Average Error: 31.0 → 0.5
Time: 20.9s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)} \cdot \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)\right)\right)\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)} \cdot \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)\right)\right)\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}
double f(double re, double im) {
        double r826537 = re;
        double r826538 = r826537 * r826537;
        double r826539 = im;
        double r826540 = r826539 * r826539;
        double r826541 = r826538 + r826540;
        double r826542 = sqrt(r826541);
        double r826543 = log(r826542);
        double r826544 = 10.0;
        double r826545 = log(r826544);
        double r826546 = r826543 / r826545;
        return r826546;
}
↓
double f(double re, double im) {
        double r826547 = 1.0;
        double r826548 = 10.0;
        double r826549 = log(r826548);
        double r826550 = sqrt(r826549);
        double r826551 = r826547 / r826550;
        double r826552 = sqrt(r826551);
        double r826553 = re;
        double r826554 = im;
        double r826555 = hypot(r826553, r826554);
        double r826556 = cbrt(r826555);
        double r826557 = r826556 * r826556;
        double r826558 = r826556 * r826557;
        double r826559 = log(r826558);
        double r826560 = r826551 * r826559;
        double r826561 = r826552 * r826560;
        double r826562 = r826561 * r826552;
        return r826562;
}

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 31.0

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

  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 *-un-lft-identity0.6

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

  6. Applied times-frac0.6

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

  8. Applied div-inv0.4

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

  10. Applied add-sqr-sqrt0.4

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

  11. Applied associate-*l*0.5

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

  13. Applied add-cube-cbrt0.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))