Average Error: 0.8 → 0.1
Time: 22.7s
Precision: 64
Internal Precision: 384
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt[3]{\log 10 \cdot \log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt[3]{\sqrt{\log 10}}}\]

Error

Bits error versus re

Bits error versus im

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-cbrt1.5

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

  8. Applied *-un-lft-identity1.5

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

  9. Applied times-frac1.5

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

  10. Applied associate-*r*1.5

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

  11. Applied simplify1.0

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

  13. Applied add-cbrt-cube1.0

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

  14. Applied cbrt-unprod0.1

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

  15. Applied simplify0.1

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

Runtime

Time bar (total: 22.7s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))