Average Error: 31.1 → 17.4
Time: 20.6s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.427484018494741 \cdot 10^{+134}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{elif}\;re \le 1.5824798583418597 \cdot 10^{+66}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\left|\sqrt[3]{im \cdot im + re \cdot re}\right| \cdot \sqrt{\sqrt[3]{im \cdot im + re \cdot re}}\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.427484018494741 \cdot 10^{+134}:\\
\;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\\

\mathbf{elif}\;re \le 1.5824798583418597 \cdot 10^{+66}:\\
\;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\left|\sqrt[3]{im \cdot im + re \cdot re}\right| \cdot \sqrt{\sqrt[3]{im \cdot im + re \cdot re}}\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log re}{\log 10}\\

\end{array}
double f(double re, double im) {
        double r1698811 = re;
        double r1698812 = r1698811 * r1698811;
        double r1698813 = im;
        double r1698814 = r1698813 * r1698813;
        double r1698815 = r1698812 + r1698814;
        double r1698816 = sqrt(r1698815);
        double r1698817 = log(r1698816);
        double r1698818 = 10.0;
        double r1698819 = log(r1698818);
        double r1698820 = r1698817 / r1698819;
        return r1698820;
}
↓
double f(double re, double im) {
        double r1698821 = re;
        double r1698822 = -1.427484018494741e+134;
        bool r1698823 = r1698821 <= r1698822;
        double r1698824 = -1.0;
        double r1698825 = r1698824 / r1698821;
        double r1698826 = log(r1698825);
        double r1698827 = -r1698826;
        double r1698828 = 10.0;
        double r1698829 = log(r1698828);
        double r1698830 = r1698827 / r1698829;
        double r1698831 = 1.5824798583418597e+66;
        bool r1698832 = r1698821 <= r1698831;
        double r1698833 = 1.0;
        double r1698834 = sqrt(r1698829);
        double r1698835 = r1698833 / r1698834;
        double r1698836 = im;
        double r1698837 = r1698836 * r1698836;
        double r1698838 = r1698821 * r1698821;
        double r1698839 = r1698837 + r1698838;
        double r1698840 = cbrt(r1698839);
        double r1698841 = fabs(r1698840);
        double r1698842 = sqrt(r1698840);
        double r1698843 = r1698841 * r1698842;
        double r1698844 = log(r1698843);
        double r1698845 = r1698835 * r1698844;
        double r1698846 = r1698845 * r1698835;
        double r1698847 = log(r1698821);
        double r1698848 = r1698847 / r1698829;
        double r1698849 = r1698832 ? r1698846 : r1698848;
        double r1698850 = r1698823 ? r1698830 : r1698849;
        return r1698850;
}

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 < -1.427484018494741e+134

    1. Initial program 56.7

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

    2. Taylor expanded around -inf 7.8

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

    if -1.427484018494741e+134 < re < 1.5824798583418597e+66

    1. Initial program 21.5

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

    3. Applied add-cube-cbrt21.5

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

    4. Applied sqrt-prod21.5

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

    5. Simplified21.5

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

    7. Applied add-sqr-sqrt21.5

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

    8. Applied pow121.5

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

    9. Applied pow121.5

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

    10. Applied pow-prod-down21.5

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

    11. Applied log-pow21.5

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

    12. Applied times-frac21.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\sqrt{\log 10}}}\]
    13. Using strategy rm

    14. Applied div-inv21.3

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

    if 1.5824798583418597e+66 < re

    1. Initial program 46.3

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

    2. Taylor expanded around inf 10.5

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.427484018494741 \cdot 10^{+134}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{elif}\;re \le 1.5824798583418597 \cdot 10^{+66}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\left|\sqrt[3]{im \cdot im + re \cdot re}\right| \cdot \sqrt{\sqrt[3]{im \cdot im + re \cdot re}}\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))