\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 11.7 s
Input Error: 25.5
Output Error: 25.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(b \cdot c - a \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2} & \text{when } \frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2} \le 0.0 \\ {\left(\frac{\sqrt{c \cdot b - d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2 & \text{otherwise} \end{cases}\)

    if (/ (- (* b c) (* a d)) (+ (sqr c) (sqr d))) < 0.0

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      19.1
    2. Using strategy rm
      19.1
    3. Applied div-inv to get
      \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\left(b \cdot c - a \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2}}\]
      19.2

    if 0.0 < (/ (- (* b c) (* a d)) (+ (sqr c) (sqr d)))

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      33.0
    2. Using strategy rm
      33.0
    3. Applied add-sqr-sqrt to get
      \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
      33.0
    4. Applied add-sqr-sqrt to get
      \[\frac{\color{red}{b \cdot c - a \cdot d}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{\color{blue}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}\]
      33.1
    5. Applied square-undiv to get
      \[\color{red}{\frac{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}} \leadsto \color{blue}{{\left(\frac{\sqrt{b \cdot c - a \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}^2}\]
      33.2
    6. Applied simplify to get
      \[{\color{red}{\left(\frac{\sqrt{b \cdot c - a \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}}^2 \leadsto {\color{blue}{\left(\frac{\sqrt{c \cdot b - d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}}^2\]
      31.9
  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))))))