\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 8.6 s
Input Error: 25.4
Output Error: 16.0
Log: ⚲
Profile: 🕒
\(\frac{b}{(\left(\frac{d}{c}\right) * d + c)_*} - \frac{d \cdot a}{(c * c + \left(d \cdot d\right))_*}\)
  1. Started with
    \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
    25.4
  2. Using strategy rm
    25.4
  3. Applied div-sub to get
    \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{{c}^2 + {d}^2}}\]
    25.4
  4. Using strategy rm
    25.4
  5. Applied associate-/l* to get
    \[\color{red}{\frac{b \cdot c}{{c}^2 + {d}^2}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \color{blue}{\frac{b}{\frac{{c}^2 + {d}^2}{c}}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
    24.3
  6. Applied taylor to get
    \[\frac{b}{\frac{{c}^2 + {d}^2}{c}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \frac{b}{c + \frac{{d}^2}{c}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
    17.5
  7. Taylor expanded around 0 to get
    \[\frac{b}{\color{red}{c + \frac{{d}^2}{c}}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \frac{b}{\color{blue}{c + \frac{{d}^2}{c}}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
    17.5
  8. Applied simplify to get
    \[\frac{b}{c + \frac{{d}^2}{c}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \frac{b}{\frac{d}{\frac{c}{d}} + c} - \frac{a \cdot d}{(c * c + \left(d \cdot d\right))_*}\]
    16.0
  9. Applied final simplification
  10. Applied simplify to get
    \[\color{red}{\frac{b}{\frac{d}{\frac{c}{d}} + c} - \frac{a \cdot d}{(c * c + \left(d \cdot d\right))_*}} \leadsto \color{blue}{\frac{b}{(\left(\frac{d}{c}\right) * d + c)_*} - \frac{d \cdot a}{(c * c + \left(d \cdot d\right))_*}}\]
    16.0

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))))))