Average Error: 0.8 → 0.7
Time: 15.2s
Precision: 64
Internal Precision: 384
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{\frac{1}{\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \left(\sqrt[3]{\frac{\sqrt{\log 10}}{1}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}}}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}}\right)\right)}}{\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}}\]

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 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

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

  9. Applied add-sqr-sqrt0.7

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

  10. Applied times-frac0.7

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

  11. Applied cbrt-prod0.8

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

  13. Applied add-cube-cbrt0.7

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

  14. Applied add-cube-cbrt0.8

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

  15. Applied times-frac0.8

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

  16. Applied cbrt-prod0.7

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

Runtime

Time bar (total: 15.2s)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))