\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
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: 11.4 s
Input Error: 25.3
Output Error: 13.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{x.im}{y.re} & \text{when } y.re \le -5.419122227930595 \cdot 10^{+127} \\ \left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \frac{1}{y.re \cdot y.re + y.im \cdot y.im} & \text{when } y.re \le 2.092832717516812 \cdot 10^{+135} \\ \frac{x.im}{y.re} & \text{otherwise} \end{cases}\)

    if y.re < -5.419122227930595e+127 or 2.092832717516812e+135 < y.re

    1. Started with
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      41.1
    2. Using strategy rm
      41.1
    3. Applied add-cbrt-cube to get
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{\color{red}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt[3]{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^3}}}\]
      43.0
    4. Applied add-cbrt-cube to get
      \[\frac{\color{red}{x.im \cdot y.re - x.re \cdot y.im}}{\sqrt[3]{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^3}} \leadsto \frac{\color{blue}{\sqrt[3]{{\left(x.im \cdot y.re - x.re \cdot y.im\right)}^3}}}{\sqrt[3]{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^3}}\]
      52.1
    5. Applied cbrt-undiv to get
      \[\color{red}{\frac{\sqrt[3]{{\left(x.im \cdot y.re - x.re \cdot y.im\right)}^3}}{\sqrt[3]{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^3}}} \leadsto \color{blue}{\sqrt[3]{\frac{{\left(x.im \cdot y.re - x.re \cdot y.im\right)}^3}{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^3}}}\]
      52.1
    6. Applied simplify to get
      \[\sqrt[3]{\color{red}{\frac{{\left(x.im \cdot y.re - x.re \cdot y.im\right)}^3}{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^3}}} \leadsto \sqrt[3]{\color{blue}{{\left(\frac{y.re \cdot x.im - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\right)}^3}}\]
      42.0
    7. Applied taylor to get
      \[\sqrt[3]{{\left(\frac{y.re \cdot x.im - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\right)}^3} \leadsto \frac{x.im}{y.re}\]
      0
    8. Taylor expanded around 0 to get
      \[\color{red}{\frac{x.im}{y.re}} \leadsto \color{blue}{\frac{x.im}{y.re}}\]
      0
    9. Applied simplify to get
      \[\frac{x.im}{y.re} \leadsto \frac{x.im}{y.re}\]
      0
    10. Applied final simplification

    if -5.419122227930595e+127 < y.re < 2.092832717516812e+135

    1. Started with
      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      18.5
    2. Using strategy rm
      18.5
    3. Applied div-inv to get
      \[\color{red}{\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \frac{1}{y.re \cdot y.re + y.im \cdot y.im}}\]
      18.7
  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))))