Average Error: 0.9 → 0.8
Time: 12.6s
Precision: 64
Internal Precision: 128
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \left(\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \sqrt[3]{\frac{\sqrt[3]{\log 10}}{\tan^{-1}_* \frac{im}{re}}}\right)}\]

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 *-un-lft-identity0.9

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

  4. Applied associate-/l*1.0

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

  6. Applied add-cube-cbrt0.8

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

  8. Applied *-un-lft-identity0.8

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

  9. Applied add-sqr-sqrt0.8

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

  10. Applied times-frac0.8

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

  11. Applied cbrt-prod0.8

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

  12. Simplified0.8

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

  14. Applied *-un-lft-identity0.8

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

  15. Applied add-cube-cbrt1.1

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

  16. Applied times-frac1.1

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

  17. Applied cbrt-prod0.8

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

  18. Simplified0.8

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

  19. Final simplification0.8

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

Runtime

Time bar (total: 12.6s)Debug logProfile

herbie shell --seed 2018278 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))