Average Error: 0.8 → 0.1
Time: 3.1s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\sqrt{\frac{\sqrt{1}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\sqrt{\frac{\sqrt{1}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)\right)
double code(double re, double im) {
	return (atan2(im, re) / log(10.0));
}
↓
double code(double re, double im) {
	return (sqrt((sqrt(1.0) / (cbrt(sqrt(log(10.0))) * cbrt(sqrt(log(10.0)))))) * (sqrt((1.0 / cbrt(sqrt(log(10.0))))) * (sqrt((1.0 / sqrt(log(10.0)))) * (atan2(im, re) / sqrt(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-sqr-sqrt0.8

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

  8. Applied associate-*l*0.9

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

  10. Applied add-cube-cbrt0.1

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

  11. Applied add-sqr-sqrt0.1

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

  12. Applied times-frac0.1

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

  13. Applied sqrt-prod0.1

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

  14. Applied associate-*l*0.1

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

  15. Simplified0.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2020079 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))