\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 12.2 s
Input Error: 29.5
Output Error: 5.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{b}{d} & \text{when } d \le -4.152185822643052 \cdot 10^{+136} \\ \frac{1}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}} & \text{when } d \le -1.1022654491558348 \cdot 10^{-91} \\ \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c} & \text{when } d \le 1.336475333079889 \cdot 10^{-113} \\ \frac{1}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}} & \text{when } d \le 1.0431026568679888 \cdot 10^{-46} \\ \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c} & \text{when } d \le 5.500657540493106 \cdot 10^{+125} \\ \frac{b}{d} & \text{otherwise} \end{cases}\)

    if d < -4.152185822643052e+136 or 5.500657540493106e+125 < d

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      41.0
    2. Using strategy rm
      41.0
    3. Applied add-exp-log to get
      \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{e^{\log \left({c}^2 + {d}^2\right)}}}\]
      41.3
    4. Applied add-exp-log to get
      \[\frac{\color{red}{a \cdot c + b \cdot d}}{e^{\log \left({c}^2 + {d}^2\right)}} \leadsto \frac{\color{blue}{e^{\log \left(a \cdot c + b \cdot d\right)}}}{e^{\log \left({c}^2 + {d}^2\right)}}\]
      52.7
    5. Applied div-exp to get
      \[\color{red}{\frac{e^{\log \left(a \cdot c + b \cdot d\right)}}{e^{\log \left({c}^2 + {d}^2\right)}}} \leadsto \color{blue}{e^{\log \left(a \cdot c + b \cdot d\right) - \log \left({c}^2 + {d}^2\right)}}\]
      52.7
    6. Applied taylor to get
      \[e^{\log \left(a \cdot c + b \cdot d\right) - \log \left({c}^2 + {d}^2\right)} \leadsto e^{\log b - \log d}\]
      48.9
    7. Taylor expanded around 0 to get
      \[\color{red}{e^{\log b - \log d}} \leadsto \color{blue}{e^{\log b - \log d}}\]
      48.9
    8. Applied simplify to get
      \[e^{\log b - \log d} \leadsto \frac{b}{d}\]
      0
    9. Applied final simplification

    if -4.152185822643052e+136 < d < -1.1022654491558348e-91 or 1.336475333079889e-113 < d < 1.0431026568679888e-46

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      17.6
    2. Using strategy rm
      17.6
    3. Applied clear-num to get
      \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}}\]
      17.9

    if -1.1022654491558348e-91 < d < 1.336475333079889e-113 or 1.0431026568679888e-46 < d < 5.500657540493106e+125

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      27.8
    2. Using strategy rm
      27.8
    3. Applied add-cube-cbrt to get
      \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt[3]{{c}^2 + {d}^2}\right)}^3}}\]
      28.2
    4. Applied taylor to get
      \[\frac{a \cdot c + b \cdot d}{{\left(\sqrt[3]{{c}^2 + {d}^2}\right)}^3} \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
      7.6
    5. Taylor expanded around inf to get
      \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
      7.6
    6. Applied simplify to get
      \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\]
      2.3
    7. 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))))))