Average Error: 0.9 → 0.1
Time: 2.1m
Precision: 64
Internal Precision: 320
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[{\left(\sqrt{\log 10}\right)}^{\left(\left(-\frac{1}{3}\right) - \left(\frac{1}{3} + 1\right)\right)} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt[3]{\sqrt{\log 10}}}\]

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 0.9

    \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
  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 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

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

  13. Applied pow11.0

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

  14. Applied pow1/30.1

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

  15. Applied pow-prod-up0.9

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

  16. Applied pow1/30.1

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

  17. Applied pow-flip0.1

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

  18. Applied pow-div0.1

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1072743783 989954326 4239155542 3782239461 3602631542 1719177920)' 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))