Average Error: 26.5 → 24.0
Time: 3.8s
Precision: binary64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le -6.181451911957164 \cdot 10^{293} \lor \neg \left(\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 4.5387220413702261 \cdot 10^{-178}\right):\\ \;\;\;\;c \cdot \frac{b}{c \cdot c + d \cdot d} - \frac{d}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\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} \le -6.181451911957164 \cdot 10^{293} \lor \neg \left(\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 4.5387220413702261 \cdot 10^{-178}\right):\\
\;\;\;\;c \cdot \frac{b}{c \cdot c + d \cdot d} - \frac{d}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\\

\end{array}
double code(double a, double b, double c, double d) {
	return (((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) (b * c)) - ((double) (a * d)))) / ((double) (((double) (c * c)) + ((double) (d * d))))) <= -6.1814519119571635e+293) || !((((double) (((double) (b * c)) - ((double) (a * d)))) / ((double) (((double) (c * c)) + ((double) (d * d))))) <= 4.538722041370226e-178))) {
		VAR = ((double) (((double) (c * (b / ((double) (((double) (c * c)) + ((double) (d * d))))))) - ((double) ((d / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d))))))) * (a / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d)))))))))));
	} else {
		VAR = ((double) ((1.0 / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d))))))) * (((double) (((double) (b * c)) - ((double) (a * 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

Original26.5
Target0.4
Herbie24.0
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < -6.181451911957164e293 or 4.5387220413702261e-178 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))

    1. Initial program 39.2

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

    3. Applied div-sub39.2

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

    4. Simplified37.4

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

    5. Simplified34.9

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

    7. Applied add-sqr-sqrt34.9

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

    8. Applied *-un-lft-identity34.9

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

    9. Applied times-frac34.9

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

    10. Applied associate-*r*33.7

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

    11. Simplified33.7

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

    if -6.181451911957164e293 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < 4.5387220413702261e-178

    1. Initial program 15.3

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

    3. Applied add-sqr-sqrt15.3

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

    4. Applied *-un-lft-identity15.3

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

    5. Applied times-frac15.3

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification24.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le -6.181451911957164 \cdot 10^{293} \lor \neg \left(\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 4.5387220413702261 \cdot 10^{-178}\right):\\ \;\;\;\;c \cdot \frac{b}{c \cdot c + d \cdot d} - \frac{d}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))))