Average Error: 0.9 → 0.9
Time: 38.6s
Precision: 64
Internal Precision: 320
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{\frac{\frac{1}{\sqrt{\log 10}}}{\sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}}}}{\sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}}}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.9

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

  3. Applied add-sqr-sqrt0.9

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

  4. Applied *-un-lft-identity0.9

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

  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 clear-num1.0

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

  9. Applied add-cube-cbrt0.9

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

  10. Applied associate-/r*0.9

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

  11. Taylor expanded around 0 31.1

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

  12. Applied simplify0.9

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

Runtime

Time bar (total: 38.6s)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))