Average Error: 0.8 → 0.4
Time: 23.6s
Precision: 64
Internal Precision: 320
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{\frac{1}{\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}}}{\sqrt[3]{\sqrt{\log 10}} \cdot \left(\sqrt[3]{\frac{\left|\sqrt[3]{\log 10}\right|}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}}} \cdot \sqrt[3]{\frac{\sqrt{\sqrt[3]{\log 10}}}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}}\right)}\]

Error

Bits error versus re

Bits error versus im

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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-num1.0

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

  5. Applied add-cube-cbrt0.8

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

  6. Applied associate-/r*0.7

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

  8. Applied *-un-lft-identity0.7

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

  9. Applied add-sqr-sqrt0.7

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

  10. Applied times-frac0.7

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

  11. Applied cbrt-prod0.7

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

  12. Applied simplify0.7

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

  14. Applied add-cube-cbrt0.6

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

  15. Applied add-cube-cbrt0.6

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

  16. Applied sqrt-prod0.6

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

  17. Applied times-frac0.6

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

  18. Applied cbrt-prod0.4

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

  19. Applied simplify0.4

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

Runtime

Time bar (total: 23.6s)Debug logProfile

herbie shell --seed 2018193 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))