if y.re < -1936816090422238.5

1. Started with
   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
   40.2
2. Applied simplify to get
   \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\]
   15.5
3. Applied taylor to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \cos \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
   2.5
4. Taylor expanded around 0 to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{red}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}} \cdot \cos \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}} \cdot \cos \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
   2.5
5. Applied taylor to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \cos \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \cos \left((\left(\frac{-1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)\right) + \left(-1 \cdot \frac{\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}}{y.re}\right))_*\right)\]
   2.4
6. Taylor expanded around -inf to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \color{red}{\cos \left((\left(\frac{-1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)\right) + \left(-1 \cdot \frac{\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}}{y.re}\right))_*\right)} \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \color{blue}{\cos \left((\left(\frac{-1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)\right) + \left(-1 \cdot \frac{\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}}{y.re}\right))_*\right)}\]
   2.4
7. Applied simplify to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \cos \left((\left(\frac{-1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)\right) + \left(-1 \cdot \frac{\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}}{y.re}\right))_*\right) \leadsto \cos \left((\left(\frac{-1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)\right) + \left(\frac{\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}}{\frac{y.re}{-1}}\right))_*\right) \cdot \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{(\frac{1}{2} * \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left({y.im}^2 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) + \left((y.im * \left(\tan^{-1}_* \frac{x.im}{x.re}\right) + 1)_*\right))_*}\]
   1.3

---

8. Applied final simplification
9. Applied simplify to get
   \[\color{red}{\cos \left((\left(\frac{-1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)\right) + \left(\frac{\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}}{\frac{y.re}{-1}}\right))_*\right) \cdot \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{(\frac{1}{2} * \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left({y.im}^2 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) + \left((y.im * \left(\tan^{-1}_* \frac{x.im}{x.re}\right) + 1)_*\right))_*}} \leadsto \color{blue}{\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{(\frac{1}{2} * \left({\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^2\right) + \left((y.im * \left(\tan^{-1}_* \frac{x.im}{x.re}\right) + 1)_*\right))_*} \cdot \cos \left((\left(\frac{-1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)\right) + \left(\frac{-1}{y.re} \cdot \tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right))_*\right)}\]
   0.1

if -1936816090422238.5 < y.re

1. Started with
   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
   29.9
2. Applied simplify to get
   \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\]
   1.5
3. Using strategy rm
   1.5
4. Applied fma-udef to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \color{red}{\left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)} \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \cos \color{blue}{\left(y.im \cdot \log \left(\sqrt{x.im^2 + x.re^2}^*\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\]
   1.5
5. Applied cos-sum to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \color{red}{\cos \left(y.im \cdot \log \left(\sqrt{x.im^2 + x.re^2}^*\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \color{blue}{\left(\cos \left(y.im \cdot \log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) - \sin \left(y.im \cdot \log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\]
   1.5
6. Using strategy rm
   1.5
7. Applied add-log-exp to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) - \sin \left(y.im \cdot \log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot \color{red}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) - \sin \left(y.im \cdot \log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) \cdot \color{blue}{\log \left(e^{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}\right)\]
   1.5