if y.re < -2.9335963728403624e+168

1. Started with
   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
   40.1
2. Applied simplify to get
   \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\]
   18.5
3. Using strategy rm
   18.5
4. Applied add-exp-log to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \color{red}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)})_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \color{blue}{\left(e^{\log \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)})_*\right)\]
   38.7
5. Applied taylor to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(e^{\log \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{e^{\frac{\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}}{y.im}}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(e^{\log \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right))_*\right)\]
   29.9
6. Taylor expanded around inf to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{red}{e^{\frac{\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}}{y.im}}}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(e^{\log \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{blue}{e^{\frac{\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}}{y.im}}}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(e^{\log \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right))_*\right)\]
   29.9
7. Applied simplify to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{e^{\frac{\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}}{y.im}}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(e^{\log \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\frac{e^{\frac{\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}}{y.im}}}{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}}\]
   0.9

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8. Applied final simplification

if -2.9335963728403624e+168 < y.re

1. Started with
   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
   31.3
2. Applied simplify to get
   \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\]
   1.7
3. Using strategy rm
   1.7
4. Applied fma-udef to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \color{red}{\left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)} \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{x.im^2 + x.re^2}^*\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\]
   1.7
5. Applied sin-sum to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \color{red}{\sin \left(y.im \cdot \log \left(\sqrt{x.im^2 + x.re^2}^*\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \color{blue}{\left(\sin \left(y.im \cdot \log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \cos \left(y.im \cdot \log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\]
   1.7