

    23. Simplified21.6

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

    if 1.8767613226768584e+119 < re

    1. Initial program 56.0

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

    3. Applied add-sqr-sqrt56.0

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

    4. Applied pow1/256.0

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

    5. Applied log-pow56.0

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

    6. Applied times-frac56.0

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

    8. Applied div-inv56.0

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

    9. Applied associate-*r*56.0

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

    10. Simplified56.0

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

    11. Taylor expanded around inf 7.2

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

    12. Simplified7.2

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.040563023697554117134626098062771221449 \cdot 10^{108}:\\ \;\;\;\;\left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right) \cdot \frac{-1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 1.876761322676858408683984671658326382422 \cdot 10^{119}:\\ \;\;\;\;\log \left({\left({\left(im \cdot im + re \cdot re\right)}^{\left(\frac{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}}}{\left|\sqrt[3]{\log 10}\right|}\right)}\right)}^{\left(\frac{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}}}{\sqrt{\sqrt[3]{\log 10}}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log re \cdot 2\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))