\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
Test:
Complex division, real part
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Bits error versus d
Time: 7.9 s
Input Error: 12.7
Output Error: 10.9
Log: ⚲
Profile: 🕒
\({\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}\right)}^3\)
  1. Started with
    \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
    12.7
  2. Using strategy rm
    12.7
  3. Applied add-sqr-sqrt to get
    \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
    12.7
  4. Applied simplify to get
    \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
    10.6
  5. Using strategy rm
    10.6
  6. Applied add-cube-cbrt to get
    \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}\right)}^3}\]
    10.9

Original test:


(lambda ((a default) (b default) (c default) (d default))
  #:name "Complex division, real part"
  (/ (+ (* a c) (* b d)) (+ (sqr c) (sqr d)))
  #:target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d))))))