Average Error: 0.9 → 0.8
Time: 7.5s
Precision: binary64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}{\frac{\sqrt{\log 10}}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}}}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}{\frac{\sqrt{\log 10}}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}}}
double code(double re, double im) {
	return (((double) atan2(im, re)) / ((double) log(10.0)));
}
↓
double code(double re, double im) {
	return ((double) (((((double) cbrt(((double) atan2(im, re)))) / ((double) (((double) cbrt(((double) sqrt(((double) log(10.0)))))) * ((double) cbrt(((double) sqrt(((double) log(10.0))))))))) / (((double) sqrt(((double) log(10.0)))) / ((double) cbrt(((double) atan2(im, re)))))) * (((double) cbrt(((double) atan2(im, re)))) / ((double) cbrt(((double) sqrt(((double) log(10.0)))))))));
}

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 add-cube-cbrt1.5

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

  8. Applied add-cube-cbrt0.8

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

  9. Applied times-frac1.0

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

  10. Applied associate-*r*1.0

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

  11. Simplified0.8

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

  12. Final simplification0.8

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

Reproduce

herbie shell --seed 2020182 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))