Average Error: 0.9 → 0.2
Time: 17.3s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\left(\left|\frac{1}{\sqrt[3]{\sqrt{\log 10}}}\right| \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot \tan^{-1}_* \frac{im}{re}\right)\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\left(\left|\frac{1}{\sqrt[3]{\sqrt{\log 10}}}\right| \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot \tan^{-1}_* \frac{im}{re}\right)\right)
double f(double re, double im) {
        double r1494712 = im;
        double r1494713 = re;
        double r1494714 = atan2(r1494712, r1494713);
        double r1494715 = 10.0;
        double r1494716 = log(r1494715);
        double r1494717 = r1494714 / r1494716;
        return r1494717;
}
↓
double f(double re, double im) {
        double r1494718 = 1.0;
        double r1494719 = 10.0;
        double r1494720 = log(r1494719);
        double r1494721 = sqrt(r1494720);
        double r1494722 = r1494718 / r1494721;
        double r1494723 = cbrt(r1494721);
        double r1494724 = r1494718 / r1494723;
        double r1494725 = sqrt(r1494724);
        double r1494726 = fabs(r1494724);
        double r1494727 = sqrt(r1494722);
        double r1494728 = r1494726 * r1494727;
        double r1494729 = im;
        double r1494730 = re;
        double r1494731 = atan2(r1494729, r1494730);
        double r1494732 = r1494728 * r1494731;
        double r1494733 = r1494725 * r1494732;
        double r1494734 = r1494722 * r1494733;
        return r1494734;
}

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 0.9

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

  3. Applied add-sqr-sqrt0.9

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

  4. Applied *-un-lft-identity0.9

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

  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 div-inv0.8

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

  9. Applied add-sqr-sqrt0.8

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

  10. Applied associate-*r*0.8

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

  12. Applied add-cube-cbrt0.1

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

  13. Applied *-un-lft-identity0.1

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

  14. Applied times-frac0.1

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

  15. Applied sqrt-prod0.1

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

  16. Applied associate-*r*1.3

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

  17. Simplified0.2

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

  18. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))