Average Error: 0.8 → 0.8
Time: 18.2s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \sqrt[3]{\frac{\sqrt[3]{\log 10}}{\tan^{-1}_* \frac{im}{re}}}\right)}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \sqrt[3]{\frac{\sqrt[3]{\log 10}}{\tan^{-1}_* \frac{im}{re}}}\right)}
double f(double re, double im) {
        double r32716 = im;
        double r32717 = re;
        double r32718 = atan2(r32716, r32717);
        double r32719 = 10.0;
        double r32720 = log(r32719);
        double r32721 = r32718 / r32720;
        return r32721;
}
↓
double f(double re, double im) {
        double r32722 = 1.0;
        double r32723 = 10.0;
        double r32724 = log(r32723);
        double r32725 = im;
        double r32726 = re;
        double r32727 = atan2(r32725, r32726);
        double r32728 = r32724 / r32727;
        double r32729 = cbrt(r32728);
        double r32730 = r32729 * r32729;
        double r32731 = cbrt(r32724);
        double r32732 = r32731 * r32731;
        double r32733 = cbrt(r32732);
        double r32734 = r32731 / r32727;
        double r32735 = cbrt(r32734);
        double r32736 = r32733 * r32735;
        double r32737 = r32730 * r32736;
        double r32738 = r32722 / r32737;
        return r32738;
}

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 pow10.8

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

  4. Applied log-pow0.8

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

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

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

  6. Applied times-frac0.8

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

  7. Simplified0.8

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

  9. Applied clear-num1.0

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

  11. Applied add-cube-cbrt0.8

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

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

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

  14. Applied add-cube-cbrt0.9

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

  15. Applied times-frac0.9

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

  16. Applied cbrt-prod0.8

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

  17. Simplified0.8

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

  18. Final simplification0.8

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))