\[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
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: 25.5
Output Error: 14.8
Log: ⚲
Profile: 🕒
\(\frac{b \cdot c}{(d * d + \left({c}^2\right))_*} - \frac{a}{\sqrt{c^2 + d^2}^*} \cdot \frac{d}{\sqrt{c^2 + d^2}^*}\)
  1. Started with
    \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
    25.5
  2. Using strategy rm
    25.5
  3. Applied div-sub to get
    \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{{c}^2 + {d}^2}}\]
    25.6
  4. Using strategy rm
    25.6
  5. Applied add-sqr-sqrt to get
    \[\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
    25.6
  6. Applied add-sqr-sqrt to get
    \[\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{\color{red}{a \cdot d}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - \frac{\color{blue}{{\left(\sqrt{a \cdot d}\right)}^2}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}\]
    42.1
  7. Applied square-undiv to get
    \[\frac{b \cdot c}{{c}^2 + {d}^2} - \color{red}{\frac{{\left(\sqrt{a \cdot d}\right)}^2}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}} \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - \color{blue}{{\left(\frac{\sqrt{a \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}^2}\]
    42.1
  8. Applied simplify to get
    \[\frac{b \cdot c}{{c}^2 + {d}^2} - {\color{red}{\left(\frac{\sqrt{a \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}}^2 \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - {\color{blue}{\left(\frac{\sqrt{d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}}^2\]
    40.1
  9. Applied taylor to get
    \[\frac{b \cdot c}{{c}^2 + {d}^2} - {\left(\frac{\sqrt{d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto \frac{b \cdot c}{{d}^2 + {c}^2} - {\left(\frac{\sqrt{d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2\]
    40.1
  10. Taylor expanded around 0 to get
    \[\frac{b \cdot c}{\color{red}{{d}^2 + {c}^2}} - {\left(\frac{\sqrt{d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto \frac{b \cdot c}{\color{blue}{{d}^2 + {c}^2}} - {\left(\frac{\sqrt{d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2\]
    40.1
  11. Applied simplify to get
    \[\frac{b \cdot c}{{d}^2 + {c}^2} - {\left(\frac{\sqrt{d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto \frac{c \cdot b}{(d * d + \left(c \cdot c\right))_*} - \frac{\frac{d \cdot a}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\]
    21.6
  12. Applied final simplification
  13. Applied simplify to get
    \[\color{red}{\frac{c \cdot b}{(d * d + \left(c \cdot c\right))_*} - \frac{\frac{d \cdot a}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}} \leadsto \color{blue}{\frac{b \cdot c}{(d * d + \left({c}^2\right))_*} - \frac{a}{\sqrt{c^2 + d^2}^*} \cdot \frac{d}{\sqrt{c^2 + d^2}^*}}\]
    14.8

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))))))