Average Error: 26.4 → 15.5
Time: 11.8s
Precision: binary64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;d \leq -6.1667146794926305 \cdot 10^{+118}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{elif}\;d \leq -3.0902158100781017 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;d \leq 3.1719901827793896 \cdot 10^{-93}:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot a}{{c}^{2}}\\ \mathbf{elif}\;d \leq 1.1649363628593953 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{d}\\ \end{array}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;d \leq -6.1667146794926305 \cdot 10^{+118}:\\
\;\;\;\;-\frac{a}{d}\\

\mathbf{elif}\;d \leq -3.0902158100781017 \cdot 10^{-56}:\\
\;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{elif}\;d \leq 3.1719901827793896 \cdot 10^{-93}:\\
\;\;\;\;\frac{b}{c} - \frac{d \cdot a}{{c}^{2}}\\

\mathbf{elif}\;d \leq 1.1649363628593953 \cdot 10^{+112}:\\
\;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{else}:\\
\;\;\;\;-\frac{a}{d}\\

\end{array}
(FPCore (a b c d)
 :precision binary64
 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
↓
(FPCore (a b c d)
 :precision binary64
 (if (<= d -6.1667146794926305e+118)
   (- (/ a d))
   (if (<= d -3.0902158100781017e-56)
     (/
      (/ (- (* b c) (* d a)) (sqrt (+ (* c c) (* d d))))
      (sqrt (+ (* c c) (* d d))))
     (if (<= d 3.1719901827793896e-93)
       (- (/ b c) (/ (* d a) (pow c 2.0)))
       (if (<= d 1.1649363628593953e+112)
         (/
          (/ (- (* b c) (* d a)) (sqrt (+ (* c c) (* d d))))
          (sqrt (+ (* c c) (* d d))))
         (- (/ a d)))))))
double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}
↓
double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -6.1667146794926305e+118) {
		tmp = -(a / d);
	} else if (d <= -3.0902158100781017e-56) {
		tmp = (((b * c) - (d * a)) / sqrt((c * c) + (d * d))) / sqrt((c * c) + (d * d));
	} else if (d <= 3.1719901827793896e-93) {
		tmp = (b / c) - ((d * a) / pow(c, 2.0));
	} else if (d <= 1.1649363628593953e+112) {
		tmp = (((b * c) - (d * a)) / sqrt((c * c) + (d * d))) / sqrt((c * c) + (d * d));
	} else {
		tmp = -(a / d);
	}
	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

Original26.4
Target0.3
Herbie15.5
\[\begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if d < -6.16671467949263054e118 or 1.1649363628593953e112 < d

    1. Initial program 41.0

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

    2. Taylor expanded around 0 17.1

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}}\]

    3. Simplified17.1

      \[\leadsto \color{blue}{-\frac{a}{d}}\]

    if -6.16671467949263054e118 < d < -3.0902158100781017e-56 or 3.1719901827793896e-93 < d < 1.1649363628593953e112

    1. Initial program 16.9

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

    3. Applied add-sqr-sqrt_binary64_78216.9

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

    4. Applied associate-/r*_binary64_70416.8

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

    if -3.0902158100781017e-56 < d < 3.1719901827793896e-93

    1. Initial program 20.8

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

    2. Taylor expanded around inf 12.8

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.1667146794926305 \cdot 10^{+118}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{elif}\;d \leq -3.0902158100781017 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;d \leq 3.1719901827793896 \cdot 10^{-93}:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot a}{{c}^{2}}\\ \mathbf{elif}\;d \leq 1.1649363628593953 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{d}\\ \end{array}\]

Reproduce

herbie shell --seed 2021096 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

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

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