\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
Test:
Complex division, real part
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Bits error versus d
Time: 13.1 s
Input Error: 26.6
Output Error: 10.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c} & \text{when } c \le -1.083599015752864 \cdot 10^{+73} \\ \frac{1}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}} & \text{when } c \le 3.586178138959661 \cdot 10^{+58} \\ \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c} & \text{otherwise} \end{cases}\)

    if c < -1.083599015752864e+73 or 3.586178138959661e+58 < c

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      39.4
    2. Using strategy rm
      39.4
    3. Applied div-inv to get
      \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\left(a \cdot c + b \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2}}\]
      39.4
    4. Using strategy rm
      39.4
    5. Applied add-cube-cbrt to get
      \[\left(a \cdot c + b \cdot d\right) \cdot \color{red}{\frac{1}{{c}^2 + {d}^2}} \leadsto \left(a \cdot c + b \cdot d\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{1}{{c}^2 + {d}^2}}\right)}^3}\]
      39.6
    6. Applied taylor to get
      \[\left(a \cdot c + b \cdot d\right) \cdot {\left(\sqrt[3]{\frac{1}{{c}^2 + {d}^2}}\right)}^3 \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
      10.9
    7. Taylor expanded around inf to get
      \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
      10.9
    8. Applied simplify to get
      \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\]
      0.8
    9. Applied final simplification

    if -1.083599015752864e+73 < c < 3.586178138959661e+58

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      17.5
    2. Using strategy rm
      17.5
    3. Applied clear-num to get
      \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}}\]
      17.7
  1. Removed slow pow expressions

Original test:


(lambda ((a default) (b default) (c default) (d default))
  #:name "Complex division, real part"
  (/ (+ (* a c) (* b d)) (+ (sqr c) (sqr d)))
  #:target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d))))))