\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
Test:
powComplex, imaginary part
Bits:
128 bits
Bits error versus x.re
Bits error versus x.im
Bits error versus y.re
Bits error versus y.im
Time: 23.6 s
Input Error: 32.3
Output Error: 4.6
Log: ⚲
Profile: 🕒
\(e^{\log \left(\sqrt{x.im^2 + x.re^2}^*\right) \cdot y.re - \sqrt[3]{{\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^3}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\)
  1. Started with
    \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
    32.3
  2. Applied simplify to get
    \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\]
    9.6
  3. Using strategy rm
    9.6
  4. Applied pow-exp to get
    \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{red}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
    8.8
  5. Applied add-exp-log to get
    \[\frac{{\color{red}{\left(\sqrt{x.im^2 + x.re^2}^*\right)}}^{y.re}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \frac{{\color{blue}{\left(e^{\log \left(\sqrt{x.im^2 + x.re^2}^*\right)}\right)}}^{y.re}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
    8.8
  6. Applied pow-exp to get
    \[\frac{\color{red}{{\left(e^{\log \left(\sqrt{x.im^2 + x.re^2}^*\right)}\right)}^{y.re}}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \frac{\color{blue}{e^{\log \left(\sqrt{x.im^2 + x.re^2}^*\right) \cdot y.re}}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
    8.8
  7. Applied div-exp to get
    \[\color{red}{\frac{e^{\log \left(\sqrt{x.im^2 + x.re^2}^*\right) \cdot y.re}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \color{blue}{e^{\log \left(\sqrt{x.im^2 + x.re^2}^*\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
    3.5
  8. Using strategy rm
    3.5
  9. Applied add-cbrt-cube to get
    \[e^{\log \left(\sqrt{x.im^2 + x.re^2}^*\right) \cdot y.re - \color{red}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto e^{\log \left(\sqrt{x.im^2 + x.re^2}^*\right) \cdot y.re - \color{blue}{\sqrt[3]{{\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^3}}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
    4.6

Original test:


(lambda ((x.re default) (x.im default) (y.re default) (y.im default))
  #: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)))))