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Derivation

1. Initial program 32.6

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

2. Initial simplification9.2

   \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\]
3. Using strategy rm

4. Applied pow-to-exp9.2

   \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{\color{blue}{e^{\log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}}}\]

5. Applied pow-exp8.5

   \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}}{e^{\log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}}\]

6. Applied div-exp3.3

   \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}}\]
7. Using strategy rm

8. Applied add-cube-cbrt3.3

   \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{e^{y.im \cdot \color{blue}{\left(\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}\]

9. Applied associate-*r*3.3

   \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{e^{\color{blue}{\left(y.im \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}\]
10. Using strategy rm

11. Applied fma-udef3.3

    \[\leadsto \frac{\sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}{e^{\left(y.im \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}\]

12. Applied sin-sum3.3

    \[\leadsto \frac{\color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}{e^{\left(y.im \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}\]
13. Using strategy rm

14. Applied add-cube-cbrt3.3

    \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(\left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)}\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\left(y.im \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}\]

15. Applied log-prod3.3

    \[\leadsto \frac{\sin \left(y.im \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) + \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right)}\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\left(y.im \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}\]

16. Applied distribute-rgt-in3.3

    \[\leadsto \frac{\sin \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im + \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right)} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\left(y.im \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}\]

17. Applied sin-sum3.3

    \[\leadsto \frac{\color{blue}{\left(\sin \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right) \cdot \cos \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right) + \cos \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right) \cdot \sin \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right)\right)} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\left(y.im \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}\]

18. Simplified3.3

    \[\leadsto \frac{\left(\color{blue}{\cos \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right) \cdot \sin \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot \left(y.im + y.im\right)\right)} + \cos \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right) \cdot \sin \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right)\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\left(y.im \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}\]

19. Final simplification3.3

    \[\leadsto \frac{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \cos \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.im\right) + \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right) \cdot \sin \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right) + \cos \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot y.im\right) \cdot \sin \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot \left(y.im + y.im\right)\right)\right)}{e^{\left(\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot y.im\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}\]

Runtime

Time bar (total: 55.3s)Debug logProfile

herbie shell --seed 2018242 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))