Average Error: 0.9 → 0.1
Time: 5.6s
Precision: binary64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)\right)
(FPCore (re im) :precision binary64 (/ (atan2 im re) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (*
  (sqrt (/ 1.0 (* (cbrt (sqrt (log 10.0))) (cbrt (sqrt (log 10.0))))))
  (*
   (sqrt (/ 1.0 (cbrt (sqrt (log 10.0)))))
   (* (/ (atan2 im re) (sqrt (log 10.0))) (sqrt (/ 1.0 (sqrt (log 10.0))))))))
double code(double re, double im) {
	return atan2(im, re) / log(10.0);
}
↓
double code(double re, double im) {
	return sqrt(1.0 / (cbrt(sqrt(log(10.0))) * cbrt(sqrt(log(10.0))))) * (sqrt(1.0 / cbrt(sqrt(log(10.0)))) * ((atan2(im, re) / sqrt(log(10.0))) * sqrt(1.0 / 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 0.9

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

  3. Applied add-sqr-sqrt_binary64_1000.9

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

  4. Applied *-un-lft-identity_binary64_780.9

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

  5. Applied times-frac_binary64_840.8

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

  7. Applied add-sqr-sqrt_binary64_1000.8

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

  8. Applied associate-*l*_binary64_190.9

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

  9. Simplified0.9

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

  11. Applied add-cube-cbrt_binary64_1130.1

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

  12. Applied *-un-lft-identity_binary64_780.1

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

  13. Applied times-frac_binary64_840.1

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

  14. Applied sqrt-prod_binary64_940.1

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

  15. Applied associate-*l*_binary64_190.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2021060 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))