\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 5.7 s
Input Error: 12.5
Output Error: 4.6
Log: ⚲
Profile: 🕒
\((\left(\frac{b}{\sqrt{c^2 + d^2}^*}\right) * \left(\frac{d}{\sqrt{c^2 + d^2}^*}\right) + \left(\frac{a \cdot c}{\sqrt{c^2 + d^2}^* \cdot \sqrt{c^2 + d^2}^*}\right))_*\)
  1. Started with
    \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
    12.5
  2. Using strategy rm
    12.5
  3. Applied add-sqr-sqrt to get
    \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
    12.4
  4. Applied simplify to get
    \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2 + {d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\sqrt{c^2 + d^2}^*\right)}}^2}\]
    7.9
  5. Applied taylor to get
    \[\frac{a \cdot c + b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
    7.9
  6. Taylor expanded around 0 to get
    \[\color{red}{\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \color{blue}{\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
    7.9
  7. Applied simplify to get
    \[\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto (\left(\frac{b}{\sqrt{c^2 + d^2}^*}\right) * \left(\frac{d}{\sqrt{c^2 + d^2}^*}\right) + \left(\frac{a \cdot c}{\sqrt{c^2 + d^2}^* \cdot \sqrt{c^2 + d^2}^*}\right))_*\]
    4.6
  8. Applied final simplification

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))))))