\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 12.9 s
Input Error: 25.4
Output Error: 15.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{b}{(\left(\frac{d}{c}\right) * d + c)_*} - \frac{d \cdot a}{(c * c + \left(d \cdot d\right))_*} & \text{when } c \le -8.763808252046891 \cdot 10^{-141} \\ \frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{1} \cdot \frac{d}{{c}^2 + {d}^2} & \text{when } c \le 1.0222346456089473 \cdot 10^{+104} \\ \frac{b}{(\left(\frac{d}{c}\right) * d + c)_*} - \frac{d \cdot a}{(c * c + \left(d \cdot d\right))_*} & \text{otherwise} \end{cases}\)

    if c < -8.763808252046891e-141 or 1.0222346456089473e+104 < c

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      30.7
    2. Using strategy rm
      30.7
    3. Applied div-sub to get
      \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{{c}^2 + {d}^2}}\]
      30.7
    4. Using strategy rm
      30.7
    5. Applied associate-/l* to get
      \[\color{red}{\frac{b \cdot c}{{c}^2 + {d}^2}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \color{blue}{\frac{b}{\frac{{c}^2 + {d}^2}{c}}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
      28.8
    6. Applied taylor to get
      \[\frac{b}{\frac{{c}^2 + {d}^2}{c}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \frac{b}{c + \frac{{d}^2}{c}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
      16.4
    7. Taylor expanded around 0 to get
      \[\frac{b}{\color{red}{c + \frac{{d}^2}{c}}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \frac{b}{\color{blue}{c + \frac{{d}^2}{c}}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
      16.4
    8. Applied simplify to get
      \[\frac{b}{c + \frac{{d}^2}{c}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \frac{b}{\frac{d}{\frac{c}{d}} + c} - \frac{a \cdot d}{(c * c + \left(d \cdot d\right))_*}\]
      13.9
    9. Applied final simplification
    10. Applied simplify to get
      \[\color{red}{\frac{b}{\frac{d}{\frac{c}{d}} + c} - \frac{a \cdot d}{(c * c + \left(d \cdot d\right))_*}} \leadsto \color{blue}{\frac{b}{(\left(\frac{d}{c}\right) * d + c)_*} - \frac{d \cdot a}{(c * c + \left(d \cdot d\right))_*}}\]
      13.9

    if -8.763808252046891e-141 < c < 1.0222346456089473e+104

    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-sub to get
      \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{{c}^2 + {d}^2}}\]
      19.1
    4. Using strategy rm
      19.1
    5. Applied *-un-lft-identity to get
      \[\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{\color{blue}{1 \cdot \left({c}^2 + {d}^2\right)}}\]
      19.1
    6. Applied times-frac to get
      \[\frac{b \cdot c}{{c}^2 + {d}^2} - \color{red}{\frac{a \cdot d}{1 \cdot \left({c}^2 + {d}^2\right)}} \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - \color{blue}{\frac{a}{1} \cdot \frac{d}{{c}^2 + {d}^2}}\]
      16.8
  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))))))