Average Error: 0.8 → 0.1
Time: 3.4s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\sqrt{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\sqrt{\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\sqrt{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\sqrt{\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)\right)
double f(double re, double im) {
        double r77834 = im;
        double r77835 = re;
        double r77836 = atan2(r77834, r77835);
        double r77837 = 10.0;
        double r77838 = log(r77837);
        double r77839 = r77836 / r77838;
        return r77839;
}
↓
double f(double re, double im) {
        double r77840 = 1.0;
        double r77841 = cbrt(r77840);
        double r77842 = r77841 * r77841;
        double r77843 = 10.0;
        double r77844 = log(r77843);
        double r77845 = sqrt(r77844);
        double r77846 = cbrt(r77845);
        double r77847 = r77846 * r77846;
        double r77848 = r77842 / r77847;
        double r77849 = sqrt(r77848);
        double r77850 = r77841 / r77846;
        double r77851 = sqrt(r77850);
        double r77852 = r77840 / r77845;
        double r77853 = sqrt(r77852);
        double r77854 = im;
        double r77855 = re;
        double r77856 = atan2(r77854, r77855);
        double r77857 = r77856 / r77845;
        double r77858 = r77853 * r77857;
        double r77859 = r77851 * r77858;
        double r77860 = r77849 * r77859;
        return r77860;
}

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 0.8

    \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.8

    \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

  4. Applied *-un-lft-identity0.8

    \[\leadsto \frac{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

  5. Applied times-frac0.8

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.8

    \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\]

  8. Applied associate-*l*0.9

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt0.1

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

  11. Applied add-cube-cbrt0.1

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

  12. Applied times-frac0.1

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

  13. Applied sqrt-prod0.1

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

  14. Applied associate-*l*0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))