\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
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: 11.2 s
Input Error: 12.5
Output Error: 2.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b}{1} \cdot \frac{d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} & \text{when } d \le -3.1052615f+11 \\ \frac{c}{1} \cdot \frac{a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} & \text{when } d \le 9.210374f+08 \\ \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b}{1} \cdot \frac{d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} & \text{otherwise} \end{cases}\)

    if d < -3.1052615f+11 or 9.210374f+08 < d

    1. Started with
      \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
      18.4
    2. Using strategy rm
      18.4
    3. Applied add-sqr-sqrt to get
      \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
      18.4
    4. Applied simplify to get
      \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2 + {d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\sqrt{c^2 + d^2}^*\right)}}^2}\]
      12.3
    5. Applied taylor to get
      \[\frac{a \cdot c + b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
      12.3
    6. Taylor expanded around 0 to get
      \[\color{red}{\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \color{blue}{\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
      12.3
    7. Using strategy rm
      12.3
    8. Applied *-un-lft-identity to get
      \[\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\color{red}{\left(\sqrt{c^2 + d^2}^*\right)}}^2} \leadsto \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\color{blue}{\left(1 \cdot \sqrt{c^2 + d^2}^*\right)}}^2}\]
      12.3
    9. Applied square-prod to get
      \[\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{\color{red}{{\left(1 \cdot \sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{\color{blue}{{1}^2 \cdot {\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
      12.3
    10. Applied times-frac to get
      \[\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \color{red}{\frac{b \cdot d}{{1}^2 \cdot {\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \color{blue}{\frac{b}{{1}^2} \cdot \frac{d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
      3.7
    11. Applied simplify to get
      \[\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \color{red}{\frac{b}{{1}^2}} \cdot \frac{d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \color{blue}{\frac{b}{1}} \cdot \frac{d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
      3.7

    if -3.1052615f+11 < d < 9.210374f+08

    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
      \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
      8.5
    4. Applied simplify to get
      \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2 + {d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\sqrt{c^2 + d^2}^*\right)}}^2}\]
      5.2
    5. Applied taylor to get
      \[\frac{a \cdot c + b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
      5.2
    6. Taylor expanded around 0 to get
      \[\color{red}{\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \color{blue}{\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
      5.2
    7. Using strategy rm
      5.2
    8. Applied *-un-lft-identity to get
      \[\frac{c \cdot a}{{\color{red}{\left(\sqrt{c^2 + d^2}^*\right)}}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{c \cdot a}{{\color{blue}{\left(1 \cdot \sqrt{c^2 + d^2}^*\right)}}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
      5.2
    9. Applied square-prod to get
      \[\frac{c \cdot a}{\color{red}{{\left(1 \cdot \sqrt{c^2 + d^2}^*\right)}^2}} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{c \cdot a}{\color{blue}{{1}^2 \cdot {\left(\sqrt{c^2 + d^2}^*\right)}^2}} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
      5.2
    10. Applied times-frac to get
      \[\color{red}{\frac{c \cdot a}{{1}^2 \cdot {\left(\sqrt{c^2 + d^2}^*\right)}^2}} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \color{blue}{\frac{c}{{1}^2} \cdot \frac{a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
      0.8
    11. Applied simplify to get
      \[\color{red}{\frac{c}{{1}^2}} \cdot \frac{a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \color{blue}{\frac{c}{1}} \cdot \frac{a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
      0.8
  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))))))