Average Error: 0.8 → 0.2
Time: 15.9s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt[3]{\sqrt{\log 10}}} \cdot \frac{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt[3]{\sqrt{\sqrt{\log 10}}} \cdot \sqrt[3]{\sqrt{\sqrt{\log 10}}}}\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt[3]{\sqrt{\log 10}}} \cdot \frac{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt[3]{\sqrt{\sqrt{\log 10}}} \cdot \sqrt[3]{\sqrt{\sqrt{\log 10}}}}\right)
double f(double re, double im) {
        double r1621687 = im;
        double r1621688 = re;
        double r1621689 = atan2(r1621687, r1621688);
        double r1621690 = 10.0;
        double r1621691 = log(r1621690);
        double r1621692 = r1621689 / r1621691;
        return r1621692;
}
↓
double f(double re, double im) {
        double r1621693 = 1.0;
        double r1621694 = 10.0;
        double r1621695 = log(r1621694);
        double r1621696 = sqrt(r1621695);
        double r1621697 = r1621693 / r1621696;
        double r1621698 = cbrt(r1621696);
        double r1621699 = r1621693 / r1621698;
        double r1621700 = im;
        double r1621701 = re;
        double r1621702 = atan2(r1621700, r1621701);
        double r1621703 = r1621702 / r1621698;
        double r1621704 = sqrt(r1621696);
        double r1621705 = cbrt(r1621704);
        double r1621706 = r1621705 * r1621705;
        double r1621707 = r1621703 / r1621706;
        double r1621708 = r1621699 * r1621707;
        double r1621709 = r1621697 * r1621708;
        return r1621709;
}

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. Using strategy rm

  9. Applied add-cube-cbrt1.5

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

  10. Applied *-un-lft-identity1.5

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

  11. Applied times-frac1.5

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

  12. Applied associate-*r*1.5

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

  13. Simplified1.5

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

  15. Applied add-sqr-sqrt1.5

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

  16. Applied sqrt-prod0.2

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

  17. Applied cbrt-prod0.2

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

  18. Final simplification0.2

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10.0)))