Average Error: 26.1 → 22.6
Time: 3.8s
Precision: binary64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} = -inf.0 \lor \neg \left(\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le -6.64002945018322003 \cdot 10^{170}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{\sqrt{c \cdot c + d \cdot d}} - d \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} = -inf.0 \lor \neg \left(\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le -6.64002945018322003 \cdot 10^{170}\right):\\
\;\;\;\;\frac{c \cdot \frac{b}{\sqrt{c \cdot c + d \cdot d}} - d \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{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 (((((double) (((double) (((double) (b * c)) - ((double) (a * d)))) / ((double) (((double) (c * c)) + ((double) (d * d)))))) <= -inf.0) || !(((double) (((double) (((double) (b * c)) - ((double) (a * d)))) / ((double) (((double) (c * c)) + ((double) (d * d)))))) <= -6.64002945018322e+170))) {
		VAR = ((double) (((double) (((double) (c * ((double) (b / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d)))))))))) - ((double) (d * ((double) (a / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d)))))))))))) / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d))))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (b * c)) - ((double) (a * d)))) / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d)))))))) / ((double) sqrt(((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

Original26.1
Target0.4
Herbie22.6
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < -inf.0 or -6.64002945018322003e170 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))

    1. Initial program 27.2

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

    3. Applied add-sqr-sqrt27.2

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

    4. Applied associate-/r*27.2

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

    6. Applied div-sub27.2

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

    7. Simplified25.3

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

    8. Simplified23.5

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

    if -inf.0 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < -6.64002945018322003e170

    1. Initial program 0.7

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

    3. Applied add-sqr-sqrt0.7

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

    4. Applied associate-/r*0.6

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

  4. Final simplification22.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} = -inf.0 \lor \neg \left(\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le -6.64002945018322003 \cdot 10^{170}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{\sqrt{c \cdot c + d \cdot d}} - d \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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))))