Average Error: 0.8 → 0.8
Time: 3.3s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \left(\sqrt[3]{\frac{\sqrt{\log 10}}{1}} \cdot \frac{\sqrt[3]{\sqrt{\log 10}}}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}\right)\right) \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \left(\sqrt[3]{\frac{\sqrt{\log 10}}{1}} \cdot \frac{\sqrt[3]{\sqrt{\log 10}}}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}\right)\right) \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}}
double f(double re, double im) {
        double r31425 = im;
        double r31426 = re;
        double r31427 = atan2(r31425, r31426);
        double r31428 = 10.0;
        double r31429 = log(r31428);
        double r31430 = r31427 / r31429;
        return r31430;
}
↓
double f(double re, double im) {
        double r31431 = 1.0;
        double r31432 = 10.0;
        double r31433 = log(r31432);
        double r31434 = im;
        double r31435 = re;
        double r31436 = atan2(r31434, r31435);
        double r31437 = r31433 / r31436;
        double r31438 = cbrt(r31437);
        double r31439 = sqrt(r31433);
        double r31440 = r31439 / r31431;
        double r31441 = cbrt(r31440);
        double r31442 = cbrt(r31439);
        double r31443 = cbrt(r31436);
        double r31444 = r31442 / r31443;
        double r31445 = r31441 * r31444;
        double r31446 = r31438 * r31445;
        double r31447 = r31446 * r31438;
        double r31448 = r31431 / r31447;
        return r31448;
}

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

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

  8. Applied add-sqr-sqrt0.8

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

  9. Applied times-frac0.8

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

  10. Applied cbrt-prod0.8

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

  12. Applied cbrt-div0.8

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

  13. Final simplification0.8

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

Reproduce

herbie shell --seed 2020003 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))