Average Error: 31.3 → 16.7
Time: 49.2s
Precision: binary64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.1269686448080675 \cdot 10^{-74}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le 7.454968088927992 \cdot 10^{-107} \lor \neg \left(x \le 1.1264816293584452 \cdot 10^{-96}\right) \land x \le 4.15936032582048784 \cdot 10^{-36}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.1269686448080675 \cdot 10^{-74}:\\
\;\;\;\;1\\

\mathbf{elif}\;x \le 7.454968088927992 \cdot 10^{-107} \lor \neg \left(x \le 1.1264816293584452 \cdot 10^{-96}\right) \land x \le 4.15936032582048784 \cdot 10^{-36}:\\
\;\;\;\;-1\\

\mathbf{else}:\\
\;\;\;\;1\\

\end{array}
double code(double x, double y) {
	return (((double) (((double) (x * x)) - ((double) (((double) (y * 4.0)) * y)))) / ((double) (((double) (x * x)) + ((double) (((double) (y * 4.0)) * y)))));
}
↓
double code(double x, double y) {
	double VAR;
	if ((x <= -1.1269686448080675e-74)) {
		VAR = 1.0;
	} else {
		double VAR_1;
		if (((x <= 7.454968088927992e-107) || (!(x <= 1.1264816293584452e-96) && (x <= 4.159360325820488e-36)))) {
			VAR_1 = -1.0;
		} else {
			VAR_1 = 1.0;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.3
Target31.0
Herbie16.7
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.974323384962678118:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.1269686448080675e-74 or 7.454968088927992e-107 < x < 1.1264816293584452e-96 or 4.15936032582048784e-36 < x

    1. Initial program 36.4

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

    2. Taylor expanded around inf 18.7

      \[\leadsto \color{blue}{1}\]

    if -1.1269686448080675e-74 < x < 7.454968088927992e-107 or 1.1264816293584452e-96 < x < 4.15936032582048784e-36

    1. Initial program 24.1

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

    2. Taylor expanded around 0 13.8

      \[\leadsto \color{blue}{-1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.1269686448080675 \cdot 10^{-74}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le 7.454968088927992 \cdot 10^{-107} \lor \neg \left(x \le 1.1264816293584452 \cdot 10^{-96}\right) \land x \le 4.15936032582048784 \cdot 10^{-36}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4.0))) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4.0)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))