\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
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: 10.3 s
Input Error: 25.0
Output Error: 13.4
Log: ⚲
Profile: 🕒
\(\begin{cases} -\frac{x.re}{y.im} & \text{when } y.im \le -2.0646128749844613 \cdot 10^{+129} \\ \left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \frac{1}{y.re \cdot y.re + y.im \cdot y.im} & \text{when } y.im \le 1.2610866829211587 \cdot 10^{+151} \\ -\frac{x.re}{y.im} & \text{otherwise} \end{cases}\)

    if y.im < -2.0646128749844613e+129 or 1.2610866829211587e+151 < y.im

    1. Started with
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      41.7
    2. Using strategy rm
      41.7
    3. Applied add-exp-log to get
      \[\color{red}{\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{e^{\log \left(\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right)}}\]
      42.7
    4. Applied taylor to get
      \[e^{\log \left(\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right)} \leadsto e^{\left(\log x.re + \log -1\right) - \log y.im}\]
      62.9
    5. Taylor expanded around 0 to get
      \[e^{\color{red}{\left(\log x.re + \log -1\right) - \log y.im}} \leadsto e^{\color{blue}{\left(\log x.re + \log -1\right) - \log y.im}}\]
      62.9
    6. 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
    7. Applied final simplification
    8. Applied simplify to get
      \[\color{red}{\frac{-1}{y.im} \cdot x.re} \leadsto \color{blue}{-\frac{x.re}{y.im}}\]
      0

    if -2.0646128749844613e+129 < y.im < 1.2610866829211587e+151

    1. Started with
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      18.4
    2. Using strategy rm
      18.4
    3. Applied div-inv to get
      \[\color{red}{\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \frac{1}{y.re \cdot y.re + y.im \cdot y.im}}\]
      18.6
  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))))