\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
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: 13.2 s
Input Error: 13.0
Output Error: 3.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re} & \text{when } y.re \le -2.19141f+06 \\ \left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \frac{1}{y.re \cdot y.re + y.im \cdot y.im} & \text{when } y.re \le -3.0482872f-20 \\ \frac{y.re}{y.im} \cdot \frac{x.re}{y.im} + \frac{x.im}{y.im} & \text{when } y.re \le 2.9389908f-22 \\ {\left(\sqrt[3]{\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}}\right)}^3 & \text{when } y.re \le 1.8772478f+13 \\ \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re} & \text{otherwise} \end{cases}\)

    if y.re < -2.19141f+06 or 1.8772478f+13 < y.re

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      19.6
    2. Using strategy rm
      19.6
    3. Applied add-sqr-sqrt to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{\color{red}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{{\left(\sqrt{y.re \cdot y.re + y.im \cdot y.im}\right)}^2}}\]
      19.6
    4. Applied simplify to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{{\color{red}{\left(\sqrt{y.re \cdot y.re + y.im \cdot y.im}\right)}}^2} \leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{{\color{blue}{\left(\sqrt{{y.re}^2 + y.im \cdot y.im}\right)}}^2}\]
      19.6
    5. Using strategy rm
      19.6
    6. Applied add-cube-cbrt to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{{\color{red}{\left(\sqrt{{y.re}^2 + y.im \cdot y.im}\right)}}^2} \leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{{\color{blue}{\left({\left(\sqrt[3]{\sqrt{{y.re}^2 + y.im \cdot y.im}}\right)}^3\right)}}^2}\]
      19.7
    7. Applied taylor to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{{\left({\left(\sqrt[3]{\sqrt{{y.re}^2 + y.im \cdot y.im}}\right)}^3\right)}^2} \leadsto \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2}\]
      5.3
    8. Taylor expanded around inf to get
      \[\color{red}{\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2}} \leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2}}\]
      5.3
    9. Applied simplify to get
      \[\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^2} \leadsto \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\]
      0.4
    10. Applied final simplification

    if -2.19141f+06 < y.re < -3.0482872f-20

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      7.2
    2. Using strategy rm
      7.2
    3. Applied div-inv to get
      \[\color{red}{\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \frac{1}{y.re \cdot y.re + y.im \cdot y.im}}\]
      7.3

    if -3.0482872f-20 < y.re < 2.9389908f-22

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      10.7
    2. Using strategy rm
      10.7
    3. Applied add-sqr-sqrt to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{\color{red}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{{\left(\sqrt{y.re \cdot y.re + y.im \cdot y.im}\right)}^2}}\]
      10.7
    4. Applied simplify to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{{\color{red}{\left(\sqrt{y.re \cdot y.re + y.im \cdot y.im}\right)}}^2} \leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{{\color{blue}{\left(\sqrt{{y.re}^2 + y.im \cdot y.im}\right)}}^2}\]
      10.7
    5. Applied taylor to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{{\left(\sqrt{{y.re}^2 + y.im \cdot y.im}\right)}^2} \leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{{y.im}^2}\]
      10.7
    6. Taylor expanded around 0 to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{{\color{red}{y.im}}^2} \leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{{\color{blue}{y.im}}^2}\]
      10.7
    7. Applied taylor to get
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{{y.im}^2} \leadsto \frac{y.re \cdot x.re}{{y.im}^2} + \frac{x.im}{y.im}\]
      3.3
    8. Taylor expanded around 0 to get
      \[\color{red}{\frac{y.re \cdot x.re}{{y.im}^2} + \frac{x.im}{y.im}} \leadsto \color{blue}{\frac{y.re \cdot x.re}{{y.im}^2} + \frac{x.im}{y.im}}\]
      3.3
    9. Applied simplify to get
      \[\frac{y.re \cdot x.re}{{y.im}^2} + \frac{x.im}{y.im} \leadsto \frac{y.re}{y.im} \cdot \frac{x.re}{y.im} + \frac{x.im}{y.im}\]
      1.3
    10. Applied final simplification

    if 2.9389908f-22 < y.re < 1.8772478f+13

    1. Started with
      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
      7.7
    2. Using strategy rm
      7.7
    3. Applied add-cube-cbrt to get
      \[\color{red}{\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}}\right)}^3}\]
      8.0
  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))))