Average Error: 0.9 → 0.8
Time: 3.9s
Precision: binary64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}}}{1}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}}}{\tan^{-1}_* \frac{im}{re}}}\right)\right)}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}}}{1}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}}}{\tan^{-1}_* \frac{im}{re}}}\right)\right)}
double code(double re, double im) {
	return ((double) (((double) atan2(im, re)) / ((double) log(10.0))));
}
↓
double code(double re, double im) {
	return ((double) (1.0 / ((double) (((double) (((double) cbrt(((double) (((double) log(10.0)) / ((double) atan2(im, re)))))) * ((double) cbrt(((double) (((double) log(10.0)) / ((double) atan2(im, re)))))))) * ((double) (((double) cbrt(((double) (((double) (((double) cbrt(((double) log(10.0)))) * ((double) cbrt(((double) log(10.0)))))) / 1.0)))) * ((double) (((double) cbrt(((double) (((double) cbrt(((double) sqrt(((double) log(10.0)))))) / 1.0)))) * ((double) cbrt(((double) (((double) cbrt(((double) sqrt(((double) log(10.0)))))) / ((double) atan2(im, re))))))))))))));
}

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 clear-num1.0

    \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.8

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

  7. Applied *-un-lft-identity0.8

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

  8. Applied add-cube-cbrt0.9

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

  9. Applied times-frac0.9

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

  10. Applied cbrt-prod0.8

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

  12. Applied *-un-lft-identity0.8

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

  13. Applied add-sqr-sqrt0.8

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

  14. Applied cbrt-prod0.8

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

  15. Applied times-frac0.8

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

  16. Applied cbrt-prod0.8

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

  17. Final simplification0.8

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

Reproduce

herbie shell --seed 2020150 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))