Average Error: 0.9 → 0.1
Time: 13.8s
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 r481922 = im;
        double r481923 = re;
        double r481924 = atan2(r481922, r481923);
        double r481925 = 10.0;
        double r481926 = log(r481925);
        double r481927 = r481924 / r481926;
        return r481927;
}
↓
double f(double re, double im) {
        double r481928 = 1.0;
        double r481929 = 10.0;
        double r481930 = log(r481929);
        double r481931 = sqrt(r481930);
        double r481932 = cbrt(r481931);
        double r481933 = r481928 / r481932;
        double r481934 = fabs(r481933);
        double r481935 = im;
        double r481936 = re;
        double r481937 = atan2(r481935, r481936);
        double r481938 = r481937 / r481931;
        double r481939 = r481934 * r481938;
        double r481940 = sqrt(r481933);
        double r481941 = r481939 * r481940;
        double r481942 = r481928 / r481931;
        double r481943 = sqrt(r481942);
        double r481944 = r481941 * r481943;
        return r481944;
}

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. 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 2019152 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))