\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 12.1 s
Input Error: 24.3
Output Error: 21.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{1}{\frac{{c}^2 + {d}^2}{b \cdot c - a \cdot d}} & \text{when } d \le 1.8512012586360278 \cdot 10^{+143} \\ \frac{1}{{\left(\frac{\sqrt{c^2 + d^2}^*}{\sqrt{c \cdot b - a \cdot d}}\right)}^2} & \text{when } d \le 2.1269122109662698 \cdot 10^{+212} \\ 0 & \text{otherwise} \end{cases}\)

    if d < 1.8512012586360278e+143

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      22.9
    2. Using strategy rm
      22.9
    3. Applied clear-num to get
      \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{b \cdot c - a \cdot d}}}\]
      23.0

    if 1.8512012586360278e+143 < d < 2.1269122109662698e+212

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      48.1
    2. Using strategy rm
      48.1
    3. Applied clear-num to get
      \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{b \cdot c - a \cdot d}}}\]
      48.1
    4. Using strategy rm
      48.1
    5. Applied add-sqr-sqrt to get
      \[\frac{1}{\frac{{c}^2 + {d}^2}{\color{red}{b \cdot c - a \cdot d}}} \leadsto \frac{1}{\frac{{c}^2 + {d}^2}{\color{blue}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}}}\]
      48.1
    6. Applied add-sqr-sqrt to get
      \[\frac{1}{\frac{\color{red}{{c}^2 + {d}^2}}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}} \leadsto \frac{1}{\frac{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}}\]
      48.1
    7. Applied square-undiv to get
      \[\frac{1}{\color{red}{\frac{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}}} \leadsto \frac{1}{\color{blue}{{\left(\frac{\sqrt{{c}^2 + {d}^2}}{\sqrt{b \cdot c - a \cdot d}}\right)}^2}}\]
      48.2
    8. Applied simplify to get
      \[\frac{1}{{\color{red}{\left(\frac{\sqrt{{c}^2 + {d}^2}}{\sqrt{b \cdot c - a \cdot d}}\right)}}^2} \leadsto \frac{1}{{\color{blue}{\left(\frac{\sqrt{c^2 + d^2}^*}{\sqrt{c \cdot b - a \cdot d}}\right)}}^2}\]
      27.7

    if 2.1269122109662698e+212 < d

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      31.2
    2. Applied taylor to get
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2} \leadsto 0\]
      0
    3. Taylor expanded around 0 to get
      \[\color{red}{0} \leadsto \color{blue}{0}\]
      0
  1. 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))))))