\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 26.1 s
Input Error: 27.0
Output Error: 19.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\frac{b}{d} + \frac{\frac{a}{c}}{d \cdot d}\right) - \frac{b}{{d}^3 \cdot \left(c \cdot c\right)} & \text{when } d \le -1.3574531607557133 \cdot 10^{+150} \\ \frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2} & \text{when } d \le 1.1593976875307301 \cdot 10^{+138} \\ \left(\frac{b}{d} + \frac{\frac{a}{c}}{d \cdot d}\right) - \frac{b}{{d}^3 \cdot \left(c \cdot c\right)} & \text{otherwise} \end{cases}\)

    if d < -1.3574531607557133e+150 or 1.1593976875307301e+138 < d

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      48.4
    2. Using strategy rm
      48.4
    3. Applied clear-num to get
      \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}}\]
      48.4
    4. Using strategy rm
      48.4
    5. Applied add-log-exp to get
      \[\frac{1}{\color{red}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}} \leadsto \frac{1}{\color{blue}{\log \left(e^{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}\right)}}\]
      49.0
    6. Applied taylor to get
      \[\frac{1}{\log \left(e^{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}\right)} \leadsto \left(\frac{b}{d} + \frac{a}{c \cdot {d}^2}\right) - \frac{b}{{c}^2 \cdot {d}^{3}}\]
      19.6
    7. Taylor expanded around inf to get
      \[\color{red}{\left(\frac{b}{d} + \frac{a}{c \cdot {d}^2}\right) - \frac{b}{{c}^2 \cdot {d}^{3}}} \leadsto \color{blue}{\left(\frac{b}{d} + \frac{a}{c \cdot {d}^2}\right) - \frac{b}{{c}^2 \cdot {d}^{3}}}\]
      19.6
    8. Applied simplify to get
      \[\left(\frac{b}{d} + \frac{a}{c \cdot {d}^2}\right) - \frac{b}{{c}^2 \cdot {d}^{3}} \leadsto \left(\frac{b}{d} + \frac{\frac{a}{c}}{d \cdot d}\right) - \frac{b}{{d}^3 \cdot \left(c \cdot c\right)}\]
      20.4
    9. Applied final simplification

    if -1.3574531607557133e+150 < d < 1.1593976875307301e+138

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      18.7
  1. Removed slow pow expressions

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))))))