\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
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: 40.3 s
Input Error: 39.5
Output Error: 7.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(-x.re\right)}^{y.re}}} & \text{when } x.re \le -8.13670534637118 \cdot 10^{-05} \\ \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.re\right) \cdot y.im\right)}{\frac{\left(1 + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) + \left(\frac{1}{2} \cdot \left(y.im \cdot y.im\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}}} & \text{when } x.re \le 4.577564681758249 \cdot 10^{-297} \\ \frac{\sin \left(y.im \cdot \log x.re + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}} & \text{otherwise} \end{cases}\)

    if x.re < -8.13670534637118e-05

    1. Started with
      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
      40.3
    2. Applied simplify to get
      \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}}\]
      40.4
    3. Applied taylor to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}\]
      21.4
    4. Taylor expanded around -inf to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \color{red}{\left(-1 \cdot x.re\right)} \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \color{blue}{\left(-1 \cdot x.re\right)} \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}\]
      21.4
    5. Applied taylor to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(-1 \cdot x.re\right)}^{y.re}}}\]
      0.5
    6. Taylor expanded around -inf to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\color{red}{\left(-1 \cdot x.re\right)}}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\color{blue}{\left(-1 \cdot x.re\right)}}^{y.re}}}\]
      0.5
    7. Applied simplify to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(-1 \cdot x.re\right)}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(-x.re\right)}^{y.re}}}\]
      0.5
    8. Applied final simplification

    if -8.13670534637118e-05 < x.re < 4.577564681758249e-297

    1. Started with
      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
      39.9
    2. Applied simplify to get
      \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}}\]
      41.5
    3. Applied taylor to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}\]
      34.0
    4. Taylor expanded around 0 to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{\color{red}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}\]
      34.0
    5. Applied taylor to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}\]
      19.6
    6. Taylor expanded around -inf to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \color{red}{\left(-1 \cdot x.re\right)} \cdot y.im\right)}{\frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \color{blue}{\left(-1 \cdot x.re\right)} \cdot y.im\right)}{\frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}\]
      19.6
    7. Applied simplify to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.re\right) \cdot y.im\right)}{\frac{\left(1 + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) + \left(\frac{1}{2} \cdot \left(y.im \cdot y.im\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}}}\]
      19.6
    8. Applied final simplification

    if 4.577564681758249e-297 < x.re

    1. Started with
      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
      38.9
    2. Applied simplify to get
      \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}}\]
      38.0
    3. Applied taylor to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}}\]
      35.3
    4. Taylor expanded around inf to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\color{red}{x.re}}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\color{blue}{x.re}}^{y.re}}}\]
      35.3
    5. Applied taylor to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log x.re \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}}\]
      2.3
    6. Taylor expanded around inf to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \color{red}{x.re} \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}} \leadsto \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \color{blue}{x.re} \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}}\]
      2.3
    7. Applied simplify to get
      \[\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log x.re \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}} \leadsto \frac{\sin \left(y.im \cdot \log x.re + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}}\]
      2.3
    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)))))