\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 10.0 s
Input Error: 12.8
Output Error: 6.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c} & \text{when } c \le -2.668639f+09 \\ \frac{b \cdot c - a \cdot d}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} & \text{when } c \le 3.5788512f+20 \\ \frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c} & \text{otherwise} \end{cases}\)

    if c < -2.668639f+09 or 3.5788512f+20 < c

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      20.8
    2. Using strategy rm
      20.8
    3. Applied add-sqr-sqrt to get
      \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
      20.8
    4. Applied taylor to get
      \[\frac{b \cdot c - a \cdot d}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\left(-1 \cdot c\right)}^2}\]
      19.7
    5. Taylor expanded around -inf to get
      \[\frac{b \cdot c - a \cdot d}{{\color{red}{\left(-1 \cdot c\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\color{blue}{\left(-1 \cdot c\right)}}^2}\]
      19.7
    6. Applied simplify to get
      \[\color{red}{\frac{b \cdot c - a \cdot d}{{\left(-1 \cdot c\right)}^2}} \leadsto \color{blue}{\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}}\]
      0.3

    if -2.668639f+09 < c < 3.5788512f+20

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      9.1
    2. Using strategy rm
      9.1
    3. Applied add-sqr-sqrt to get
      \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
      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))))))