\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
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: 43.5 s
Input Error: 34.1
Output Error: 4.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot {\left(\sqrt[3]{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\right)}^3 & \text{when } y.im \le 1.3305672087666973 \cdot 10^{+181} \\ \left(2 \cdot \frac{e^{\frac{\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)}{\frac{y.re}{-1}}}}{\left(\left(y.im \cdot y.im\right) \cdot \tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right) \cdot \tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}} - \frac{\frac{e^{\frac{\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)}{\frac{y.re}{-1}}}}{\frac{{\left(\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right)}^3}{4}}}{{y.im}^3}\right) \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) & \text{when } y.im \le 5.787612046082071 \cdot 10^{+268} \\ \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot {\left(\sqrt[3]{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\right)}^3 & \text{otherwise} \end{cases}\)

    if y.im < 1.3305672087666973e+181 or 5.787612046082071e+268 < y.im

    1. Started with
      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
      32.6
    2. Applied simplify to get
      \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\]
      4.5
    3. Using strategy rm
      4.5
    4. Applied add-cube-cbrt to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \color{red}{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)} \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \color{blue}{{\left(\sqrt[3]{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\right)}^3}\]
      5.1
    5. Using strategy rm
      5.1
    6. Applied pow-exp to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{red}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}} \cdot {\left(\sqrt[3]{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\right)}^3 \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot {\left(\sqrt[3]{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\right)}^3\]
      4.2

    if 1.3305672087666973e+181 < y.im < 5.787612046082071e+268

    1. Started with
      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
      49.8
    2. Applied simplify to get
      \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\]
      43.3
    3. Applied taylor to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
      29.8
    4. Taylor expanded around 0 to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{red}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
      29.8
    5. Applied taylor to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \left(2 \cdot \frac{e^{-1 \cdot \frac{\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)}{y.re}}}{{y.im}^2 \cdot {\left(\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right)}^2} - 4 \cdot \frac{e^{-1 \cdot \frac{\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)}{y.re}}}{{y.im}^{3} \cdot {\left(\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right)}^{3}}\right) \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
      9.7
    6. Taylor expanded around -inf to get
      \[\color{red}{\left(2 \cdot \frac{e^{-1 \cdot \frac{\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)}{y.re}}}{{y.im}^2 \cdot {\left(\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right)}^2} - 4 \cdot \frac{e^{-1 \cdot \frac{\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)}{y.re}}}{{y.im}^{3} \cdot {\left(\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right)}^{3}}\right)} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \color{blue}{\left(2 \cdot \frac{e^{-1 \cdot \frac{\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)}{y.re}}}{{y.im}^2 \cdot {\left(\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right)}^2} - 4 \cdot \frac{e^{-1 \cdot \frac{\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)}{y.re}}}{{y.im}^{3} \cdot {\left(\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right)}^{3}}\right)} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
      9.7
    7. Applied simplify to get
      \[\left(2 \cdot \frac{e^{-1 \cdot \frac{\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)}{y.re}}}{{y.im}^2 \cdot {\left(\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right)}^2} - 4 \cdot \frac{e^{-1 \cdot \frac{\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)}{y.re}}}{{y.im}^{3} \cdot {\left(\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right)}^{3}}\right) \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \left(2 \cdot \frac{e^{\frac{\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)}{\frac{y.re}{-1}}}}{\left(\left(y.im \cdot y.im\right) \cdot \tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right) \cdot \tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}} - \frac{\frac{e^{\frac{\log \left(\sqrt{\left(\frac{-1}{x.im}\right)^2 + \left(\frac{-1}{x.re}\right)^2}^*\right)}{\frac{y.re}{-1}}}}{\frac{{\left(\tan^{-1}_* \frac{\frac{-1}{x.im}}{\frac{-1}{x.re}}\right)}^3}{4}}}{{y.im}^3}\right) \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
      9.8
    8. Applied final simplification
  1. Removed slow pow expressions

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)))))