\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 12.5 s
Input Error: 25.9
Output Error: 12.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{a}{c} & \text{when } c \le -4.770922648494208 \cdot 10^{+101} \\ \left(a \cdot c + b \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2} & \text{when } c \le 1.1692787636589298 \cdot 10^{+124} \\ \frac{a}{c} & \text{otherwise} \end{cases}\)

    if c < -4.770922648494208e+101 or 1.1692787636589298e+124 < c

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      41.0
    2. Using strategy rm
      41.0
    3. Applied clear-num to get
      \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}}\]
      41.0
    4. Using strategy rm
      41.0
    5. Applied add-cube-cbrt to get
      \[\frac{1}{\color{red}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}} \leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}\right)}^3}}\]
      41.1
    6. Applied taylor to get
      \[\frac{1}{{\left(\sqrt[3]{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}\right)}^3} \leadsto \frac{a}{c}\]
      0
    7. Taylor expanded around 0 to get
      \[\color{red}{\frac{a}{c}} \leadsto \color{blue}{\frac{a}{c}}\]
      0
    8. Applied simplify to get
      \[\frac{a}{c} \leadsto \frac{a}{c}\]
      0
    9. Applied final simplification

    if -4.770922648494208e+101 < c < 1.1692787636589298e+124

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      18.3
    2. Using strategy rm
      18.3
    3. Applied div-inv to get
      \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\left(a \cdot c + b \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2}}\]
      18.6
  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))))))