\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 10.3 s
Input Error: 25.4
Output Error: 23.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{b}{\frac{{c}^2 + {d}^2}{c}} - \frac{a \cdot d}{{c}^2 + {d}^2} & \text{when } c \le -2.0245393610925515 \cdot 10^{-226} \\ \frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{1} \cdot \frac{d}{{c}^2 + {d}^2} & \text{when } c \le 1.0222346456089473 \cdot 10^{+104} \\ \frac{b}{\frac{{c}^2 + {d}^2}{c}} - \frac{a \cdot d}{{c}^2 + {d}^2} & \text{otherwise} \end{cases}\)

    if c < -2.0245393610925515e-226 or 1.0222346456089473e+104 < c

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      29.4
    2. Using strategy rm
      29.4
    3. Applied div-sub to get
      \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{{c}^2 + {d}^2}}\]
      29.5
    4. Using strategy rm
      29.5
    5. Applied associate-/l* to get
      \[\color{red}{\frac{b \cdot c}{{c}^2 + {d}^2}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \color{blue}{\frac{b}{\frac{{c}^2 + {d}^2}{c}}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
      27.9

    if -2.0245393610925515e-226 < 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))))))