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Derivation

1. Split input into 2 regimes

2. if (* (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)))) < 0.06980966285566138

   1. Initial program 2.0

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

   if 0.06980966285566138 < (* (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))))

   1. Initial program 62.6

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

   2. Initial simplification10.4

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

   4. Applied fma-udef10.4

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

   5. Applied sin-sum10.4

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

   7. Applied add-cube-cbrt10.4

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

   8. Applied log-prod10.4

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

   9. Applied distribute-lft-in10.4

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

   10. Applied sin-sum10.5

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

   11. Simplified10.5

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

   13. Applied add-cube-cbrt10.5

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

   14. Applied cbrt-prod10.5

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

   15. Applied log-prod10.4

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

   16. Applied distribute-lft-in10.4

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

   17. Applied sin-sum10.4

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

   19. Applied add-cube-cbrt10.4

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

4. Final simplification6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \le 0.06980966285566138:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \cos \left(y.im \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\sin \left(\log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right) \cdot \left(y.im + y.im\right)\right) \cdot \cos \left(y.im \cdot \log \left(\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}\right)\right) + \left(\cos \left(\log \left(\sqrt[3]{\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}}\right) \cdot y.im\right) \cdot \sin \left(\log \left(\sqrt[3]{\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*} \cdot \sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}}\right) \cdot y.im\right) + \cos \left(\log \left(\sqrt[3]{\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 \left(\left(\sqrt[3]{\sin \left(\log \left(\sqrt[3]{\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}}\right) \cdot y.im\right)} \cdot \sqrt[3]{\sin \left(\log \left(\sqrt[3]{\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}}\right) \cdot y.im\right)}\right) \cdot \sqrt[3]{\sin \left(\log \left(\sqrt[3]{\sqrt[3]{\sqrt{x.re^2 + x.im^2}^*}}\right) \cdot y.im\right)}\right)\right) \cdot \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)\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\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}}}\\ \end{array}\]

Runtime

Time bar (total: 46.2s)Debug logProfile

herbie shell --seed 2018221 +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)))))