\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 9.5 s
Input Error: 24.8
Output Error: 13.8
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{a}{c} & \text{when } c \le -1.8410844372587685 \cdot 10^{+146} \\ \frac{1}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}} & \text{when } c \le 6.566722189442364 \cdot 10^{+163} \\ \frac{a}{c} & \text{otherwise} \end{cases}\)

    if c < -1.8410844372587685e+146 or 6.566722189442364e+163 < c

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      42.1
    2. Using strategy rm
      42.1
    3. Applied clear-num to get
      \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}}\]
      42.1
    4. Using strategy rm
      42.1
    5. Applied add-cube-cbrt to get
      \[\frac{1}{\color{red}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}} \leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}\right)}^3}}\]
      42.1
    6. Applied taylor to get
      \[\frac{1}{{\left(\sqrt[3]{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}\right)}^3} \leadsto \frac{a}{c}\]
      0
    7. Taylor expanded around 0 to get
      \[\color{red}{\frac{a}{c}} \leadsto \color{blue}{\frac{a}{c}}\]
      0
    8. Applied simplify to get
      \[\frac{a}{c} \leadsto \frac{a}{c}\]
      0
    9. Applied final simplification

    if -1.8410844372587685e+146 < c < 6.566722189442364e+163

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      18.6
    2. Using strategy rm
      18.6
    3. Applied clear-num to get
      \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}}\]
      18.7
  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))))))