\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
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: 12.4 s
Input Error: 26.3
Output Error: 11.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re} & \text{when } y.re \le -5.671665100714436 \cdot 10^{+89} \\ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} & \text{when } y.re \le 6.209412377058245 \cdot 10^{+96} \\ \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re} & \text{otherwise} \end{cases}\)

    if y.re < -5.671665100714436e+89 or 6.209412377058245e+96 < y.re

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      40.9
    2. Using strategy rm
      40.9
    3. Applied div-inv to get
      \[\color{red}{\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \frac{1}{y.re \cdot y.re + y.im \cdot y.im}}\]
      40.9
    4. Using strategy rm
      40.9
    5. Applied add-cbrt-cube to get
      \[\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \color{red}{\frac{1}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \color{blue}{\sqrt[3]{{\left(\frac{1}{y.re \cdot y.re + y.im \cdot y.im}\right)}^3}}\]
      48.6
    6. Applied taylor to get
      \[\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \sqrt[3]{{\left(\frac{1}{y.re \cdot y.re + y.im \cdot y.im}\right)}^3} \leadsto \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2}\]
      11.8
    7. Taylor expanded around inf to get
      \[\color{red}{\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2}} \leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2}}\]
      11.8
    8. Applied simplify to get
      \[\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2} \leadsto \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\]
      0.9
    9. Applied final simplification

    if -5.671665100714436e+89 < y.re < 6.209412377058245e+96

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      17.7
  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))))