\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
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: 10.0 s
Input Error: 24.8
Output Error: 13.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{x.re}{y.re} & \text{when } y.re \le -1.8410844372587685 \cdot 10^{+146} \\ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} & \text{when } y.re \le 6.566722189442364 \cdot 10^{+163} \\ \frac{x.re}{y.re} & \text{otherwise} \end{cases}\)

    if y.re < -1.8410844372587685e+146 or 6.566722189442364e+163 < y.re

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      42.1
    2. Using strategy rm
      42.1
    3. Applied clear-num to get
      \[\color{red}{\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.re \cdot y.re + x.im \cdot y.im}}}\]
      42.1
    4. Applied simplify to get
      \[\frac{1}{\color{red}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.re \cdot y.re + x.im \cdot y.im}}} \leadsto \frac{1}{\color{blue}{\frac{{y.re}^2 + y.im \cdot y.im}{y.re \cdot x.re + x.im \cdot y.im}}}\]
      42.1
    5. Using strategy rm
      42.1
    6. Applied add-cube-cbrt to get
      \[\frac{1}{\color{red}{\frac{{y.re}^2 + y.im \cdot y.im}{y.re \cdot x.re + x.im \cdot y.im}}} \leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{\frac{{y.re}^2 + y.im \cdot y.im}{y.re \cdot x.re + x.im \cdot y.im}}\right)}^3}}\]
      42.1
    7. Applied taylor to get
      \[\frac{1}{{\left(\sqrt[3]{\frac{{y.re}^2 + y.im \cdot y.im}{y.re \cdot x.re + x.im \cdot y.im}}\right)}^3} \leadsto \frac{x.re}{y.re}\]
      0
    8. Taylor expanded around 0 to get
      \[\color{red}{\frac{x.re}{y.re}} \leadsto \color{blue}{\frac{x.re}{y.re}}\]
      0
    9. Applied simplify to get
      \[\frac{x.re}{y.re} \leadsto \frac{x.re}{y.re}\]
      0
    10. Applied final simplification

    if -1.8410844372587685e+146 < y.re < 6.566722189442364e+163

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      18.6
  1. Removed slow pow expressions

Original test:


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