\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 11.5 s
Input Error: 25.5
Output Error: 1.5
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
    \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
    25.5
  2. Using strategy rm
    25.5
  3. Applied div-inv to get
    \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\left(b \cdot c - a \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2}}\]
    25.7
  4. Using strategy rm
    25.7
  5. Applied add-sqr-sqrt to get
    \[\left(b \cdot c - a \cdot d\right) \cdot \color{red}{\frac{1}{{c}^2 + {d}^2}} \leadsto \left(b \cdot c - a \cdot d\right) \cdot \color{blue}{{\left(\sqrt{\frac{1}{{c}^2 + {d}^2}}\right)}^2}\]
    25.8
  6. Using strategy rm
    25.8
  7. Applied sqrt-div to get
    \[\left(b \cdot c - a \cdot d\right) \cdot {\color{red}{\left(\sqrt{\frac{1}{{c}^2 + {d}^2}}\right)}}^2 \leadsto \left(b \cdot c - a \cdot d\right) \cdot {\color{blue}{\left(\frac{\sqrt{1}}{\sqrt{{c}^2 + {d}^2}}\right)}}^2\]
    25.8
  8. Applied simplify to get
    \[\left(b \cdot c - a \cdot d\right) \cdot {\left(\frac{\sqrt{1}}{\color{red}{\sqrt{{c}^2 + {d}^2}}}\right)}^2 \leadsto \left(b \cdot c - a \cdot d\right) \cdot {\left(\frac{\sqrt{1}}{\color{blue}{\sqrt{c^2 + d^2}^*}}\right)}^2\]
    25.4
  9. Applied taylor to get
    \[\left(b \cdot c - a \cdot d\right) \cdot {\left(\frac{\sqrt{1}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto \frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
    25.6
  10. Taylor expanded around 0 to get
    \[\color{red}{\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
    25.6
  11. Applied simplify to get
    \[\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{b}{\sqrt{c^2 + d^2}^*} \cdot \frac{c}{\sqrt{c^2 + d^2}^*} - \frac{a}{\sqrt{c^2 + d^2}^*} \cdot \frac{d}{\sqrt{c^2 + d^2}^*}\]
    1.5
  12. Applied final simplification
  13. 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))))))