\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
Test:
_divideComplex, real 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: 14.0 s
Input Error: 26.6
Output Error: 10.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re} & \text{when } y.re \le -1.083599015752864 \cdot 10^{+73} \\ \frac{1}{\frac{{y.re}^2 + y.im \cdot y.im}{y.re \cdot x.re + x.im \cdot y.im}} & \text{when } y.re \le 3.586178138959661 \cdot 10^{+58} \\ \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re} & \text{otherwise} \end{cases}\)

    if y.re < -1.083599015752864e+73 or 3.586178138959661e+58 < y.re

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      39.4
    2. Using strategy rm
      39.4
    3. Applied div-inv to get
      \[\color{red}{\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \frac{1}{y.re \cdot y.re + y.im \cdot y.im}}\]
      39.4
    4. Using strategy rm
      39.4
    5. Applied add-cube-cbrt to get
      \[\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \color{red}{\frac{1}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{1}{y.re \cdot y.re + y.im \cdot y.im}}\right)}^3}\]
      39.6
    6. Applied taylor to get
      \[\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot {\left(\sqrt[3]{\frac{1}{y.re \cdot y.re + y.im \cdot y.im}}\right)}^3 \leadsto \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2}\]
      10.9
    7. Taylor expanded around inf to get
      \[\color{red}{\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2}} \leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2}}\]
      10.9
    8. Applied simplify to get
      \[\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2} \leadsto \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\]
      0.8
    9. Applied final simplification

    if -1.083599015752864e+73 < y.re < 3.586178138959661e+58

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      17.5
    2. Using strategy rm
      17.5
    3. Applied clear-num to get
      \[\color{red}{\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.re \cdot y.re + x.im \cdot y.im}}}\]
      17.7
    4. Applied simplify to get
      \[\frac{1}{\color{red}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.re \cdot y.re + x.im \cdot y.im}}} \leadsto \frac{1}{\color{blue}{\frac{{y.re}^2 + y.im \cdot y.im}{y.re \cdot x.re + x.im \cdot y.im}}}\]
      17.7
  1. Removed slow pow expressions

Original test:


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