Average Error: 0.9 → 0.1
Time: 28.6s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{\left(\left(\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{1}{\sqrt{\sqrt{\log 10}}}}\right) \cdot \sqrt{\frac{1}{\left|\sqrt[3]{\log 10}\right|}}}{\sqrt{\sqrt{\sqrt[3]{\log 10}}} \cdot \left(\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt{\sqrt{\sqrt{\log 10}}}\right)}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{\left(\left(\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{1}{\sqrt{\sqrt{\log 10}}}}\right) \cdot \sqrt{\frac{1}{\left|\sqrt[3]{\log 10}\right|}}}{\sqrt{\sqrt{\sqrt[3]{\log 10}}} \cdot \left(\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt{\sqrt{\sqrt{\log 10}}}\right)}
double f(double re, double im) {
        double r1826582 = im;
        double r1826583 = re;
        double r1826584 = atan2(r1826582, r1826583);
        double r1826585 = 10.0;
        double r1826586 = log(r1826585);
        double r1826587 = r1826584 / r1826586;
        return r1826587;
}
↓
double f(double re, double im) {
        double r1826588 = im;
        double r1826589 = re;
        double r1826590 = atan2(r1826588, r1826589);
        double r1826591 = 1.0;
        double r1826592 = 10.0;
        double r1826593 = log(r1826592);
        double r1826594 = sqrt(r1826593);
        double r1826595 = cbrt(r1826594);
        double r1826596 = r1826595 * r1826595;
        double r1826597 = r1826591 / r1826596;
        double r1826598 = r1826590 * r1826597;
        double r1826599 = sqrt(r1826594);
        double r1826600 = r1826591 / r1826599;
        double r1826601 = sqrt(r1826600);
        double r1826602 = r1826598 * r1826601;
        double r1826603 = cbrt(r1826593);
        double r1826604 = fabs(r1826603);
        double r1826605 = r1826591 / r1826604;
        double r1826606 = sqrt(r1826605);
        double r1826607 = r1826602 * r1826606;
        double r1826608 = sqrt(r1826603);
        double r1826609 = sqrt(r1826608);
        double r1826610 = sqrt(r1826599);
        double r1826611 = r1826595 * r1826610;
        double r1826612 = r1826609 * r1826611;
        double r1826613 = r1826607 / r1826612;
        return r1826613;
}

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. Applied associate-*r*0.8

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

  13. Applied add-cube-cbrt1.4

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

  14. Applied sqrt-prod1.4

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

  15. Applied associate-/r*1.4

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

  16. Applied sqrt-div1.4

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

  17. Applied add-sqr-sqrt1.4

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

  18. Applied sqrt-prod1.5

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

  19. Applied associate-/r*1.5

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

  20. Applied sqrt-div1.5

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

  21. Applied associate-*r/1.5

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

  22. Applied add-cube-cbrt0.1

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

  23. Applied associate-/r*0.1

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

  24. Applied frac-times0.1

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

  25. Applied frac-times0.1

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

  26. Simplified0.1

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

  27. Final simplification0.1

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

Reproduce

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