\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 8.7 s
Input Error: 25.5
Output Error: 17.1
Log: ⚲
Profile: 🕒
\(\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{d + \frac{{c}^2}{d}}\)
  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. Applied taylor to get
    \[\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{\frac{{c}^2 + {d}^2}{d}} \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{d + \frac{{c}^2}{d}}\]
    17.1
  7. Taylor expanded around 0 to get
    \[\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{\color{red}{d + \frac{{c}^2}{d}}} \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{\color{blue}{d + \frac{{c}^2}{d}}}\]
    17.1

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))))))