\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
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: 11.7 s
Input Error: 26.2
Output Error: 16.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{x.im}{y.re} - \frac{x.re \cdot y.im}{y.re \cdot y.re} & \text{when } y.re \le -2.2155360606085996 \cdot 10^{+134} \\ \frac{1}{\frac{{y.re}^2 + y.im \cdot y.im}{y.re \cdot x.im - x.re \cdot y.im}} & \text{when } y.re \le 7.70075503365186 \cdot 10^{+104} \\ \frac{x.im}{y.re} - \frac{x.re \cdot y.im}{y.re \cdot y.re} & \text{otherwise} \end{cases}\)

    if y.re < -2.2155360606085996e+134 or 7.70075503365186e+104 < y.re

    1. Started with
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      42.7
    2. Using strategy rm
      42.7
    3. Applied div-inv to get
      \[\color{red}{\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \frac{1}{y.re \cdot y.re + y.im \cdot y.im}}\]
      42.7
    4. Using strategy rm
      42.7
    5. Applied add-cube-cbrt to get
      \[\left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \color{red}{\frac{1}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{1}{y.re \cdot y.re + y.im \cdot y.im}}\right)}^3}\]
      42.8
    6. Applied taylor to get
      \[\left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot {\left(\sqrt[3]{\frac{1}{y.re \cdot y.re + y.im \cdot y.im}}\right)}^3 \leadsto \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^2}\]
      12.0
    7. Taylor expanded around inf to get
      \[\color{red}{\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^2}} \leadsto \color{blue}{\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^2}}\]
      12.0
    8. Applied simplify to get
      \[\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^2} \leadsto \frac{x.im}{y.re} - \frac{x.re \cdot y.im}{y.re \cdot y.re}\]
      12.0
    9. Applied final simplification

    if -2.2155360606085996e+134 < y.re < 7.70075503365186e+104

    1. Started with
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      18.3
    2. Using strategy rm
      18.3
    3. Applied clear-num to get
      \[\color{red}{\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}}\]
      18.4
    4. Applied simplify to get
      \[\frac{1}{\color{red}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}} \leadsto \frac{1}{\color{blue}{\frac{{y.re}^2 + y.im \cdot y.im}{y.re \cdot x.im - x.re \cdot y.im}}}\]
      18.4
  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))))