\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 11.6 s
Input Error: 25.0
Output Error: 22.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(a \cdot c + b \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2} & \text{when } d \le 1.5225514705194093 \cdot 10^{+142} \\ {\left(\frac{\sqrt{(c * a + \left(d \cdot b\right))_*}}{\sqrt{c^2 + d^2}^*}\right)}^2 & \text{when } d \le 6.041405597054045 \cdot 10^{+263} \\ 0 & \text{otherwise} \end{cases}\)

    if d < 1.5225514705194093e+142

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      23.2
    2. Using strategy rm
      23.2
    3. Applied div-inv to get
      \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\left(a \cdot c + b \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2}}\]
      23.4

    if 1.5225514705194093e+142 < d < 6.041405597054045e+263

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      47.0
    2. Using strategy rm
      47.0
    3. Applied add-sqr-sqrt to get
      \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
      47.0
    4. Applied add-sqr-sqrt to get
      \[\frac{\color{red}{a \cdot c + b \cdot d}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{\color{blue}{{\left(\sqrt{a \cdot c + b \cdot d}\right)}^2}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}\]
      47.0
    5. Applied square-undiv to get
      \[\color{red}{\frac{{\left(\sqrt{a \cdot c + b \cdot d}\right)}^2}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}} \leadsto \color{blue}{{\left(\frac{\sqrt{a \cdot c + b \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}^2}\]
      47.0
    6. Applied simplify to get
      \[{\color{red}{\left(\frac{\sqrt{a \cdot c + b \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}}^2 \leadsto {\color{blue}{\left(\frac{\sqrt{(c * a + \left(d \cdot b\right))_*}}{\sqrt{c^2 + d^2}^*}\right)}}^2\]
      26.0

    if 6.041405597054045e+263 < d

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      37.0
    2. Applied taylor to get
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2} \leadsto 0\]
      0
    3. Taylor expanded around 0 to get
      \[\color{red}{0} \leadsto \color{blue}{0}\]
      0
  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))))))