\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 6.6 s
Input Error: 12.5
Output Error: 7.9
Log: ⚲
Profile: 🕒
\(\frac{c \cdot b}{\left|d\right| \cdot \left|d\right| + {\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{\left|d\right| \cdot \left|d\right| + {\left(\left|c\right|\right)}^2}\)
  1. Started with
    \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
    12.5
  2. Using strategy rm
    12.5
  3. Applied add-sqr-sqrt to get
    \[\frac{b \cdot c - a \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
    12.5
  4. Applied simplify to get
    \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
    10.5
  5. Using strategy rm
    10.5
  6. Applied add-sqr-sqrt to get
    \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2} + {\left(\left|d\right|\right)}^2} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {\left(\left|d\right|\right)}^2}\]
    10.5
  7. Applied simplify to get
    \[\frac{b \cdot c - a \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {\left(\left|d\right|\right)}^2}\]
    7.9
  8. Applied taylor to get
    \[\frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{b \cdot c}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2}\]
    7.9
  9. Taylor expanded around 0 to get
    \[\color{red}{\frac{b \cdot c}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2}}\]
    7.9
  10. Applied simplify to get
    \[\frac{b \cdot c}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2} \leadsto \frac{c \cdot b}{\left|d\right| \cdot \left|d\right| + {\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{\left|d\right| \cdot \left|d\right| + {\left(\left|c\right|\right)}^2}\]
    7.9
  11. Applied final simplification

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))))))