\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
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: 38.3 s
Input Error: 34.5
Output Error: 19.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} & \text{when } x.re \le -1.1403171900823759 \cdot 10^{-10} \\ \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} & \text{when } x.re \le -5.738570526864902 \cdot 10^{-249} \\ \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} & \text{when } x.re \le 2.2804430027321976 \cdot 10^{-308} \\ \frac{y.im \cdot \log x.re + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} & \text{otherwise} \end{cases}\)

    if x.re < -1.1403171900823759e-10 or -5.738570526864902e-249 < x.re < 2.2804430027321976e-308

    1. Started with
      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
      39.0
    2. Applied simplify to get
      \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}}\]
      39.3
    3. Applied taylor to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}\]
      21.2
    4. Taylor expanded around -inf to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \color{red}{\left(-1 \cdot x.re\right)} \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \color{blue}{\left(-1 \cdot x.re\right)} \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}\]
      21.2

    if -1.1403171900823759e-10 < x.re < -5.738570526864902e-249

    1. Started with
      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
      21.6
    2. Applied simplify to get
      \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}}\]
      21.7

    if 2.2804430027321976e-308 < x.re

    1. Started with
      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
      37.2
    2. Applied simplify to get
      \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}}\]
      36.0
    3. Applied taylor to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{y.im \cdot \log x.re + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}\]
      17.9
    4. Taylor expanded around 0 to get
      \[\frac{\color{red}{y.im \cdot \log x.re + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\color{blue}{y.im \cdot \log x.re + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}\]
      17.9
  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)))))