\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
Test:
_divideComplex, 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: 15.3 s
Input Error: 13.1
Output Error: 3.9
Log: ⚲
Profile: 🕒
\(\begin{cases} -\frac{x.re}{y.im} & \text{when } y.im \le -3.3116914f+16 \\ \frac{1}{\frac{{y.re}^2 + y.im \cdot y.im}{y.re \cdot x.im - x.re \cdot y.im}} & \text{when } y.im \le -1.9157653f-23 \\ \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^2} & \text{when } y.im \le 6.0254553f-15 \\ \frac{1}{\frac{{y.re}^2 + y.im \cdot y.im}{y.re \cdot x.im - x.re \cdot y.im}} & \text{when } y.im \le 151413.64f0 \\ -\frac{x.re}{y.im} & \text{otherwise} \end{cases}\)

    if y.im < -3.3116914f+16 or 151413.64f0 < y.im

    1. Started with
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      19.8
    2. Using strategy rm
      19.8
    3. Applied add-exp-log to get
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{\color{red}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{e^{\log \left(y.re \cdot y.re + y.im \cdot y.im\right)}}}\]
      20.1
    4. Applied add-exp-log to get
      \[\frac{\color{red}{x.im \cdot y.re - x.re \cdot y.im}}{e^{\log \left(y.re \cdot y.re + y.im \cdot y.im\right)}} \leadsto \frac{\color{blue}{e^{\log \left(x.im \cdot y.re - x.re \cdot y.im\right)}}}{e^{\log \left(y.re \cdot y.re + y.im \cdot y.im\right)}}\]
      25.6
    5. Applied div-exp to get
      \[\color{red}{\frac{e^{\log \left(x.im \cdot y.re - x.re \cdot y.im\right)}}{e^{\log \left(y.re \cdot y.re + y.im \cdot y.im\right)}}} \leadsto \color{blue}{e^{\log \left(x.im \cdot y.re - x.re \cdot y.im\right) - \log \left(y.re \cdot y.re + y.im \cdot y.im\right)}}\]
      25.6
    6. Applied taylor to get
      \[e^{\log \left(x.im \cdot y.re - x.re \cdot y.im\right) - \log \left(y.re \cdot y.re + y.im \cdot y.im\right)} \leadsto e^{\left(\log x.re + \log -1\right) - \log y.im}\]
      30.9
    7. Taylor expanded around 0 to get
      \[\color{red}{e^{\left(\log x.re + \log -1\right) - \log y.im}} \leadsto \color{blue}{e^{\left(\log x.re + \log -1\right) - \log y.im}}\]
      30.9
    8. Applied simplify to get
      \[e^{\left(\log x.re + \log -1\right) - \log y.im} \leadsto \frac{-1}{y.im} \cdot x.re\]
      0.2
    9. Applied final simplification
    10. Applied simplify to get
      \[\color{red}{\frac{-1}{y.im} \cdot x.re} \leadsto \color{blue}{-\frac{x.re}{y.im}}\]
      0

    if -3.3116914f+16 < y.im < -1.9157653f-23 or 6.0254553f-15 < y.im < 151413.64f0

    1. Started with
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      7.9
    2. Using strategy rm
      7.9
    3. Applied clear-num to get
      \[\color{red}{\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}}\]
      8.1
    4. Applied simplify to get
      \[\frac{1}{\color{red}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}} \leadsto \frac{1}{\color{blue}{\frac{{y.re}^2 + y.im \cdot y.im}{y.re \cdot x.im - x.re \cdot y.im}}}\]
      8.1

    if -1.9157653f-23 < y.im < 6.0254553f-15

    1. Started with
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      10.5
    2. Using strategy rm
      10.5
    3. Applied add-cube-cbrt to get
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{\color{red}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{{\left(\sqrt[3]{y.re \cdot y.re + y.im \cdot y.im}\right)}^3}}\]
      10.8
    4. Applied simplify to get
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{{\color{red}{\left(\sqrt[3]{y.re \cdot y.re + y.im \cdot y.im}\right)}}^3} \leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{{\color{blue}{\left(\sqrt[3]{{y.re}^2 + y.im \cdot y.im}\right)}}^3}\]
      10.8
    5. Applied taylor to get
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{{\left(\sqrt[3]{{y.re}^2 + y.im \cdot y.im}\right)}^3} \leadsto \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^2}\]
      3.0
    6. Taylor expanded around inf to get
      \[\color{red}{\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^2}} \leadsto \color{blue}{\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^2}}\]
      3.0
  1. Removed slow pow expressions

Original test:


(lambda ((x.re default) (x.im default) (y.re default) (y.im default))
  #:name "_divideComplex, imaginary part"
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))