\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
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: 26.7
Output Error: 11.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{y.im}{y.re} \cdot \frac{x.im}{y.re} + \frac{x.re}{y.re} & \text{when } y.re \le -1.3770151188228676 \cdot 10^{+30} \\ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} & \text{when } y.re \le 4.6559433524913966 \cdot 10^{+129} \\ \frac{y.im}{y.re} \cdot \frac{x.im}{y.re} + \frac{x.re}{y.re} & \text{otherwise} \end{cases}\)

    if y.re < -1.3770151188228676e+30 or 4.6559433524913966e+129 < y.re

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      39.8
    2. Using strategy rm
      39.8
    3. Applied clear-num to get
      \[\color{red}{\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.re \cdot y.re + x.im \cdot y.im}}}\]
      39.9
    4. Applied simplify to get
      \[\frac{1}{\color{red}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.re \cdot y.re + x.im \cdot y.im}}} \leadsto \frac{1}{\color{blue}{\frac{{y.re}^2 + y.im \cdot y.im}{y.re \cdot x.re + x.im \cdot y.im}}}\]
      39.9
    5. Using strategy rm
      39.9
    6. Applied add-cbrt-cube to get
      \[\frac{1}{\frac{{y.re}^2 + y.im \cdot y.im}{\color{red}{y.re \cdot x.re + x.im \cdot y.im}}} \leadsto \frac{1}{\frac{{y.re}^2 + y.im \cdot y.im}{\color{blue}{\sqrt[3]{{\left(y.re \cdot x.re + x.im \cdot y.im\right)}^3}}}}\]
      51.0
    7. Applied add-cbrt-cube to get
      \[\frac{1}{\frac{\color{red}{{y.re}^2 + y.im \cdot y.im}}{\sqrt[3]{{\left(y.re \cdot x.re + x.im \cdot y.im\right)}^3}}} \leadsto \frac{1}{\frac{\color{blue}{\sqrt[3]{{\left({y.re}^2 + y.im \cdot y.im\right)}^3}}}{\sqrt[3]{{\left(y.re \cdot x.re + x.im \cdot y.im\right)}^3}}}\]
      54.2
    8. Applied cbrt-undiv to get
      \[\frac{1}{\color{red}{\frac{\sqrt[3]{{\left({y.re}^2 + y.im \cdot y.im\right)}^3}}{\sqrt[3]{{\left(y.re \cdot x.re + x.im \cdot y.im\right)}^3}}}} \leadsto \frac{1}{\color{blue}{\sqrt[3]{\frac{{\left({y.re}^2 + y.im \cdot y.im\right)}^3}{{\left(y.re \cdot x.re + x.im \cdot y.im\right)}^3}}}}\]
      54.5
    9. Applied simplify to get
      \[\frac{1}{\sqrt[3]{\color{red}{\frac{{\left({y.re}^2 + y.im \cdot y.im\right)}^3}{{\left(y.re \cdot x.re + x.im \cdot y.im\right)}^3}}}} \leadsto \frac{1}{\sqrt[3]{\color{blue}{{\left(\frac{{y.re}^2 + y.im \cdot y.im}{x.im \cdot y.im + y.re \cdot x.re}\right)}^3}}}\]
      44.8
    10. Applied taylor to get
      \[\frac{1}{\sqrt[3]{{\left(\frac{{y.re}^2 + y.im \cdot y.im}{x.im \cdot y.im + y.re \cdot x.re}\right)}^3}} \leadsto \frac{y.im \cdot x.im}{{y.re}^2} + \frac{x.re}{y.re}\]
      11.0
    11. Taylor expanded around 0 to get
      \[\color{red}{\frac{y.im \cdot x.im}{{y.re}^2} + \frac{x.re}{y.re}} \leadsto \color{blue}{\frac{y.im \cdot x.im}{{y.re}^2} + \frac{x.re}{y.re}}\]
      11.0
    12. Applied simplify to get
      \[\frac{y.im \cdot x.im}{{y.re}^2} + \frac{x.re}{y.re} \leadsto \frac{y.im}{y.re} \cdot \frac{x.im}{y.re} + \frac{x.re}{y.re}\]
      0.7
    13. Applied final simplification

    if -1.3770151188228676e+30 < y.re < 4.6559433524913966e+129

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      18.3
  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))))