Average Error: 31.1 → 0.5
Time: 18.4s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\sqrt{\sqrt[3]{\frac{1}{\sqrt{\log 10}}}} \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \left(\left|\sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right| \cdot \frac{1}{\sqrt{\log 10}}\right)\right)\right)\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\sqrt{\sqrt[3]{\frac{1}{\sqrt{\log 10}}}} \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \left(\left|\sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right| \cdot \frac{1}{\sqrt{\log 10}}\right)\right)\right)
double f(double re, double im) {
        double r1196832 = re;
        double r1196833 = r1196832 * r1196832;
        double r1196834 = im;
        double r1196835 = r1196834 * r1196834;
        double r1196836 = r1196833 + r1196835;
        double r1196837 = sqrt(r1196836);
        double r1196838 = log(r1196837);
        double r1196839 = 10.0;
        double r1196840 = log(r1196839);
        double r1196841 = r1196838 / r1196840;
        return r1196841;
}
↓
double f(double re, double im) {
        double r1196842 = 1.0;
        double r1196843 = 10.0;
        double r1196844 = log(r1196843);
        double r1196845 = sqrt(r1196844);
        double r1196846 = r1196842 / r1196845;
        double r1196847 = sqrt(r1196846);
        double r1196848 = cbrt(r1196846);
        double r1196849 = sqrt(r1196848);
        double r1196850 = re;
        double r1196851 = im;
        double r1196852 = hypot(r1196850, r1196851);
        double r1196853 = log(r1196852);
        double r1196854 = fabs(r1196848);
        double r1196855 = r1196854 * r1196846;
        double r1196856 = r1196853 * r1196855;
        double r1196857 = r1196849 * r1196856;
        double r1196858 = r1196847 * r1196857;
        return r1196858;
}

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. Initial program 31.1

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

  2. Simplified0.6

    \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.6

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

  5. Applied pow10.6

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

  6. Applied log-pow0.6

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

  7. Applied times-frac0.6

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}}\]
  8. Using strategy rm

  9. Applied div-inv0.4

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

  10. Applied associate-*r*0.4

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

  12. Applied add-sqr-sqrt0.4

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

  13. Applied associate-*r*0.5

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

  15. Applied add-cube-cbrt0.5

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

  16. Applied sqrt-prod1.0

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

  17. Applied associate-*r*0.8

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

  18. Simplified0.5

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

  19. Final simplification0.5

    \[\leadsto \sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\sqrt{\sqrt[3]{\frac{1}{\sqrt{\log 10}}}} \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \left(\left|\sqrt[3]{\frac{1}{\sqrt{\log 10}}}\right| \cdot \frac{1}{\sqrt{\log 10}}\right)\right)\right)\]

Reproduce

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