Average Error: 26.4 → 23.8
Time: 3.3s
Precision: binary64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} = -inf.0 \lor \neg \left(\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le -2.4410351 \cdot 10^{-318}\right):\\ \;\;\;\;c \cdot \frac{b}{c \cdot c + d \cdot d} - \frac{d}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\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 -2.4410351 \cdot 10^{-318}\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{\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)))))) <= -2.4410350770643e-318))) {
		VAR = ((double) (((double) (c * ((double) (b / ((double) (((double) (c * c)) + ((double) (d * d)))))))) - ((double) (((double) (d / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d)))))))) * ((double) (a / ((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

Original26.4
Target0.4
Herbie23.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < -inf.0 or -2.4410351e-318 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))

    1. Initial program 33.8

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

    3. Applied div-sub33.8

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

    4. Simplified32.6

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

    5. Simplified31.2

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

    7. Applied add-sqr-sqrt31.2

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

    8. Applied *-un-lft-identity31.2

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

    9. Applied times-frac31.2

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

    10. Applied associate-*r*30.4

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

    11. Simplified30.4

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

    if -inf.0 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < -2.4410351e-318

    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*0.5

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

  4. Final simplification23.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} = -inf.0 \lor \neg \left(\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le -2.4410351 \cdot 10^{-318}\right):\\ \;\;\;\;c \cdot \frac{b}{c \cdot c + d \cdot d} - \frac{d}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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))))