Average Error: 31.7 → 18.2
Time: 22.4s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.96812445344144669 \cdot 10^{65}:\\ \;\;\;\;\frac{-1}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \le -4.5165339976767707 \cdot 10^{-174}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -1.2339565047327783 \cdot 10^{-286}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \le 2.40573116711001939 \cdot 10^{87}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.96812445344144669 \cdot 10^{65}:\\
\;\;\;\;\frac{-1}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\

\mathbf{elif}\;re \le -4.5165339976767707 \cdot 10^{-174}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\\

\mathbf{elif}\;re \le -1.2339565047327783 \cdot 10^{-286}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\\

\mathbf{elif}\;re \le 2.40573116711001939 \cdot 10^{87}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\\

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

\end{array}
double f(double re, double im) {
        double r39828 = re;
        double r39829 = r39828 * r39828;
        double r39830 = im;
        double r39831 = r39830 * r39830;
        double r39832 = r39829 + r39831;
        double r39833 = sqrt(r39832);
        double r39834 = log(r39833);
        double r39835 = 10.0;
        double r39836 = log(r39835);
        double r39837 = r39834 / r39836;
        return r39837;
}
↓
double f(double re, double im) {
        double r39838 = re;
        double r39839 = -6.968124453441447e+65;
        bool r39840 = r39838 <= r39839;
        double r39841 = -1.0;
        double r39842 = 10.0;
        double r39843 = log(r39842);
        double r39844 = sqrt(r39843);
        double r39845 = r39841 / r39844;
        double r39846 = r39841 / r39838;
        double r39847 = log(r39846);
        double r39848 = 1.0;
        double r39849 = r39848 / r39843;
        double r39850 = sqrt(r39849);
        double r39851 = r39847 * r39850;
        double r39852 = r39845 * r39851;
        double r39853 = -4.516533997676771e-174;
        bool r39854 = r39838 <= r39853;
        double r39855 = r39848 / r39844;
        double r39856 = r39838 * r39838;
        double r39857 = im;
        double r39858 = r39857 * r39857;
        double r39859 = r39856 + r39858;
        double r39860 = sqrt(r39859);
        double r39861 = log(r39860);
        double r39862 = r39861 / r39844;
        double r39863 = r39855 * r39862;
        double r39864 = -1.2339565047327783e-286;
        bool r39865 = r39838 <= r39864;
        double r39866 = log(r39857);
        double r39867 = r39866 * r39850;
        double r39868 = r39855 * r39867;
        double r39869 = 2.4057311671100194e+87;
        bool r39870 = r39838 <= r39869;
        double r39871 = log(r39838);
        double r39872 = r39850 * r39871;
        double r39873 = r39872 * r39855;
        double r39874 = r39870 ? r39863 : r39873;
        double r39875 = r39865 ? r39868 : r39874;
        double r39876 = r39854 ? r39863 : r39875;
        double r39877 = r39840 ? r39852 : r39876;
        return r39877;
}

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 re < -6.968124453441447e+65

    1. Initial program 45.9

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

    3. Applied add-sqr-sqrt45.9

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

    4. Applied pow145.9

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

    5. Applied log-pow45.9

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

    6. Applied times-frac45.8

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

    7. Taylor expanded around -inf 10.3

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

    8. Simplified10.3

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

    if -6.968124453441447e+65 < re < -4.516533997676771e-174 or -1.2339565047327783e-286 < re < 2.4057311671100194e+87

    1. Initial program 20.5

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

    3. Applied add-sqr-sqrt20.5

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

    4. Applied pow120.5

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

    5. Applied log-pow20.5

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

    6. Applied times-frac20.4

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

    if -4.516533997676771e-174 < re < -1.2339565047327783e-286

    1. Initial program 31.8

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

    3. Applied add-sqr-sqrt31.8

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

    4. Applied pow131.8

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

    5. Applied log-pow31.8

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

    6. Applied times-frac31.7

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

    7. Taylor expanded around 0 36.8

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

    if 2.4057311671100194e+87 < re

    1. Initial program 50.4

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

    3. Applied add-sqr-sqrt50.4

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

    4. Applied pow150.4

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

    5. Applied log-pow50.4

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

    6. Applied times-frac50.4

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

    7. Taylor expanded around inf 9.9

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

    8. Simplified9.9

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

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.96812445344144669 \cdot 10^{65}:\\ \;\;\;\;\frac{-1}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \le -4.5165339976767707 \cdot 10^{-174}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -1.2339565047327783 \cdot 10^{-286}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \le 2.40573116711001939 \cdot 10^{87}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))