Average Error: 0.8 → 0.2
Time: 35.3s
Precision: 64
Internal Precision: 576
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\left(\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right) \cdot \left(\sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right)\right)\]

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 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 div-inv0.8

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

  8. Applied associate-*r*0.8

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

  10. Applied add-sqr-sqrt0.8

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

  11. Applied associate-*l*0.8

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

  13. Applied add-cube-cbrt0.1

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

  14. Applied associate-*l*0.2

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

  15. Final simplification0.2

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

Runtime

Time bar (total: 35.3s)Debug logProfile

herbie shell --seed 2018217 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))