\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
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: 11.0 s
Input Error: 25.3
Output Error: 13.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{x.re}{y.re} & \text{when } y.re \le -2.709243247914685 \cdot 10^{+149} \\ \left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \frac{1}{y.re \cdot y.re + y.im \cdot y.im} & \text{when } y.re \le 1.0663382734753051 \cdot 10^{+99} \\ \frac{x.re}{y.re} & \text{otherwise} \end{cases}\)

    if y.re < -2.709243247914685e+149 or 1.0663382734753051e+99 < y.re

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      40.4
    2. Using strategy rm
      40.4
    3. Applied add-cbrt-cube to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{\color{red}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt[3]{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^3}}}\]
      43.1
    4. Applied add-cbrt-cube to get
      \[\frac{\color{red}{x.re \cdot y.re + x.im \cdot y.im}}{\sqrt[3]{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^3}} \leadsto \frac{\color{blue}{\sqrt[3]{{\left(x.re \cdot y.re + x.im \cdot y.im\right)}^3}}}{\sqrt[3]{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^3}}\]
      52.3
    5. Applied cbrt-undiv to get
      \[\color{red}{\frac{\sqrt[3]{{\left(x.re \cdot y.re + x.im \cdot y.im\right)}^3}}{\sqrt[3]{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^3}}} \leadsto \color{blue}{\sqrt[3]{\frac{{\left(x.re \cdot y.re + x.im \cdot y.im\right)}^3}{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^3}}}\]
      52.3
    6. Applied simplify to get
      \[\sqrt[3]{\color{red}{\frac{{\left(x.re \cdot y.re + x.im \cdot y.im\right)}^3}{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^3}}} \leadsto \sqrt[3]{\color{blue}{{\left(\frac{y.re \cdot x.re + y.im \cdot x.im}{y.im \cdot y.im + y.re \cdot y.re}\right)}^3}}\]
      41.5
    7. Applied taylor to get
      \[\sqrt[3]{{\left(\frac{y.re \cdot x.re + y.im \cdot x.im}{y.im \cdot y.im + y.re \cdot y.re}\right)}^3} \leadsto \frac{x.re}{y.re}\]
      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 -2.709243247914685e+149 < y.re < 1.0663382734753051e+99

    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
    2. Using strategy rm
      18.6
    3. Applied div-inv to get
      \[\color{red}{\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \frac{1}{y.re \cdot y.re + y.im \cdot y.im}}\]
      18.9
  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))))