\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 13.4 s
Input Error: 13.2
Output Error: 2.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c} & \text{when } c \le -2.2840177f+18 \\ \frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2} & \text{when } c \le -7.427552f-08 \\ \frac{b}{\left|d\right|} \cdot \frac{d}{\left|d\right|} + \frac{a}{\left|d\right|} \cdot \frac{c}{\left|d\right|} & \text{when } c \le 1.1737455f-15 \\ \frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2} & \text{when } c \le 2.8363323f+14 \\ \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c} & \text{otherwise} \end{cases}\)

    if c < -2.2840177f+18 or 2.8363323f+14 < c

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      21.2
    2. Using strategy rm
      21.2
    3. Applied add-sqr-sqrt to get
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
      21.2
    4. Applied simplify to get
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
      21.2
    5. Using strategy rm
      21.2
    6. Applied add-cube-cbrt to get
      \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}\right)}^3}\]
      21.2
    7. Applied taylor to get
      \[{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}\right)}^3 \leadsto {\left(\sqrt[3]{\left(\frac{a}{c} + \frac{b \cdot d}{{c}^2}\right) - \frac{b \cdot \left(d \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}}\right)}^3\]
      6.6
    8. Taylor expanded around inf to get
      \[{\left(\sqrt[3]{\color{red}{\left(\frac{a}{c} + \frac{b \cdot d}{{c}^2}\right) - \frac{b \cdot \left(d \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\left(\frac{a}{c} + \frac{b \cdot d}{{c}^2}\right) - \frac{b \cdot \left(d \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}}}\right)}^3\]
      6.6
    9. Applied simplify to get
      \[\color{red}{{\left(\sqrt[3]{\left(\frac{a}{c} + \frac{b \cdot d}{{c}^2}\right) - \frac{b \cdot \left(d \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}}\right)}^3} \leadsto \color{blue}{\left(\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\right) - \frac{{\left(\left|\frac{1}{d}\right|\right)}^2 \cdot \left(d \cdot b\right)}{{c}^{4}}}\]
      4.3
    10. Applied taylor to get
      \[\left(\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\right) - \frac{{\left(\left|\frac{1}{d}\right|\right)}^2 \cdot \left(d \cdot b\right)}{{c}^{4}} \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
      6.2
    11. Taylor expanded around inf to get
      \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
      6.2
    12. 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
    13. Applied final simplification

    if -2.2840177f+18 < c < -7.427552f-08 or 1.1737455f-15 < c < 2.8363323f+14

    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

    if -7.427552f-08 < c < 1.1737455f-15

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      10.9
    2. Using strategy rm
      10.9
    3. Applied add-sqr-sqrt to get
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
      10.9
    4. Applied simplify to get
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
      7.2
    5. Using strategy rm
      7.2
    6. Applied add-cube-cbrt to get
      \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}\right)}^3}\]
      7.5
    7. Using strategy rm
      7.5
    8. Applied add-cube-cbrt to get
      \[{\color{red}{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}\right)}}^3 \leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}}\right)}^3\right)}}^3\]
      8.0
    9. Applied taylor to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}}}\right)}^3\right)}^3\]
      6.7
    10. Taylor expanded around 0 to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\color{red}{\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\color{blue}{\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}}}}\right)}^3\right)}^3\]
      6.7
    11. Applied simplify to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}}}\right)}^3\right)}^3 \leadsto \frac{b}{\left|d\right|} \cdot \frac{d}{\left|d\right|} + \frac{a}{\left|d\right|} \cdot \frac{c}{\left|d\right|}\]
      0
    12. 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))))))