Average Error: 0.8 → 0.2
Time: 3.1s
Precision: binary64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot {\left(\sqrt{\log 10}\right)}^{\frac{1}{3}}} \cdot \frac{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}{\sqrt[3]{\sqrt{\log 10}}}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot {\left(\sqrt{\log 10}\right)}^{\frac{1}{3}}} \cdot \frac{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}{\sqrt[3]{\sqrt{\log 10}}}
double code(double re, double im) {
	return ((double) (((double) atan2(im, re)) / ((double) log(10.0))));
}
↓
double code(double re, double im) {
	return ((double) (((double) (1.0 / ((double) (((double) cbrt(((double) sqrt(((double) log(10.0)))))) * ((double) pow(((double) sqrt(((double) log(10.0)))), 0.3333333333333333)))))) * ((double) (((double) (((double) atan2(im, re)) / ((double) sqrt(((double) log(10.0)))))) / ((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 0.8

    \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
  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 add-cube-cbrt1.0

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

  8. Applied *-un-lft-identity1.0

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

  9. Applied times-frac1.0

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

  10. Applied associate-*l*1.0

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

  11. Simplified1.0

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

  13. Applied pow1/30.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2020148 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))