\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 8.5 s
Input Error: 25.5
Output Error: 25.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{b \cdot d}{(c * c + \left({d}^2\right))_*} + \frac{c \cdot a}{(c * c + \left({d}^2\right))_*} & \text{when } b \le -9.364561599243466 \cdot 10^{+17} \\ {\left(\frac{\sqrt{(c * a + \left(d \cdot b\right))_*}}{\sqrt{c^2 + d^2}^*}\right)}^2 & \text{when } b \le -5.047869208437734 \cdot 10^{-09} \\ \frac{b \cdot d}{(c * c + \left({d}^2\right))_*} + \frac{c \cdot a}{(c * c + \left({d}^2\right))_*} & \text{otherwise} \end{cases}\)

    if b < -9.364561599243466e+17 or -5.047869208437734e-09 < b

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      25.6
    2. Using strategy rm
      25.6
    3. Applied square-mult to get
      \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{c \cdot c} + {d}^2}\]
      25.6
    4. Applied fma-def to get
      \[\frac{a \cdot c + b \cdot d}{\color{red}{c \cdot c + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{(c * c + \left({d}^2\right))_*}}\]
      25.6
    5. Applied taylor to get
      \[\frac{a \cdot c + b \cdot d}{(c * c + \left({d}^2\right))_*} \leadsto \frac{b \cdot d}{(c * c + \left({d}^2\right))_*} + \frac{c \cdot a}{(c * c + \left({d}^2\right))_*}\]
      25.6
    6. Taylor expanded around 0 to get
      \[\color{red}{\frac{b \cdot d}{(c * c + \left({d}^2\right))_*} + \frac{c \cdot a}{(c * c + \left({d}^2\right))_*}} \leadsto \color{blue}{\frac{b \cdot d}{(c * c + \left({d}^2\right))_*} + \frac{c \cdot a}{(c * c + \left({d}^2\right))_*}}\]
      25.6

    if -9.364561599243466e+17 < b < -5.047869208437734e-09

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      22.1
    2. Using strategy rm
      22.1
    3. Applied add-sqr-sqrt to get
      \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
      22.1
    4. Applied add-sqr-sqrt to get
      \[\frac{\color{red}{a \cdot c + b \cdot d}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{\color{blue}{{\left(\sqrt{a \cdot c + b \cdot d}\right)}^2}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}\]
      22.3
    5. Applied square-undiv to get
      \[\color{red}{\frac{{\left(\sqrt{a \cdot c + b \cdot d}\right)}^2}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}} \leadsto \color{blue}{{\left(\frac{\sqrt{a \cdot c + b \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}^2}\]
      22.3
    6. Applied simplify to get
      \[{\color{red}{\left(\frac{\sqrt{a \cdot c + b \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}}^2 \leadsto {\color{blue}{\left(\frac{\sqrt{(c * a + \left(d \cdot b\right))_*}}{\sqrt{c^2 + d^2}^*}\right)}}^2\]
      9.0
  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))))))