\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
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: 8.7 s
Input Error: 25.5
Output Error: 25.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{y.re \cdot x.im}{(y.im * y.im + \left({y.re}^2\right))_*} - \frac{y.im}{\frac{(y.im * y.im + \left({y.re}^2\right))_*}{x.re}} & \text{when } x.re \le -3.746428604476787 \cdot 10^{+212} \\ \frac{1}{\frac{(y.im * y.im + \left(y.re \cdot y.re\right))_*}{y.re \cdot x.im - y.im \cdot x.re}} & \text{otherwise} \end{cases}\)

    if x.re < -3.746428604476787e+212

    1. Started with
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      39.3
    2. Applied simplify to get
      \[\color{red}{\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{\frac{y.re \cdot x.im - y.im \cdot x.re}{(y.im * y.im + \left(y.re \cdot y.re\right))_*}}\]
      39.3
    3. Applied taylor to get
      \[\frac{y.re \cdot x.im - y.im \cdot x.re}{(y.im * y.im + \left(y.re \cdot y.re\right))_*} \leadsto \frac{y.re \cdot x.im}{(y.im * y.im + \left({y.re}^2\right))_*} - \frac{y.im \cdot x.re}{(y.im * y.im + \left({y.re}^2\right))_*}\]
      39.3
    4. Taylor expanded around 0 to get
      \[\color{red}{\frac{y.re \cdot x.im}{(y.im * y.im + \left({y.re}^2\right))_*} - \frac{y.im \cdot x.re}{(y.im * y.im + \left({y.re}^2\right))_*}} \leadsto \color{blue}{\frac{y.re \cdot x.im}{(y.im * y.im + \left({y.re}^2\right))_*} - \frac{y.im \cdot x.re}{(y.im * y.im + \left({y.re}^2\right))_*}}\]
      39.3
    5. Using strategy rm
      39.3
    6. Applied associate-/l* to get
      \[\frac{y.re \cdot x.im}{(y.im * y.im + \left({y.re}^2\right))_*} - \color{red}{\frac{y.im \cdot x.re}{(y.im * y.im + \left({y.re}^2\right))_*}} \leadsto \frac{y.re \cdot x.im}{(y.im * y.im + \left({y.re}^2\right))_*} - \color{blue}{\frac{y.im}{\frac{(y.im * y.im + \left({y.re}^2\right))_*}{x.re}}}\]
      35.2

    if -3.746428604476787e+212 < x.re

    1. Started with
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      24.5
    2. Applied simplify to get
      \[\color{red}{\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{\frac{y.re \cdot x.im - y.im \cdot x.re}{(y.im * y.im + \left(y.re \cdot y.re\right))_*}}\]
      24.5
    3. Using strategy rm
      24.5
    4. Applied clear-num to get
      \[\color{red}{\frac{y.re \cdot x.im - y.im \cdot x.re}{(y.im * y.im + \left(y.re \cdot y.re\right))_*}} \leadsto \color{blue}{\frac{1}{\frac{(y.im * y.im + \left(y.re \cdot y.re\right))_*}{y.re \cdot x.im - y.im \cdot x.re}}}\]
      24.6
  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))))