\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 7.1 s
Input Error: 12.6
Output Error: 10.7
Log: ⚲
Profile: 🕒
\({\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {d}^2}}\right)}^3\)
  1. Started with
    \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
    12.6
  2. Using strategy rm
    12.6
  3. Applied add-sqr-sqrt to get
    \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {d}^2}\]
    12.6
  4. Applied simplify to get
    \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
    10.4
  5. Using strategy rm
    10.4
  6. Applied add-cube-cbrt to get
    \[\color{red}{\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {d}^2}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {d}^2}}\right)}^3}\]
    10.7

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))))))