Average Error: 31.6 → 19.0
Time: 6.9s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.30410728802987954 \cdot 10^{54}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log \left(-1 \cdot re\right)}}\\ \mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}\\ \mathbf{elif}\;re \le 2.15180429374657097 \cdot 10^{-295}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(\log im\right)}^{3}}{{\left(\log 10\right)}^{3}}}\\ \mathbf{elif}\;re \le 2.55703958874010118 \cdot 10^{92}:\\ \;\;\;\;\sqrt[3]{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right) \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log re}{\log 10}\right)}^{3}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.30410728802987954 \cdot 10^{54}:\\
\;\;\;\;\frac{1}{\frac{\log 10}{\log \left(-1 \cdot re\right)}}\\

\mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\
\;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}\\

\mathbf{elif}\;re \le 2.15180429374657097 \cdot 10^{-295}:\\
\;\;\;\;\sqrt[3]{\frac{{\left(\log im\right)}^{3}}{{\left(\log 10\right)}^{3}}}\\

\mathbf{elif}\;re \le 2.55703958874010118 \cdot 10^{92}:\\
\;\;\;\;\sqrt[3]{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right) \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{\log re}{\log 10}\right)}^{3}}\\

\end{array}
double f(double re, double im) {
        double r37810 = re;
        double r37811 = r37810 * r37810;
        double r37812 = im;
        double r37813 = r37812 * r37812;
        double r37814 = r37811 + r37813;
        double r37815 = sqrt(r37814);
        double r37816 = log(r37815);
        double r37817 = 10.0;
        double r37818 = log(r37817);
        double r37819 = r37816 / r37818;
        return r37819;
}
↓
double f(double re, double im) {
        double r37820 = re;
        double r37821 = -5.3041072880298795e+54;
        bool r37822 = r37820 <= r37821;
        double r37823 = 1.0;
        double r37824 = 10.0;
        double r37825 = log(r37824);
        double r37826 = -1.0;
        double r37827 = r37826 * r37820;
        double r37828 = log(r37827);
        double r37829 = r37825 / r37828;
        double r37830 = r37823 / r37829;
        double r37831 = -3.8099669373079583e-103;
        bool r37832 = r37820 <= r37831;
        double r37833 = 3.0;
        double r37834 = r37820 * r37820;
        double r37835 = im;
        double r37836 = r37835 * r37835;
        double r37837 = r37834 + r37836;
        double r37838 = sqrt(r37837);
        double r37839 = cbrt(r37838);
        double r37840 = log(r37839);
        double r37841 = r37825 / r37840;
        double r37842 = r37833 / r37841;
        double r37843 = 2.151804293746571e-295;
        bool r37844 = r37820 <= r37843;
        double r37845 = log(r37835);
        double r37846 = pow(r37845, r37833);
        double r37847 = pow(r37825, r37833);
        double r37848 = r37846 / r37847;
        double r37849 = cbrt(r37848);
        double r37850 = 2.5570395887401012e+92;
        bool r37851 = r37820 <= r37850;
        double r37852 = log(r37838);
        double r37853 = r37852 / r37825;
        double r37854 = r37853 * r37853;
        double r37855 = r37854 * r37853;
        double r37856 = cbrt(r37855);
        double r37857 = log(r37820);
        double r37858 = r37857 / r37825;
        double r37859 = pow(r37858, r37833);
        double r37860 = cbrt(r37859);
        double r37861 = r37851 ? r37856 : r37860;
        double r37862 = r37844 ? r37849 : r37861;
        double r37863 = r37832 ? r37842 : r37862;
        double r37864 = r37822 ? r37830 : r37863;
        return r37864;
}

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 5 regimes

  2. if re < -5.3041072880298795e+54

    1. Initial program 44.6

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

    3. Applied add-cube-cbrt44.6

      \[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}{\log 10}\]
    4. Using strategy rm

    5. Applied pow1/344.6

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

    6. Applied pow1/344.6

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

    7. Applied pow1/344.6

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

    8. Applied pow-sqr44.6

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

    9. Applied pow-prod-up44.6

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

    10. Applied log-pow44.6

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

    11. Applied associate-/l*44.6

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

    12. Taylor expanded around -inf 11.1

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

    if -5.3041072880298795e+54 < re < -3.8099669373079583e-103

    1. Initial program 17.0

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

    3. Applied add-cube-cbrt16.9

      \[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}{\log 10}\]
    4. Using strategy rm

    5. Applied pow116.9

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

    6. Applied pow116.9

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

    7. Applied pow116.9

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

    8. Applied pow-prod-up16.9

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

    9. Applied pow-prod-up16.9

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

    10. Applied log-pow17.0

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

    11. Applied associate-/l*16.9

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

    if -3.8099669373079583e-103 < re < 2.151804293746571e-295

    1. Initial program 28.8

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

    3. Applied add-cbrt-cube29.3

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

    4. Applied add-cbrt-cube29.2

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

    5. Applied cbrt-undiv28.8

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

    6. Simplified28.8

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

    7. Taylor expanded around 0 36.1

      \[\leadsto \sqrt[3]{\color{blue}{\frac{{\left(\log im\right)}^{3}}{{\left(\log 10\right)}^{3}}}}\]

    if 2.151804293746571e-295 < re < 2.5570395887401012e+92

    1. Initial program 21.0

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

    3. Applied add-cbrt-cube21.6

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

    4. Applied add-cbrt-cube21.5

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

    5. Applied cbrt-undiv21.0

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

    6. Simplified21.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}}\]
    7. Using strategy rm

    8. Applied add-cbrt-cube21.0

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

    9. Applied rem-cube-cbrt21.0

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

    if 2.5570395887401012e+92 < re

    1. Initial program 49.9

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

    3. Applied add-cbrt-cube50.1

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

    4. Applied add-cbrt-cube50.1

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

    5. Applied cbrt-undiv50.0

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

    6. Simplified49.9

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

    7. Taylor expanded around inf 9.4

      \[\leadsto \sqrt[3]{{\left(\frac{\log \color{blue}{re}}{\log 10}\right)}^{3}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification19.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.30410728802987954 \cdot 10^{54}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log \left(-1 \cdot re\right)}}\\ \mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}\\ \mathbf{elif}\;re \le 2.15180429374657097 \cdot 10^{-295}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(\log im\right)}^{3}}{{\left(\log 10\right)}^{3}}}\\ \mathbf{elif}\;re \le 2.55703958874010118 \cdot 10^{92}:\\ \;\;\;\;\sqrt[3]{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right) \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log re}{\log 10}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))