\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 11.8 s
Input Error: 12.8
Output Error: 2.8
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{b}{c} - \frac{b \cdot {\left(\left|\frac{1}{d}\right|\right)}^2}{{c}^{3}} & \text{when } c \le -1.2816005f+19 \\ \frac{b \cdot c - {\left(\sqrt[3]{a} \cdot \sqrt[3]{d}\right)}^3}{{c}^2 + {\left(\left|d\right|\right)}^2} & \text{when } c \le 2.1933226f+17 \\ \frac{b}{c} - \frac{b \cdot {\left(\left|\frac{1}{d}\right|\right)}^2}{{c}^{3}} & \text{otherwise} \end{cases}\)

    if c < -1.2816005f+19 or 2.1933226f+17 < c

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      21.8
    2. Using strategy rm
      21.8
    3. Applied add-sqr-sqrt to get
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
      21.8
    4. Applied simplify to get
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
      21.8
    5. Using strategy rm
      21.8
    6. Applied add-cube-cbrt to get
      \[\frac{b \cdot c - \color{red}{a \cdot d}}{{c}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{b \cdot c - \color{blue}{{\left(\sqrt[3]{a \cdot d}\right)}^3}}{{c}^2 + {\left(\left|d\right|\right)}^2}\]
      21.8
    7. Applied taylor to get
      \[\frac{b \cdot c - {\left(\sqrt[3]{a \cdot d}\right)}^3}{{c}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{b}{c} - \frac{b \cdot {\left(\left|\frac{1}{d}\right|\right)}^2}{{c}^{3}}\]
      3.5
    8. Taylor expanded around inf to get
      \[\color{red}{\frac{b}{c} - \frac{b \cdot {\left(\left|\frac{1}{d}\right|\right)}^2}{{c}^{3}}} \leadsto \color{blue}{\frac{b}{c} - \frac{b \cdot {\left(\left|\frac{1}{d}\right|\right)}^2}{{c}^{3}}}\]
      3.5

    if -1.2816005f+19 < c < 2.1933226f+17

    1. Started with
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
      9.2
    2. Using strategy rm
      9.2
    3. Applied add-sqr-sqrt to get
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
      9.1
    4. Applied simplify to get
      \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
      6.3
    5. Using strategy rm
      6.3
    6. Applied add-cube-cbrt to get
      \[\frac{b \cdot c - \color{red}{a \cdot d}}{{c}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{b \cdot c - \color{blue}{{\left(\sqrt[3]{a \cdot d}\right)}^3}}{{c}^2 + {\left(\left|d\right|\right)}^2}\]
      6.5
    7. Using strategy rm
      6.5
    8. Applied cbrt-prod to get
      \[\frac{b \cdot c - {\color{red}{\left(\sqrt[3]{a \cdot d}\right)}}^3}{{c}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{b \cdot c - {\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{d}\right)}}^3}{{c}^2 + {\left(\left|d\right|\right)}^2}\]
      2.6
  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))))))