\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 11.5 s
Input Error: 12.5
Output Error: 6.9
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\frac{{\left(\left|\frac{-1}{d}\right|\right)}^2 \cdot \left(d \cdot a\right)}{{c}^{4}} + \frac{b}{c}\right) - \frac{d \cdot a}{{c}^2} & \text{when } c \le -9.155065f+15 \\ \frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {\left(\left|d\right|\right)}^2} & \text{otherwise} \end{cases}\)

    if c < -9.155065f+15

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      21.2
    2. Using strategy rm
      21.2
    3. Applied add-sqr-sqrt to get
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
      21.2
    4. Applied simplify to get
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
      21.2
    5. Applied taylor to get
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2} \leadsto \left(\frac{{\left(\left|\frac{-1}{d}\right|\right)}^2 \cdot \left(d \cdot a\right)}{{c}^{4}} + \frac{b}{c}\right) - \frac{d \cdot a}{{c}^2}\]
      6.2
    6. Taylor expanded around -inf to get
      \[\color{red}{\left(\frac{{\left(\left|\frac{-1}{d}\right|\right)}^2 \cdot \left(d \cdot a\right)}{{c}^{4}} + \frac{b}{c}\right) - \frac{d \cdot a}{{c}^2}} \leadsto \color{blue}{\left(\frac{{\left(\left|\frac{-1}{d}\right|\right)}^2 \cdot \left(d \cdot a\right)}{{c}^{4}} + \frac{b}{c}\right) - \frac{d \cdot a}{{c}^2}}\]
      6.2

    if -9.155065f+15 < c

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      10.9
    2. Using strategy rm
      10.9
    3. Applied add-sqr-sqrt to get
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
      10.9
    4. Applied simplify to get
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
      8.6
    5. Using strategy rm
      8.6
    6. Applied add-sqr-sqrt to get
      \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2} + {\left(\left|d\right|\right)}^2} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {\left(\left|d\right|\right)}^2}\]
      8.6
    7. Applied simplify to get
      \[\frac{b \cdot c - a \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {\left(\left|d\right|\right)}^2}\]
      7.0
  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))))))