\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 11.9 s
Input Error: 26.9
Output Error: 11.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c} & \text{when } c \le -9.478361925812184 \cdot 10^{+101} \\ \left(a \cdot c + b \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2} & \text{when } c \le 6.1232819804241275 \cdot 10^{+28} \\ \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c} & \text{otherwise} \end{cases}\)

    if c < -9.478361925812184e+101 or 6.1232819804241275e+28 < c

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      38.7
    2. Using strategy rm
      38.7
    3. Applied div-inv to get
      \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\left(a \cdot c + b \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2}}\]
      38.7
    4. Using strategy rm
      38.7
    5. Applied add-cube-cbrt to get
      \[\left(a \cdot c + b \cdot d\right) \cdot \color{red}{\frac{1}{{c}^2 + {d}^2}} \leadsto \left(a \cdot c + b \cdot d\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{1}{{c}^2 + {d}^2}}\right)}^3}\]
      38.9
    6. Applied taylor to get
      \[\left(a \cdot c + b \cdot d\right) \cdot {\left(\sqrt[3]{\frac{1}{{c}^2 + {d}^2}}\right)}^3 \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
      10.2
    7. Taylor expanded around inf to get
      \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
      10.2
    8. Applied simplify to get
      \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\]
      0.8
    9. Applied final simplification

    if -9.478361925812184e+101 < c < 6.1232819804241275e+28

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      18.6
    2. Using strategy rm
      18.6
    3. Applied div-inv to get
      \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\left(a \cdot c + b \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2}}\]
      18.9
  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))))))