\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 16.1 s
Input Error: 12.8
Output Error: 3.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c} & \text{when } c \le -1.7422738f+19 \\ \frac{a \cdot c + {\left(\sqrt[3]{b \cdot d}\right)}^3}{{c}^2 + {\left(\left|d\right|\right)}^2} & \text{when } c \le -1.1200846f-20 \\ \frac{b}{\left|d\right|} \cdot \frac{d}{\left|d\right|} + \frac{c}{\left|d\right|} \cdot \frac{a}{\left|d\right|} & \text{when } c \le 2.7818448f-09 \\ \frac{a \cdot c + {\left(\sqrt[3]{b \cdot d}\right)}^3}{{c}^2 + {\left(\left|d\right|\right)}^2} & \text{when } c \le 5.2291484f+19 \\ \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c} & \text{otherwise} \end{cases}\)

    if c < -1.7422738f+19 or 5.2291484f+19 < c

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      22.6
    2. Using strategy rm
      22.6
    3. Applied add-sqr-sqrt to get
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
      22.6
    4. Applied simplify to get
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
      22.6
    5. Applied taylor to get
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2} \leadsto \left(\frac{a}{c} + \frac{b \cdot d}{{c}^2}\right) - \frac{b \cdot \left(d \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}\]
      6.6
    6. Taylor expanded around inf to get
      \[\color{red}{\left(\frac{a}{c} + \frac{b \cdot d}{{c}^2}\right) - \frac{b \cdot \left(d \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}} \leadsto \color{blue}{\left(\frac{a}{c} + \frac{b \cdot d}{{c}^2}\right) - \frac{b \cdot \left(d \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}}\]
      6.6
    7. Applied taylor to get
      \[\left(\frac{a}{c} + \frac{b \cdot d}{{c}^2}\right) - \frac{b \cdot \left(d \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}} \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
      6.6
    8. Taylor expanded around inf to get
      \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
      6.6
    9. Applied simplify to get
      \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\]
      0.3
    10. Applied final simplification

    if -1.7422738f+19 < c < -1.1200846f-20 or 2.7818448f-09 < c < 5.2291484f+19

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      8.5
    2. Using strategy rm
      8.5
    3. Applied add-sqr-sqrt to get
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
      8.5
    4. Applied simplify to get
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
      6.1
    5. Using strategy rm
      6.1
    6. Applied add-cube-cbrt to get
      \[\frac{a \cdot c + \color{red}{b \cdot d}}{{c}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{a \cdot c + \color{blue}{{\left(\sqrt[3]{b \cdot d}\right)}^3}}{{c}^2 + {\left(\left|d\right|\right)}^2}\]
      6.3

    if -1.1200846f-20 < c < 2.7818448f-09

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      10.7
    2. Using strategy rm
      10.7
    3. Applied add-sqr-sqrt to get
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
      10.7
    4. Applied simplify to get
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
      6.6
    5. Using strategy rm
      6.6
    6. Applied add-cbrt-cube to get
      \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}} \leadsto \color{blue}{\sqrt[3]{{\left(\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}\right)}^3}}\]
      18.5
    7. Applied taylor to get
      \[\sqrt[3]{{\left(\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}\right)}^3} \leadsto \sqrt[3]{{\left(\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}\right)}^3}\]
      18.3
    8. Taylor expanded around 0 to get
      \[\sqrt[3]{{\color{red}{\left(\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}\right)}}^3} \leadsto \sqrt[3]{{\color{blue}{\left(\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}\right)}}^3}\]
      18.3
    9. Applied simplify to get
      \[\sqrt[3]{{\left(\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}\right)}^3} \leadsto \frac{b}{\left|d\right|} \cdot \frac{d}{\left|d\right|} + \frac{c}{\left|d\right|} \cdot \frac{a}{\left|d\right|}\]
      0
    10. 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))))))