\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 9.9 s
Input Error: 12.6
Output Error: 8.0
Log: ⚲
Profile: 🕒
\(\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^* \cdot \sqrt{c^2 + d^2}^*}\)
  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
    \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
    12.6
  4. Applied add-sqr-sqrt to get
    \[\frac{\color{red}{b \cdot c - a \cdot d}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{\color{blue}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}\]
    22.0
  5. Applied square-undiv to get
    \[\color{red}{\frac{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}} \leadsto \color{blue}{{\left(\frac{\sqrt{b \cdot c - a \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}^2}\]
    22.0
  6. Applied simplify to get
    \[{\color{red}{\left(\frac{\sqrt{b \cdot c - a \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}}^2 \leadsto {\color{blue}{\left(\frac{\sqrt{c \cdot b - d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}}^2\]
    19.8
  7. Applied taylor to get
    \[{\left(\frac{\sqrt{c \cdot b - d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto {\left(\frac{\sqrt{c \cdot b - d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2\]
    19.8
  8. Taylor expanded around 0 to get
    \[{\left(\frac{\sqrt{c \cdot b - \color{red}{d \cdot a}}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto {\left(\frac{\sqrt{c \cdot b - \color{blue}{d \cdot a}}}{\sqrt{c^2 + d^2}^*}\right)}^2\]
    19.8
  9. Applied simplify to get
    \[{\left(\frac{\sqrt{c \cdot b - d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto \frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^* \cdot \sqrt{c^2 + d^2}^*}\]
    8.0
  10. 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))))))