Average Error: 30.5 → 0.4
Time: 35.8s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\left(\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot 3\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\]
double f(double re, double im) {
        double r1435691 = re;
        double r1435692 = r1435691 * r1435691;
        double r1435693 = im;
        double r1435694 = r1435693 * r1435693;
        double r1435695 = r1435692 + r1435694;
        double r1435696 = sqrt(r1435695);
        double r1435697 = log(r1435696);
        double r1435698 = 10.0;
        double r1435699 = log(r1435698);
        double r1435700 = r1435697 / r1435699;
        return r1435700;
}
↓
double f(double re, double im) {
        double r1435701 = re;
        double r1435702 = im;
        double r1435703 = hypot(r1435701, r1435702);
        double r1435704 = cbrt(r1435703);
        double r1435705 = log(r1435704);
        double r1435706 = 1.0;
        double r1435707 = 10.0;
        double r1435708 = log(r1435707);
        double r1435709 = sqrt(r1435708);
        double r1435710 = r1435706 / r1435709;
        double r1435711 = 3.0;
        double r1435712 = r1435710 * r1435711;
        double r1435713 = r1435705 * r1435712;
        double r1435714 = r1435713 * r1435710;
        return r1435714;
}


\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\left(\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot 3\right)\right) \cdot \frac{1}{\sqrt{\log 10}}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 30.5

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

  2. Simplified0.6

    \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log 10}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.6

    \[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}}{\log 10}\]
  5. Using strategy rm

  6. Applied pow10.6

    \[\leadsto \frac{\log \left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{\log \color{blue}{\left({10}^{1}\right)}}\]

  7. Applied log-pow0.6

    \[\leadsto \frac{\log \left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{\color{blue}{1 \cdot \log 10}}\]

  8. Applied pow10.6

    \[\leadsto \frac{\log \left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}^{1}}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{1 \cdot \log 10}\]

  9. Applied pow10.6

    \[\leadsto \frac{\log \left(\left(\color{blue}{{\left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}^{1}} \cdot {\left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}^{1}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{1 \cdot \log 10}\]

  10. Applied pow-sqr0.6

    \[\leadsto \frac{\log \left(\color{blue}{{\left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}^{\left(2 \cdot 1\right)}} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{1 \cdot \log 10}\]

  11. Applied pow-plus0.6

    \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}^{\left(2 \cdot 1 + 1\right)}\right)}}{1 \cdot \log 10}\]

  12. Applied log-pow0.6

    \[\leadsto \frac{\color{blue}{\left(2 \cdot 1 + 1\right) \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}}{1 \cdot \log 10}\]

  13. Applied times-frac0.6

    \[\leadsto \color{blue}{\frac{2 \cdot 1 + 1}{1} \cdot \frac{\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{\log 10}}\]

  14. Simplified0.6

    \[\leadsto \color{blue}{3} \cdot \frac{\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{\log 10}\]
  15. Using strategy rm

  16. Applied add-sqr-sqrt0.6

    \[\leadsto 3 \cdot \frac{\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

  17. Applied *-un-lft-identity0.6

    \[\leadsto 3 \cdot \frac{\color{blue}{1 \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

  18. Applied times-frac0.6

    \[\leadsto 3 \cdot \color{blue}{\left(\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{\sqrt{\log 10}}\right)}\]

  19. Applied associate-*r*0.5

    \[\leadsto \color{blue}{\left(3 \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \frac{\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{\sqrt{\log 10}}}\]
  20. Using strategy rm

  21. Applied div-inv0.4

    \[\leadsto \left(3 \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)}\]

  22. Applied associate-*r*0.4

    \[\leadsto \color{blue}{\left(\left(3 \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)\right) \cdot \frac{1}{\sqrt{\log 10}}}\]

  23. Final simplification0.4

    \[\leadsto \left(\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot 3\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))