Average Error: 26.5 → 13.3
Time: 14.4s
Precision: binary64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;d \leq -9.996605145581266 \cdot 10^{+165}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{\frac{d \cdot d}{c}}\\ \mathbf{elif}\;d \leq -7.204616623989053 \cdot 10^{-127}:\\ \;\;\;\;\frac{d}{\sqrt[3]{d \cdot d + c \cdot c} \cdot \sqrt[3]{d \cdot d + c \cdot c}} \cdot \frac{b}{\sqrt[3]{d \cdot d + c \cdot c}} + \frac{a \cdot c}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 4.838421126276139 \cdot 10^{-155}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}\\ \mathbf{elif}\;d \leq 9.135302267760445 \cdot 10^{+153}:\\ \;\;\;\;\frac{d}{\sqrt[3]{d \cdot d + c \cdot c} \cdot \sqrt[3]{d \cdot d + c \cdot c}} \cdot \frac{b}{\sqrt[3]{d \cdot d + c \cdot c}} + \frac{a \cdot c}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{\frac{d \cdot d}{c}}\\ \end{array}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;d \leq -9.996605145581266 \cdot 10^{+165}:\\
\;\;\;\;\frac{b}{d} + \frac{a}{\frac{d \cdot d}{c}}\\

\mathbf{elif}\;d \leq -7.204616623989053 \cdot 10^{-127}:\\
\;\;\;\;\frac{d}{\sqrt[3]{d \cdot d + c \cdot c} \cdot \sqrt[3]{d \cdot d + c \cdot c}} \cdot \frac{b}{\sqrt[3]{d \cdot d + c \cdot c}} + \frac{a \cdot c}{d \cdot d + c \cdot c}\\

\mathbf{elif}\;d \leq 4.838421126276139 \cdot 10^{-155}:\\
\;\;\;\;\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}\\

\mathbf{elif}\;d \leq 9.135302267760445 \cdot 10^{+153}:\\
\;\;\;\;\frac{d}{\sqrt[3]{d \cdot d + c \cdot c} \cdot \sqrt[3]{d \cdot d + c \cdot c}} \cdot \frac{b}{\sqrt[3]{d \cdot d + c \cdot c}} + \frac{a \cdot c}{d \cdot d + c \cdot c}\\

\mathbf{else}:\\
\;\;\;\;\frac{b}{d} + \frac{a}{\frac{d \cdot d}{c}}\\

\end{array}
(FPCore (a b c d)
 :precision binary64
 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
↓
(FPCore (a b c d)
 :precision binary64
 (if (<= d -9.996605145581266e+165)
   (+ (/ b d) (/ a (/ (* d d) c)))
   (if (<= d -7.204616623989053e-127)
     (+
      (*
       (/ d (* (cbrt (+ (* d d) (* c c))) (cbrt (+ (* d d) (* c c)))))
       (/ b (cbrt (+ (* d d) (* c c)))))
      (/ (* a c) (+ (* d d) (* c c))))
     (if (<= d 4.838421126276139e-155)
       (+ (/ a c) (/ (* d b) (pow c 2.0)))
       (if (<= d 9.135302267760445e+153)
         (+
          (*
           (/ d (* (cbrt (+ (* d d) (* c c))) (cbrt (+ (* d d) (* c c)))))
           (/ b (cbrt (+ (* d d) (* c c)))))
          (/ (* a c) (+ (* d d) (* c c))))
         (+ (/ b d) (/ a (/ (* d d) c))))))))
double code(double a, double b, double c, double d) {
	return ((a * c) + (b * d)) / ((c * c) + (d * d));
}
↓
double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -9.996605145581266e+165) {
		tmp = (b / d) + (a / ((d * d) / c));
	} else if (d <= -7.204616623989053e-127) {
		tmp = ((d / (cbrt((d * d) + (c * c)) * cbrt((d * d) + (c * c)))) * (b / cbrt((d * d) + (c * c)))) + ((a * c) / ((d * d) + (c * c)));
	} else if (d <= 4.838421126276139e-155) {
		tmp = (a / c) + ((d * b) / pow(c, 2.0));
	} else if (d <= 9.135302267760445e+153) {
		tmp = ((d / (cbrt((d * d) + (c * c)) * cbrt((d * d) + (c * c)))) * (b / cbrt((d * d) + (c * c)))) + ((a * c) / ((d * d) + (c * c)));
	} else {
		tmp = (b / d) + (a / ((d * d) / c));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original26.5
Target0.5
Herbie13.3
\[\begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if d < -9.9966051455812663e165 or 9.1353022677604454e153 < d

    1. Initial program 45.4

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt_binary64_44145.4

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    4. Applied *-un-lft-identity_binary64_41945.4

      \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]

    5. Applied times-frac_binary64_42545.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]

    6. Simplified45.4

      \[\leadsto \frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \color{blue}{\frac{d \cdot b + c \cdot a}{\sqrt{c \cdot c + d \cdot d}}}\]

    7. Taylor expanded around 0 14.4

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}}\]

    8. Simplified13.0

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a}{\frac{d \cdot d}{c}}}\]

    if -9.9966051455812663e165 < d < -7.20461662398905326e-127 or 4.83842112627613932e-155 < d < 9.1353022677604454e153

    1. Initial program 18.1

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]

    2. Taylor expanded around 0 18.1

      \[\leadsto \color{blue}{\frac{a \cdot c}{{c}^{2} + {d}^{2}} + \frac{d \cdot b}{{c}^{2} + {d}^{2}}}\]

    3. Simplified18.1

      \[\leadsto \color{blue}{\frac{d \cdot b}{c \cdot c + d \cdot d} + \frac{c \cdot a}{c \cdot c + d \cdot d}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt_binary64_45418.5

      \[\leadsto \frac{d \cdot b}{\color{blue}{\left(\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}\right) \cdot \sqrt[3]{c \cdot c + d \cdot d}}} + \frac{c \cdot a}{c \cdot c + d \cdot d}\]

    6. Applied times-frac_binary64_42515.4

      \[\leadsto \color{blue}{\frac{d}{\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}} \cdot \frac{b}{\sqrt[3]{c \cdot c + d \cdot d}}} + \frac{c \cdot a}{c \cdot c + d \cdot d}\]

    if -7.20461662398905326e-127 < d < 4.83842112627613932e-155

    1. Initial program 23.5

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]

    2. Taylor expanded around inf 9.4

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.996605145581266 \cdot 10^{+165}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{\frac{d \cdot d}{c}}\\ \mathbf{elif}\;d \leq -7.204616623989053 \cdot 10^{-127}:\\ \;\;\;\;\frac{d}{\sqrt[3]{d \cdot d + c \cdot c} \cdot \sqrt[3]{d \cdot d + c \cdot c}} \cdot \frac{b}{\sqrt[3]{d \cdot d + c \cdot c}} + \frac{a \cdot c}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 4.838421126276139 \cdot 10^{-155}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}\\ \mathbf{elif}\;d \leq 9.135302267760445 \cdot 10^{+153}:\\ \;\;\;\;\frac{d}{\sqrt[3]{d \cdot d + c \cdot c} \cdot \sqrt[3]{d \cdot d + c \cdot c}} \cdot \frac{b}{\sqrt[3]{d \cdot d + c \cdot c}} + \frac{a \cdot c}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{\frac{d \cdot d}{c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021027 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))