Average Error: 0.8 → 0.7
Time: 3.5s
Precision: binary64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}} \cdot \frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}}}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}}\right)}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}} \cdot \frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}}}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}}\right)}
(FPCore (re im) :precision binary64 (/ (atan2 im re) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (*
  (/
   1.0
   (* (cbrt (/ (log 10.0) (atan2 im re))) (cbrt (/ (log 10.0) (atan2 im re)))))
  (/
   1.0
   (*
    (cbrt (sqrt (log 10.0)))
    (*
     (cbrt
      (/
       (* (cbrt (sqrt (log 10.0))) (cbrt (sqrt (log 10.0))))
       (* (cbrt (atan2 im re)) (cbrt (atan2 im re)))))
     (cbrt (/ (cbrt (sqrt (log 10.0))) (cbrt (atan2 im re)))))))))
double code(double re, double im) {
	return atan2(im, re) / log(10.0);
}
↓
double code(double re, double im) {
	return (1.0 / (cbrt(log(10.0) / atan2(im, re)) * cbrt(log(10.0) / atan2(im, re)))) * (1.0 / (cbrt(sqrt(log(10.0))) * (cbrt((cbrt(sqrt(log(10.0))) * cbrt(sqrt(log(10.0)))) / (cbrt(atan2(im, re)) * cbrt(atan2(im, re)))) * cbrt(cbrt(sqrt(log(10.0))) / cbrt(atan2(im, re))))));
}

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 clear-num_binary64_7591.0

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

  5. Applied add-cube-cbrt_binary64_7950.8

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

  6. Applied add-sqr-sqrt_binary64_7820.8

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

  7. Applied times-frac_binary64_7660.7

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

  8. Simplified0.7

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

  9. Simplified0.7

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

  11. Applied *-un-lft-identity_binary64_7600.7

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

  12. Applied add-sqr-sqrt_binary64_7820.7

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

  13. Applied times-frac_binary64_7660.7

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

  14. Applied cbrt-prod_binary64_7910.8

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

  15. Simplified0.8

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

  17. Applied add-cube-cbrt_binary64_7950.7

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

  18. Applied add-cube-cbrt_binary64_7950.8

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

  19. Applied times-frac_binary64_7660.8

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

  20. Applied cbrt-prod_binary64_7910.7

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

  21. Final simplification0.7

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

Reproduce

herbie shell --seed 2020356 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))