Average Error: 31.1 → 17.4
Time: 23.1s
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 r1273885 = re;
        double r1273886 = r1273885 * r1273885;
        double r1273887 = im;
        double r1273888 = r1273887 * r1273887;
        double r1273889 = r1273886 + r1273888;
        double r1273890 = sqrt(r1273889);
        double r1273891 = log(r1273890);
        double r1273892 = 10.0;
        double r1273893 = log(r1273892);
        double r1273894 = r1273891 / r1273893;
        return r1273894;
}
↓
double f(double re, double im) {
        double r1273895 = re;
        double r1273896 = -1.427484018494741e+134;
        bool r1273897 = r1273895 <= r1273896;
        double r1273898 = -1.0;
        double r1273899 = r1273898 / r1273895;
        double r1273900 = log(r1273899);
        double r1273901 = -r1273900;
        double r1273902 = 10.0;
        double r1273903 = log(r1273902);
        double r1273904 = r1273901 / r1273903;
        double r1273905 = 1.5824798583418597e+66;
        bool r1273906 = r1273895 <= r1273905;
        double r1273907 = 1.0;
        double r1273908 = sqrt(r1273903);
        double r1273909 = r1273907 / r1273908;
        double r1273910 = im;
        double r1273911 = r1273910 * r1273910;
        double r1273912 = r1273895 * r1273895;
        double r1273913 = r1273911 + r1273912;
        double r1273914 = cbrt(r1273913);
        double r1273915 = fabs(r1273914);
        double r1273916 = sqrt(r1273914);
        double r1273917 = r1273915 * r1273916;
        double r1273918 = log(r1273917);
        double r1273919 = r1273909 * r1273918;
        double r1273920 = r1273919 * r1273909;
        double r1273921 = log(r1273895);
        double r1273922 = r1273921 / r1273903;
        double r1273923 = r1273906 ? r1273920 : r1273922;
        double r1273924 = r1273897 ? r1273904 : r1273923;
        return r1273924;
}

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)))