\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 12.6 s
Input Error: 12.4
Output Error: 6.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{b}{c} & \text{when } c \le -1.5817318f+16 \\ \frac{1}{\frac{{c}^2 + {d}^2}{b \cdot c - a \cdot d}} & \text{when } c \le 6.6680844f+14 \\ \frac{b}{c} & \text{otherwise} \end{cases}\)

    if c < -1.5817318f+16 or 6.6680844f+14 < c

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      20.2
    2. Using strategy rm
      20.2
    3. Applied clear-num to get
      \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{b \cdot c - a \cdot d}}}\]
      20.2
    4. Using strategy rm
      20.2
    5. Applied add-cube-cbrt to get
      \[\frac{1}{\color{red}{\frac{{c}^2 + {d}^2}{b \cdot c - a \cdot d}}} \leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{\frac{{c}^2 + {d}^2}{b \cdot c - a \cdot d}}\right)}^3}}\]
      20.2
    6. Applied taylor to get
      \[\frac{1}{{\left(\sqrt[3]{\frac{{c}^2 + {d}^2}{b \cdot c - a \cdot d}}\right)}^3} \leadsto \frac{b}{c}\]
      0
    7. Taylor expanded around 0 to get
      \[\color{red}{\frac{b}{c}} \leadsto \color{blue}{\frac{b}{c}}\]
      0
    8. Applied simplify to get
      \[\frac{b}{c} \leadsto \frac{b}{c}\]
      0
    9. Applied final simplification

    if -1.5817318f+16 < c < 6.6680844f+14

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      8.8
    2. Using strategy rm
      8.8
    3. Applied clear-num to get
      \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{b \cdot c - a \cdot d}}}\]
      9.0
  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))))))