\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 10.5 s
Input Error: 16.2
Output Error: 0.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d} & \text{when } d \le -344.83047f0 \\ \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c} & \text{when } d \le 1.0322674f+10 \\ \frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d} & \text{otherwise} \end{cases}\)

    if d < -344.83047f0 or 1.0322674f+10 < d

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      18.8
    2. Using strategy rm
      18.8
    3. Applied add-sqr-sqrt to get
      \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
      18.8
    4. Applied taylor to get
      \[\frac{a \cdot c + b \cdot d}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{a \cdot c + b \cdot d}{{d}^2}\]
      16.3
    5. Taylor expanded around 0 to get
      \[\frac{a \cdot c + b \cdot d}{{\color{red}{d}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{d}}^2}\]
      16.3
    6. Applied taylor to get
      \[\frac{a \cdot c + b \cdot d}{{d}^2} \leadsto \frac{c \cdot a}{{d}^2} + \frac{b}{d}\]
      5.2
    7. Taylor expanded around 0 to get
      \[\color{red}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}} \leadsto \color{blue}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}}\]
      5.2
    8. Applied simplify to get
      \[\frac{c \cdot a}{{d}^2} + \frac{b}{d} \leadsto \frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}\]
      0.4
    9. Applied final simplification

    if -344.83047f0 < d < 1.0322674f+10

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      13.8
    2. Using strategy rm
      13.8
    3. Applied add-sqr-sqrt to get
      \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
      13.7
    4. Applied taylor to get
      \[\frac{a \cdot c + b \cdot d}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2}\]
      10.7
    5. Taylor expanded around inf to get
      \[\frac{a \cdot c + b \cdot d}{{\color{red}{c}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{c}}^2}\]
      10.7
    6. Applied taylor to get
      \[\frac{a \cdot c + b \cdot d}{{c}^2} \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
      3.0
    7. Taylor expanded around 0 to get
      \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
      3.0
    8. Applied simplify to get
      \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\]
      0.7
    9. Applied final simplification
  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))))))