Average Error: 0.8 → 0.8
Time: 3.5s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt{\log 10}}{1}} \cdot \frac{\sqrt[3]{\sqrt{\log 10}}}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}\right)}\]
\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]{\frac{\sqrt{\log 10}}{1}} \cdot \frac{\sqrt[3]{\sqrt{\log 10}}}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}\right)}
double f(double re, double im) {
        double r28789 = im;
        double r28790 = re;
        double r28791 = atan2(r28789, r28790);
        double r28792 = 10.0;
        double r28793 = log(r28792);
        double r28794 = r28791 / r28793;
        return r28794;
}
↓
double f(double re, double im) {
        double r28795 = 1.0;
        double r28796 = 10.0;
        double r28797 = log(r28796);
        double r28798 = im;
        double r28799 = re;
        double r28800 = atan2(r28798, r28799);
        double r28801 = r28797 / r28800;
        double r28802 = cbrt(r28801);
        double r28803 = r28802 * r28802;
        double r28804 = sqrt(r28797);
        double r28805 = r28804 / r28795;
        double r28806 = cbrt(r28805);
        double r28807 = cbrt(r28804);
        double r28808 = cbrt(r28800);
        double r28809 = r28807 / r28808;
        double r28810 = r28806 * r28809;
        double r28811 = r28803 * r28810;
        double r28812 = r28795 / r28811;
        return r28812;
}

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 clear-num1.0

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

  5. Applied add-cube-cbrt0.8

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

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

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

  8. Applied add-sqr-sqrt0.8

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

  9. Applied times-frac0.8

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

  10. Applied cbrt-prod0.8

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

  12. Applied cbrt-div0.8

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

  13. Final simplification0.8

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

Reproduce

herbie shell --seed 2019356 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))