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Derivation

1. Split input into 2 regimes

2. if y.re < -6.864987875366157e-55

   1. Initial program 37.5

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

   2. Taylor expanded around -inf 0.2

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

   3. Simplified0.2

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

   5. Applied add-cube-cbrt0.2

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

   6. Applied associate-*l*0.2

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

   8. Applied cbrt-unprod0.2

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

   10. Applied add-cube-cbrt0.2

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

   11. Applied associate-*r*0.2

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

   if -6.864987875366157e-55 < y.re

   1. Initial program 26.3

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

   2. Taylor expanded around -inf 31.2

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

   3. Simplified31.2

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

   4. Taylor expanded around -inf 19.3

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

   5. Simplified19.3

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

4. Final simplification9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -6.864987875366157 \cdot 10^{-55}:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \left(\sqrt[3]{\log \left(-x.re\right)} \cdot y.im\right) \cdot \sqrt[3]{\left(\log \left(-x.re\right) \cdot \left(\sqrt[3]{\log \left(-x.re\right)} \cdot \sqrt[3]{\log \left(-x.re\right)}\right)\right) \cdot \sqrt[3]{\log \left(-x.re\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(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)))))

Details

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