\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 7.6 s
Input Error: 25.5
Output Error: 24.1
Log: ⚲
Profile: 🕒
\(\frac{b}{\frac{(c * c + \left({d}^2\right))_*}{d}} + \frac{c \cdot a}{(c * c + \left({d}^2\right))_*}\)
  1. Started with
    \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
    25.5
  2. Using strategy rm
    25.5
  3. Applied square-mult to get
    \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{c \cdot c} + {d}^2}\]
    25.5
  4. Applied fma-def to get
    \[\frac{a \cdot c + b \cdot d}{\color{red}{c \cdot c + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{(c * c + \left({d}^2\right))_*}}\]
    25.5
  5. Applied taylor to get
    \[\frac{a \cdot c + b \cdot d}{(c * c + \left({d}^2\right))_*} \leadsto \frac{b \cdot d}{(c * c + \left({d}^2\right))_*} + \frac{c \cdot a}{(c * c + \left({d}^2\right))_*}\]
    25.6
  6. Taylor expanded around 0 to get
    \[\color{red}{\frac{b \cdot d}{(c * c + \left({d}^2\right))_*} + \frac{c \cdot a}{(c * c + \left({d}^2\right))_*}} \leadsto \color{blue}{\frac{b \cdot d}{(c * c + \left({d}^2\right))_*} + \frac{c \cdot a}{(c * c + \left({d}^2\right))_*}}\]
    25.6
  7. Using strategy rm
    25.6
  8. Applied associate-/l* to get
    \[\color{red}{\frac{b \cdot d}{(c * c + \left({d}^2\right))_*}} + \frac{c \cdot a}{(c * c + \left({d}^2\right))_*} \leadsto \color{blue}{\frac{b}{\frac{(c * c + \left({d}^2\right))_*}{d}}} + \frac{c \cdot a}{(c * c + \left({d}^2\right))_*}\]
    24.1

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))))))