Average Error: 0.8 → 0.1
Time: 7.8s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{\left(\left|\sqrt[3]{1}\right| \cdot \sqrt{\frac{1}{\sqrt{\sqrt{\log 10}}}}\right) \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}{\sqrt{\frac{\sqrt{\log 10}}{\sqrt[3]{1}}} \cdot \left(\sqrt{\sqrt{\sqrt{\log 10}}} \cdot \sqrt[3]{\sqrt{\log 10}}\right)}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{\left(\left|\sqrt[3]{1}\right| \cdot \sqrt{\frac{1}{\sqrt{\sqrt{\log 10}}}}\right) \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}{\sqrt{\frac{\sqrt{\log 10}}{\sqrt[3]{1}}} \cdot \left(\sqrt{\sqrt{\sqrt{\log 10}}} \cdot \sqrt[3]{\sqrt{\log 10}}\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) (((double) (((double) (((double) fabs(((double) cbrt(1.0)))) * ((double) sqrt(((double) (1.0 / ((double) sqrt(((double) sqrt(((double) log(10.0)))))))))))) * ((double) (((double) atan2(im, re)) / ((double) (((double) cbrt(((double) sqrt(((double) log(10.0)))))) * ((double) cbrt(((double) sqrt(((double) log(10.0)))))))))))) / ((double) (((double) sqrt(((double) (((double) sqrt(((double) log(10.0)))) / ((double) cbrt(1.0)))))) * ((double) (((double) sqrt(((double) sqrt(((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-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-cbrt1.5

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

  11. Applied associate-/r*1.5

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

  12. Applied add-sqr-sqrt1.5

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

  13. Applied sqrt-prod1.0

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

  14. Applied associate-/r*1.0

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

  15. Applied sqrt-div1.0

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

  16. Applied frac-times0.9

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

  17. Applied add-cube-cbrt0.9

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

  18. Applied associate-/l*0.9

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

  19. Applied sqrt-div0.1

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

  20. Applied frac-times0.1

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

  21. Simplified0.1

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

  22. Final simplification0.1

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

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))