\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 8.9 s
Input Error: 25.5
Output Error: 17.2
Log: ⚲
Profile: 🕒
\(\frac{b}{\frac{d \cdot d}{c} + c} - \frac{a \cdot d}{c \cdot c + d \cdot d}\)
  1. Started with
    \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
    25.5
  2. Using strategy rm
    25.5
  3. Applied div-sub to get
    \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{{c}^2 + {d}^2}}\]
    25.6
  4. Using strategy rm
    25.6
  5. Applied associate-/l* to get
    \[\frac{b \cdot c}{{c}^2 + {d}^2} - \color{red}{\frac{a \cdot d}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - \color{blue}{\frac{a}{\frac{{c}^2 + {d}^2}{d}}}\]
    24.4
  6. Using strategy rm
    24.4
  7. Applied associate-/l* to get
    \[\color{red}{\frac{b \cdot c}{{c}^2 + {d}^2}} - \frac{a}{\frac{{c}^2 + {d}^2}{d}} \leadsto \color{blue}{\frac{b}{\frac{{c}^2 + {d}^2}{c}}} - \frac{a}{\frac{{c}^2 + {d}^2}{d}}\]
    23.0
  8. Applied taylor to get
    \[\frac{b}{\frac{{c}^2 + {d}^2}{c}} - \frac{a}{\frac{{c}^2 + {d}^2}{d}} \leadsto \frac{b}{c + \frac{{d}^2}{c}} - \frac{a}{\frac{{c}^2 + {d}^2}{d}}\]
    15.8
  9. Taylor expanded around 0 to get
    \[\frac{b}{\color{red}{c + \frac{{d}^2}{c}}} - \frac{a}{\frac{{c}^2 + {d}^2}{d}} \leadsto \frac{b}{\color{blue}{c + \frac{{d}^2}{c}}} - \frac{a}{\frac{{c}^2 + {d}^2}{d}}\]
    15.8
  10. Applied simplify to get
    \[\frac{b}{c + \frac{{d}^2}{c}} - \frac{a}{\frac{{c}^2 + {d}^2}{d}} \leadsto \frac{b}{\frac{d \cdot d}{c} + c} - \frac{a \cdot d}{c \cdot c + d \cdot d}\]
    17.2
  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))))))