\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
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: 59.1 s
Input Error: 33.2
Output Error: 3.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{(\left({y.im}^2 \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) * \frac{1}{2} + \left((y.im * \left(\tan^{-1}_* \frac{x.im}{x.re}\right) + 1)_*\right))_*} \cdot \sin \left((\left(\frac{1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{1}{x.im}\right)^2 + \left(\frac{1}{x.re}\right)^2}^*\right)\right) + \left(\frac{\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}}{y.re}\right))_*\right) & \text{when } y.re \le -4.4583597911297456 \cdot 10^{+184} \\ \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot {\left(\sqrt[3]{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\right)}^3 & \text{when } y.re \le 7.471150229908471 \cdot 10^{-51} \\ \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\frac{(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\frac{1}{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) * \left({y.im}^2\right) + \left((\left(\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}\right) * y.im + 1)_*\right))_*}{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}} & \text{otherwise} \end{cases}\)

    if y.re < -4.4583597911297456e+184

    1. Started with
      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
      38.6
    2. Applied simplify to get
      \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\]
      13.8
    3. Applied taylor to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((\left(\frac{1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{1}{x.im}\right)^2 + \left(\frac{1}{x.re}\right)^2}^*\right)\right) + \left(\frac{\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}}{y.re}\right))_*\right)\]
      13.9
    4. Taylor expanded around inf to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \color{red}{\sin \left((\left(\frac{1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{1}{x.im}\right)^2 + \left(\frac{1}{x.re}\right)^2}^*\right)\right) + \left(\frac{\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}}{y.re}\right))_*\right)} \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \color{blue}{\sin \left((\left(\frac{1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{1}{x.im}\right)^2 + \left(\frac{1}{x.re}\right)^2}^*\right)\right) + \left(\frac{\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}}{y.re}\right))_*\right)}\]
      13.9
    5. Applied taylor to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((\left(\frac{1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{1}{x.im}\right)^2 + \left(\frac{1}{x.re}\right)^2}^*\right)\right) + \left(\frac{\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}}{y.re}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left((\left(\frac{1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{1}{x.im}\right)^2 + \left(\frac{1}{x.re}\right)^2}^*\right)\right) + \left(\frac{\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}}{y.re}\right))_*\right)\]
      3.1
    6. Taylor expanded around 0 to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{red}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}} \cdot \sin \left((\left(\frac{1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{1}{x.im}\right)^2 + \left(\frac{1}{x.re}\right)^2}^*\right)\right) + \left(\frac{\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}}{y.re}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}} \cdot \sin \left((\left(\frac{1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{1}{x.im}\right)^2 + \left(\frac{1}{x.re}\right)^2}^*\right)\right) + \left(\frac{\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}}{y.re}\right))_*\right)\]
      3.1
    7. Applied simplify to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left((\left(\frac{1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{1}{x.im}\right)^2 + \left(\frac{1}{x.re}\right)^2}^*\right)\right) + \left(\frac{\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}}{y.re}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{(\left({y.im}^2 \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) * \frac{1}{2} + \left((y.im * \left(\tan^{-1}_* \frac{x.im}{x.re}\right) + 1)_*\right))_*} \cdot \sin \left((\left(\frac{1}{y.im}\right) * \left(\log \left(\sqrt{\left(\frac{1}{x.im}\right)^2 + \left(\frac{1}{x.re}\right)^2}^*\right)\right) + \left(\frac{\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}}{y.re}\right))_*\right)\]
      3.1
    8. Applied final simplification

    if -4.4583597911297456e+184 < y.re < 7.471150229908471e-51

    1. Started with
      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
      33.4
    2. Applied simplify to get
      \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\]
      1.6
    3. Using strategy rm
      1.6
    4. Applied add-cube-cbrt to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \color{red}{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)} \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \color{blue}{{\left(\sqrt[3]{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\right)}^3}\]
      2.2

    if 7.471150229908471e-51 < y.re

    1. Started with
      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
      28.5
    2. Applied simplify to get
      \[\color{red}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \leadsto \color{blue}{\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}\]
      18.9
    3. Applied taylor to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
      7.4
    4. Taylor expanded around 0 to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{red}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)}} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
      7.4
    5. Applied taylor to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{y.im \cdot \tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}} + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
      7.0
    6. Taylor expanded around inf to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{red}{y.im \cdot \tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}} + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\color{blue}{y.im \cdot \tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}} + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)\]
      7.0
    7. Applied simplify to get
      \[\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{y.im \cdot \tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}} + \left(1 + \frac{1}{2} \cdot \left({\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}^2 \cdot {y.im}^2\right)\right)} \cdot \sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right) \leadsto \frac{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\frac{1}{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) * \left(y.im \cdot y.im\right) + \left((\left(\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}\right) * y.im + 1)_*\right))_*}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}}\]
      7.0
    8. Applied final simplification
    9. Applied simplify to get
      \[\color{red}{\frac{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\frac{1}{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) * \left(y.im \cdot y.im\right) + \left((\left(\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}\right) * y.im + 1)_*\right))_*}{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}}} \leadsto \color{blue}{\frac{{\left(\sqrt{x.im^2 + x.re^2}^*\right)}^{y.re}}{\frac{(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\frac{1}{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) * \left({y.im}^2\right) + \left((\left(\tan^{-1}_* \frac{\frac{1}{x.im}}{\frac{1}{x.re}}\right) * y.im + 1)_*\right))_*}{\sin \left((y.im * \left(\log \left(\sqrt{x.im^2 + x.re^2}^*\right)\right) + \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right))_*\right)}}}\]
      7.0
  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)))))