\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 9.9 s
Input Error: 12.5
Output Error: 1.7
Log: ⚲
Profile: 🕒
\(\begin{cases} (\left(\frac{b}{c}\right) * \left(\frac{d}{c}\right) + \left(\frac{a}{c}\right))_* & \text{when } c \le -6.199415f+20 \\ (\left(\frac{b}{\sqrt{c^2 + d^2}^*}\right) * \left(\frac{d}{\sqrt{c^2 + d^2}^*}\right) + \left(\frac{a \cdot c}{\sqrt{c^2 + d^2}^* \cdot \sqrt{c^2 + d^2}^*}\right))_* & \text{when } c \le 2.6212168f+19 \\ (\left(\frac{b}{c}\right) * \left(\frac{d}{c}\right) + \left(\frac{a}{c}\right))_* & \text{otherwise} \end{cases}\)

    if c < -6.199415f+20 or 2.6212168f+19 < c

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      21.9
    2. Using strategy rm
      21.9
    3. Applied add-cube-cbrt to get
      \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt[3]{{c}^2 + {d}^2}\right)}^3}}\]
      21.9
    4. Applied taylor to get
      \[\frac{a \cdot c + b \cdot d}{{\left(\sqrt[3]{{c}^2 + {d}^2}\right)}^3} \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
      6.2
    5. Taylor expanded around inf to get
      \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
      6.2
    6. Applied simplify to get
      \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto (\left(\frac{b}{c}\right) * \left(\frac{d}{c}\right) + \left(\frac{a}{c}\right))_*\]
      0.2
    7. Applied final simplification

    if -6.199415f+20 < c < 2.6212168f+19

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      9.4
    2. Using strategy rm
      9.4
    3. Applied add-sqr-sqrt to get
      \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
      9.3
    4. Applied simplify to get
      \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2 + {d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\sqrt{c^2 + d^2}^*\right)}}^2}\]
      5.9
    5. Applied taylor to get
      \[\frac{a \cdot c + b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
      5.9
    6. Taylor expanded around 0 to get
      \[\color{red}{\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \color{blue}{\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
      5.9
    7. Applied simplify to get
      \[\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto (\left(\frac{b}{\sqrt{c^2 + d^2}^*}\right) * \left(\frac{d}{\sqrt{c^2 + d^2}^*}\right) + \left(\frac{a \cdot c}{\sqrt{c^2 + d^2}^* \cdot \sqrt{c^2 + d^2}^*}\right))_*\]
      2.2
    8. 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))))))