\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
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: 14.1 s
Input Error: 12.7
Output Error: 6.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re} & \text{when } y.re \le -1.9353409f+09 \\ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} & \text{when } y.re \le 9.809289f+19 \\ \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re} & \text{otherwise} \end{cases}\)

    if y.re < -1.9353409f+09 or 9.809289f+19 < y.re

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      20.6
    2. Using strategy rm
      20.6
    3. Applied add-cube-cbrt to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{\color{red}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{{\left(\sqrt[3]{y.re \cdot y.re + y.im \cdot y.im}\right)}^3}}\]
      20.6
    4. Applied simplify to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{{\color{red}{\left(\sqrt[3]{y.re \cdot y.re + y.im \cdot y.im}\right)}}^3} \leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{{\color{blue}{\left(\sqrt[3]{{y.re}^2 + y.im \cdot y.im}\right)}}^3}\]
      20.6
    5. Applied taylor to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{{\left(\sqrt[3]{{y.re}^2 + y.im \cdot y.im}\right)}^3} \leadsto \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2}\]
      5.6
    6. Taylor expanded around inf to get
      \[\color{red}{\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2}} \leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2}}\]
      5.6
    7. Applied simplify to get
      \[\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2} \leadsto \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\]
      0.3
    8. Applied final simplification

    if -1.9353409f+09 < y.re < 9.809289f+19

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      8.9
  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))))