\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 10.4 s
Input Error: 25.0
Output Error: 13.4
Log: ⚲
Profile: 🕒
\(\begin{cases} -\frac{a}{d} & \text{when } d \le -2.0646128749844613 \cdot 10^{+129} \\ \left(b \cdot c - a \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2} & \text{when } d \le 1.2610866829211587 \cdot 10^{+151} \\ -\frac{a}{d} & \text{otherwise} \end{cases}\)

    if d < -2.0646128749844613e+129 or 1.2610866829211587e+151 < d

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      41.7
    2. Using strategy rm
      41.7
    3. Applied add-exp-log to get
      \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{e^{\log \left(\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\right)}}\]
      42.7
    4. Applied taylor to get
      \[e^{\log \left(\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\right)} \leadsto e^{\left(\log a + \log -1\right) - \log d}\]
      62.9
    5. Taylor expanded around 0 to get
      \[e^{\color{red}{\left(\log a + \log -1\right) - \log d}} \leadsto e^{\color{blue}{\left(\log a + \log -1\right) - \log d}}\]
      62.9
    6. Applied simplify to get
      \[e^{\left(\log a + \log -1\right) - \log d} \leadsto \frac{-1}{d} \cdot a\]
      0.2
    7. Applied final simplification
    8. Applied simplify to get
      \[\color{red}{\frac{-1}{d} \cdot a} \leadsto \color{blue}{-\frac{a}{d}}\]
      0

    if -2.0646128749844613e+129 < d < 1.2610866829211587e+151

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      18.4
    2. Using strategy rm
      18.4
    3. Applied div-inv to get
      \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\left(b \cdot c - a \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2}}\]
      18.6
  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))))))