\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
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: 14.4 s
Input Error: 14.6
Output Error: 1.9
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{x.im}{y.im} & \text{when } y.im \le -3.1052615f+11 \\ \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re} & \text{when } y.im \le 9.972299f-21 \\ \frac{1}{\frac{{y.re}^2 + y.im \cdot y.im}{y.re \cdot x.re + x.im \cdot y.im}} & \text{when } y.im \le 1.1137253f+10 \\ \frac{x.im}{y.im} & \text{otherwise} \end{cases}\)

    if y.im < -3.1052615f+11 or 1.1137253f+10 < y.im

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      19.6
    2. Using strategy rm
      19.6
    3. Applied add-exp-log to get
      \[\color{red}{\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{e^{\log \left(\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right)}}\]
      21.6
    4. Applied taylor to get
      \[e^{\log \left(\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right)} \leadsto e^{\log x.im - \log y.im}\]
      23.8
    5. Taylor expanded around 0 to get
      \[e^{\color{red}{\log x.im - \log y.im}} \leadsto e^{\color{blue}{\log x.im - \log y.im}}\]
      23.8
    6. Applied simplify to get
      \[e^{\log x.im - \log y.im} \leadsto \frac{x.im}{y.im}\]
      0
    7. Applied final simplification

    if -3.1052615f+11 < y.im < 9.972299f-21

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      13.2
    2. Using strategy rm
      13.2
    3. Applied add-cube-cbrt to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{\color{red}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{{\left(\sqrt[3]{y.re \cdot y.re + y.im \cdot y.im}\right)}^3}}\]
      13.4
    4. Applied simplify to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{{\color{red}{\left(\sqrt[3]{y.re \cdot y.re + y.im \cdot y.im}\right)}}^3} \leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{{\color{blue}{\left(\sqrt[3]{{y.re}^2 + y.im \cdot y.im}\right)}}^3}\]
      13.4
    5. Applied taylor to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{{\left(\sqrt[3]{{y.re}^2 + y.im \cdot y.im}\right)}^3} \leadsto \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2}\]
      3.2
    6. Taylor expanded around inf to get
      \[\color{red}{\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2}} \leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2}}\]
      3.2
    7. Applied simplify to get
      \[\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2} \leadsto \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\]
      0.9
    8. Applied final simplification

    if 9.972299f-21 < y.im < 1.1137253f+10

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      7.1
    2. Using strategy rm
      7.1
    3. Applied clear-num to get
      \[\color{red}{\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.re \cdot y.re + x.im \cdot y.im}}}\]
      7.3
    4. Applied simplify to get
      \[\frac{1}{\color{red}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.re \cdot y.re + x.im \cdot y.im}}} \leadsto \frac{1}{\color{blue}{\frac{{y.re}^2 + y.im \cdot y.im}{y.re \cdot x.re + x.im \cdot y.im}}}\]
      7.3
  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))))