\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
Test:
powComplex, real 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: 35.0 s
Input Error: 31.9
Output Error: 7.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(-x.re\right)}^{y.re}}} & \text{when } x.re \le 2.2804430027321976 \cdot 10^{-308} \\ \frac{\cos \left(y.im \cdot \log x.re + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot {\left(\sqrt{{x.im}^2 + x.re \cdot x.re}\right)}^{y.re} & \text{when } x.re \le 3.2366201110863356 \cdot 10^{+227} \\ \frac{\cos \left(y.im \cdot \log x.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}} & \text{otherwise} \end{cases}\)

    if x.re < 2.2804430027321976e-308

    1. Started with
      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
      33.0
    2. Applied simplify to get
      \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}}\]
      33.6
    3. Applied taylor to get
      \[\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}\]
      22.9
    4. Taylor expanded around -inf to get
      \[\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \color{red}{\left(-1 \cdot x.re\right)} \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \color{blue}{\left(-1 \cdot x.re\right)} \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}\]
      22.9
    5. Applied taylor to get
      \[\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(-1 \cdot x.re\right)}^{y.re}}}\]
      0.5
    6. Taylor expanded around -inf to get
      \[\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\color{red}{\left(-1 \cdot x.re\right)}}^{y.re}}} \leadsto \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\color{blue}{\left(-1 \cdot x.re\right)}}^{y.re}}}\]
      0.5
    7. Applied simplify to get
      \[\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-1 \cdot x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(-1 \cdot x.re\right)}^{y.re}}} \leadsto \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.re\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(-x.re\right)}^{y.re}}}\]
      0.5
    8. Applied final simplification

    if 2.2804430027321976e-308 < x.re < 3.2366201110863356e+227

    1. Started with
      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
      28.5
    2. Applied simplify to get
      \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}}\]
      27.2
    3. Using strategy rm
      27.2
    4. Applied add-cube-cbrt to get
      \[\frac{\color{red}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}\right)}^3}}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}\]
      27.2
    5. Applied taylor to get
      \[\frac{{\left(\sqrt[3]{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}\right)}^3}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{{\left(\sqrt[3]{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log x.re \cdot y.im\right)}\right)}^3}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}\]
      15.4
    6. Taylor expanded around inf to get
      \[\frac{{\left(\sqrt[3]{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \color{red}{x.re} \cdot y.im\right)}\right)}^3}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{{\left(\sqrt[3]{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \color{blue}{x.re} \cdot y.im\right)}\right)}^3}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}\]
      15.4
    7. Applied simplify to get
      \[\color{red}{\frac{{\left(\sqrt[3]{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log x.re \cdot y.im\right)}\right)}^3}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}} \leadsto \color{blue}{\frac{\cos \left(y.im \cdot \log x.re + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot {\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}}\]
      15.4
    8. Applied simplify to get
      \[\frac{\cos \left(y.im \cdot \log x.re + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \color{red}{{\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}} \leadsto \frac{\cos \left(y.im \cdot \log x.re + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \color{blue}{{\left(\sqrt{{x.im}^2 + x.re \cdot x.re}\right)}^{y.re}}\]
      15.4

    if 3.2366201110863356e+227 < x.re

    1. Started with
      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
      62.6
    2. Applied simplify to get
      \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}}}\]
      62.6
    3. Applied taylor to get
      \[\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right)}^{y.re}}} \leadsto \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}}\]
      62.6
    4. Taylor expanded around inf to get
      \[\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\color{red}{x.re}}^{y.re}}} \leadsto \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\color{blue}{x.re}}^{y.re}}}\]
      62.6
    5. Applied taylor to get
      \[\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(\sqrt{{x.re}^2 + x.im \cdot x.im}\right) \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}} \leadsto \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log x.im \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}}\]
      8.6
    6. Taylor expanded around 0 to get
      \[\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \color{red}{x.im} \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}} \leadsto \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \color{blue}{x.im} \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}}\]
      8.6
    7. Applied simplify to get
      \[\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log x.im \cdot y.im\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}} \leadsto \frac{\cos \left(y.im \cdot \log x.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{x.re}^{y.re}}}\]
      8.6
    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, real part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))