Average Error: 25.8 → 15.5
Time: 3.5s
Precision: binary64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -1.3612643717146147 \cdot 10^{34} \lor \neg \left(d \le 7.96743382463118103 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{d + \frac{{c}^{2}}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{c \cdot c + d \cdot d}{c}} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\ \end{array}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;d \le -1.3612643717146147 \cdot 10^{34} \lor \neg \left(d \le 7.96743382463118103 \cdot 10^{-17}\right):\\
\;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{d + \frac{{c}^{2}}{d}}\\

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

\end{array}
double code(double a, double b, double c, double d) {
	return ((double) (((double) (((double) (b * c)) - ((double) (a * d)))) / ((double) (((double) (c * c)) + ((double) (d * d))))));
}
↓
double code(double a, double b, double c, double d) {
	double VAR;
	if (((d <= -1.3612643717146147e+34) || !(d <= 7.967433824631181e-17))) {
		VAR = ((double) (((double) (((double) (b * c)) / ((double) (((double) (c * c)) + ((double) (d * d)))))) - ((double) (a / ((double) (d + ((double) (((double) pow(c, 2.0)) / d))))))));
	} else {
		VAR = ((double) (((double) (b / ((double) (((double) (((double) (c * c)) + ((double) (d * d)))) / c)))) - ((double) (((double) (a * d)) / ((double) (((double) (c * c)) + ((double) (d * d))))))));
	}
	return VAR;
}

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

Original25.8
Target0.4
Herbie15.5
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if d < -1.3612643717146147e34 or 7.96743382463118103e-17 < d

    1. Initial program 32.9

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

    3. Applied div-sub32.8

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

    5. Applied associate-/l*29.2

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

    6. Taylor expanded around 0 15.7

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

    if -1.3612643717146147e34 < d < 7.96743382463118103e-17

    1. Initial program 18.2

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

    3. Applied div-sub18.2

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

    5. Applied associate-/l*15.3

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -1.3612643717146147 \cdot 10^{34} \lor \neg \left(d \le 7.96743382463118103 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{d + \frac{{c}^{2}}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{c \cdot c + d \cdot d}{c}} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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)))) (/ (+ (neg a) (* b (/ c d))) (+ d (* c (/ c d)))))

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