\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 6.8 s
Input Error: 12.6
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\frac{b}{\sqrt{c^2 + d^2}^*} \cdot \frac{c}{\sqrt{c^2 + d^2}^*} - \frac{a}{\sqrt{c^2 + d^2}^*} \cdot \frac{d}{\sqrt{c^2 + d^2}^*}\)
  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
    \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
    12.5
  4. Applied simplify to get
    \[\frac{b \cdot c - a \cdot d}{{\color{red}{\left(\sqrt{{c}^2 + {d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\color{blue}{\left(\sqrt{c^2 + d^2}^*\right)}}^2}\]
    8.0
  5. Using strategy rm
    8.0
  6. Applied flip-- to get
    \[\frac{\color{red}{b \cdot c - a \cdot d}}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{\color{blue}{\frac{{\left(b \cdot c\right)}^2 - {\left(a \cdot d\right)}^2}{b \cdot c + a \cdot d}}}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
    17.5
  7. Applied associate-/l/ to get
    \[\color{red}{\frac{\frac{{\left(b \cdot c\right)}^2 - {\left(a \cdot d\right)}^2}{b \cdot c + a \cdot d}}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \color{blue}{\frac{{\left(b \cdot c\right)}^2 - {\left(a \cdot d\right)}^2}{{\left(\sqrt{c^2 + d^2}^*\right)}^2 \cdot \left(b \cdot c + a \cdot d\right)}}\]
    17.5
  8. Applied taylor to get
    \[\frac{{\left(b \cdot c\right)}^2 - {\left(a \cdot d\right)}^2}{{\left(\sqrt{c^2 + d^2}^*\right)}^2 \cdot \left(b \cdot c + a \cdot d\right)} \leadsto \frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
    8.0
  9. Taylor expanded around 0 to get
    \[\color{red}{\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
    8.0
  10. Applied simplify to get
    \[\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{b}{\sqrt{c^2 + d^2}^*} \cdot \frac{c}{\sqrt{c^2 + d^2}^*} - \frac{a}{\sqrt{c^2 + d^2}^*} \cdot \frac{d}{\sqrt{c^2 + d^2}^*}\]
    0.0
  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))))))