if y.re < -7.698210638222214e+42

1. Started with
   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
   39.5
2. Applied simplify to get
   \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\]
   14.4
3. Using strategy rm
   14.4
4. Applied add-cube-cbrt to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \color{red}{\left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)} \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \color{blue}{\left({\left(\sqrt[3]{(y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*}\right)}^3\right)}\]
   14.4
5. Applied taylor to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left({\left(\sqrt[3]{(y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*}\right)}^3\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left({\left(\sqrt[3]{(y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*}\right)}^3\right)\]
   3.0
6. Taylor expanded around 0 to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{red}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}} \cdot \sin \left({\left(\sqrt[3]{(y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*}\right)}^3\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}} \cdot \sin \left({\left(\sqrt[3]{(y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*}\right)}^3\right)\]
   3.0
7. Applied taylor to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left({\left(\sqrt[3]{(y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*}\right)}^3\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{y.im \cdot \tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}} + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left({\left(\sqrt[3]{(y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*}\right)}^3\right)\]
   3.0
8. Taylor expanded around -inf to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{red}{y.im \cdot \tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}} + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left({\left(\sqrt[3]{(y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*}\right)}^3\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{blue}{y.im \cdot \tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}} + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left({\left(\sqrt[3]{(y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*}\right)}^3\right)\]
   3.0
9. Applied simplify to get
   \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{y.im \cdot \tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}} + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left({\left(\sqrt[3]{(y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*}\right)}^3\right) \leadsto \frac{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}{\frac{(\left(y.im \cdot y.im\right) * \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{1}{2}\right)\right) + \left((\left(\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right) * y.im + 1)_*\right))_*}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}}\]
   3.0

---

10. Applied final simplification
11. Applied simplify to get
    \[\color{red}{\frac{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}{\frac{(\left(y.im \cdot y.im\right) * \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{1}{2}\right)\right) + \left((\left(\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right) * y.im + 1)_*\right))_*}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}}} \leadsto \color{blue}{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right) \cdot \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{(\left({y.im}^2\right) * \left(\left(\frac{1}{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \left((\left(\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right) * y.im + 1)_*\right))_*}}\]
    3.0

if -7.698210638222214e+42 < y.re

1. Started with
   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
   30.6
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