Average Error: 0.8 → 0.7
Time: 33.5s
Precision: 64
Internal Precision: 320
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{\left(\frac{\sqrt[3]{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}\]

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 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-cube-cbrt1.1

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

  8. Applied associate-*r*1.1

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

  9. Applied simplify1.1

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

  11. Applied cbrt-div1.0

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

  12. Applied cbrt-div1.0

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

  13. Applied associate-*r/1.0

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

  14. Applied frac-times0.7

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

Runtime

Time bar (total: 33.5s)Debug logProfile

herbie shell --seed 2018296 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))