\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 6.7 s
Input Error: 12.6
Output Error: 8.0
Log: ⚲
Profile: 🕒
\(\frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {\left(\left|d\right|\right)}^2}\)
  1. Started with
    \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
    12.6
  2. Using strategy rm
    12.6
  3. Applied add-sqr-sqrt to get
    \[\frac{b \cdot c - a \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
    12.6
  4. Applied simplify to get
    \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
    10.4
  5. Using strategy rm
    10.4
  6. Applied add-sqr-sqrt to get
    \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2} + {\left(\left|d\right|\right)}^2} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {\left(\left|d\right|\right)}^2}\]
    10.4
  7. Applied simplify to get
    \[\frac{b \cdot c - a \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {\left(\left|d\right|\right)}^2}\]
    8.0
  8. 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))))))