\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 10.8 s
Input Error: 12.9
Output Error: 7.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{a}{c} + \frac{b \cdot d}{{c}^2} & \text{when } c \le -1.898596f+09 \\ \frac{a \cdot c + b \cdot d}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} & \text{when } c \le 1.8772478f+13 \\ \frac{a}{c} + \frac{b \cdot d}{{c}^2} & \text{otherwise} \end{cases}\)

    if c < -1.898596f+09 or 1.8772478f+13 < c

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      19.9
    2. Using strategy rm
      19.9
    3. Applied add-cube-cbrt to get
      \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt[3]{{c}^2 + {d}^2}\right)}^3}}\]
      20.0
    4. Applied taylor to get
      \[\frac{a \cdot c + b \cdot d}{{\left(\sqrt[3]{{c}^2 + {d}^2}\right)}^3} \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
      5.5
    5. Taylor expanded around inf to get
      \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
      5.5

    if -1.898596f+09 < c < 1.8772478f+13

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      8.7
    2. Using strategy rm
      8.7
    3. Applied add-sqr-sqrt to get
      \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
      8.6
  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))))))