Average Error: 0.8 → 0.2
Time: 23.3s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\left(\tan^{-1}_* \frac{im}{re} \cdot \left|\frac{1}{\sqrt[3]{\sqrt{\log 10}}}\right|\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\left(\tan^{-1}_* \frac{im}{re} \cdot \left|\frac{1}{\sqrt[3]{\sqrt{\log 10}}}\right|\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)\right)
double f(double re, double im) {
        double r1692889 = im;
        double r1692890 = re;
        double r1692891 = atan2(r1692889, r1692890);
        double r1692892 = 10.0;
        double r1692893 = log(r1692892);
        double r1692894 = r1692891 / r1692893;
        return r1692894;
}
↓
double f(double re, double im) {
        double r1692895 = 1.0;
        double r1692896 = 10.0;
        double r1692897 = log(r1692896);
        double r1692898 = sqrt(r1692897);
        double r1692899 = r1692895 / r1692898;
        double r1692900 = cbrt(r1692898);
        double r1692901 = r1692895 / r1692900;
        double r1692902 = sqrt(r1692901);
        double r1692903 = im;
        double r1692904 = re;
        double r1692905 = atan2(r1692903, r1692904);
        double r1692906 = fabs(r1692901);
        double r1692907 = r1692905 * r1692906;
        double r1692908 = sqrt(r1692899);
        double r1692909 = r1692907 * r1692908;
        double r1692910 = r1692902 * r1692909;
        double r1692911 = r1692899 * r1692910;
        return r1692911;
}

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-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. Using strategy rm

  12. Applied add-cube-cbrt0.1

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

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

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

  14. Applied times-frac0.1

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

  15. Applied sqrt-prod0.1

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

  16. Applied associate-*r*1.3

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

  17. Simplified0.2

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

  18. Final simplification0.2

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

Reproduce

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