\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
Test:
Complex division, imag part
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Bits error versus d
Time: 9.4 s
Input Error: 25.5
Output Error: 22.9
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{\frac{{c}^2 + {d}^2}{d}} & \text{when } d \le -7.917566533110314 \cdot 10^{+26} \\ \frac{b}{\frac{{c}^2 + {d}^2}{c}} - \frac{a \cdot d}{{c}^2 + {d}^2} & \text{when } d \le 4.576801550953589 \cdot 10^{+76} \\ \frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{\frac{{c}^2 + {d}^2}{d}} & \text{otherwise} \end{cases}\)

    if d < -7.917566533110314e+26 or 4.576801550953589e+76 < d

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      35.0
    2. Using strategy rm
      35.0
    3. Applied div-sub to get
      \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{{c}^2 + {d}^2}}\]
      35.0
    4. Using strategy rm
      35.0
    5. Applied associate-/l* to get
      \[\frac{b \cdot c}{{c}^2 + {d}^2} - \color{red}{\frac{a \cdot d}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - \color{blue}{\frac{a}{\frac{{c}^2 + {d}^2}{d}}}\]
      32.2

    if -7.917566533110314e+26 < d < 4.576801550953589e+76

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      18.1
    2. Using strategy rm
      18.1
    3. Applied div-sub to get
      \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{{c}^2 + {d}^2}}\]
      18.2
    4. Using strategy rm
      18.2
    5. Applied associate-/l* to get
      \[\color{red}{\frac{b \cdot c}{{c}^2 + {d}^2}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \color{blue}{\frac{b}{\frac{{c}^2 + {d}^2}{c}}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
      15.6
  1. Removed slow pow expressions

Original test:


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