Average Error: 0.8 → 0.1
Time: 19.6s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\left(\left(\left|\frac{1}{\sqrt[3]{\sqrt{\log 10}}}\right| \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}}\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\left(\left(\left|\frac{1}{\sqrt[3]{\sqrt{\log 10}}}\right| \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}}\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}
double f(double re, double im) {
        double r1235527 = im;
        double r1235528 = re;
        double r1235529 = atan2(r1235527, r1235528);
        double r1235530 = 10.0;
        double r1235531 = log(r1235530);
        double r1235532 = r1235529 / r1235531;
        return r1235532;
}
↓
double f(double re, double im) {
        double r1235533 = 1.0;
        double r1235534 = 10.0;
        double r1235535 = log(r1235534);
        double r1235536 = sqrt(r1235535);
        double r1235537 = cbrt(r1235536);
        double r1235538 = r1235533 / r1235537;
        double r1235539 = fabs(r1235538);
        double r1235540 = im;
        double r1235541 = re;
        double r1235542 = atan2(r1235540, r1235541);
        double r1235543 = r1235542 / r1235536;
        double r1235544 = r1235539 * r1235543;
        double r1235545 = sqrt(r1235538);
        double r1235546 = r1235544 * r1235545;
        double r1235547 = r1235533 / r1235536;
        double r1235548 = sqrt(r1235547);
        double r1235549 = r1235546 * r1235548;
        return r1235549;
}

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 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. Applied associate-*r*0.8

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

  10. Applied add-sqr-sqrt0.8

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

  11. Applied associate-*r*0.8

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

  13. Applied add-cube-cbrt0.1

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

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

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

  15. Applied times-frac0.1

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

  16. Applied sqrt-prod0.1

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

  17. Applied associate-*r*0.2

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

  18. Simplified0.1

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

  19. Final simplification0.1

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

Reproduce

herbie shell --seed 2019170 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10.0)))