Average Error: 0.8 → 0.6
Time: 16.0s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\left(\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10}}\right) \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt{\sqrt[3]{\sqrt{\log 10}}} \cdot \sqrt{\sqrt[3]{\sqrt{\log 10}}}}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\left(\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10}}\right) \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt{\sqrt[3]{\sqrt{\log 10}}} \cdot \sqrt{\sqrt[3]{\sqrt{\log 10}}}}
double f(double re, double im) {
        double r22483 = im;
        double r22484 = re;
        double r22485 = atan2(r22483, r22484);
        double r22486 = 10.0;
        double r22487 = log(r22486);
        double r22488 = r22485 / r22487;
        return r22488;
}
↓
double f(double re, double im) {
        double r22489 = im;
        double r22490 = re;
        double r22491 = atan2(r22489, r22490);
        double r22492 = cbrt(r22491);
        double r22493 = 10.0;
        double r22494 = log(r22493);
        double r22495 = sqrt(r22494);
        double r22496 = cbrt(r22495);
        double r22497 = r22496 * r22496;
        double r22498 = r22492 / r22497;
        double r22499 = r22492 / r22495;
        double r22500 = r22498 * r22499;
        double r22501 = sqrt(r22496);
        double r22502 = r22501 * r22501;
        double r22503 = r22492 / r22502;
        double r22504 = r22500 * r22503;
        return r22504;
}

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-cube-cbrt1.5

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

  8. Applied add-cube-cbrt0.8

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

  9. Applied times-frac1.0

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

  10. Applied associate-*r*1.0

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

  11. Simplified0.8

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

  13. Applied add-sqr-sqrt0.6

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

  14. Final simplification0.6

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))