Average Error: 30.1 → 12.3
Time: 1.8s
Precision: binary64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3391086476024713 \cdot 10^{154}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -1.05172085077483252 \cdot 10^{-149}:\\ \;\;\;\;x \cdot \frac{x}{x \cdot x + y \cdot \left(y \cdot 4\right)} - y \cdot \left(4 \cdot \frac{y}{x \cdot x + y \cdot \left(y \cdot 4\right)}\right)\\ \mathbf{elif}\;x \le 4.86593442952461677 \cdot 10^{-79}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 1.110210241227892 \cdot 10^{154}:\\ \;\;\;\;x \cdot \frac{x}{x \cdot x + y \cdot \left(y \cdot 4\right)} - y \cdot \left(4 \cdot \frac{y}{x \cdot x + y \cdot \left(y \cdot 4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\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 -1.3391086476024713 \cdot 10^{154}:\\
\;\;\;\;1\\

\mathbf{elif}\;x \le -1.05172085077483252 \cdot 10^{-149}:\\
\;\;\;\;x \cdot \frac{x}{x \cdot x + y \cdot \left(y \cdot 4\right)} - y \cdot \left(4 \cdot \frac{y}{x \cdot x + y \cdot \left(y \cdot 4\right)}\right)\\

\mathbf{elif}\;x \le 4.86593442952461677 \cdot 10^{-79}:\\
\;\;\;\;-1\\

\mathbf{elif}\;x \le 1.110210241227892 \cdot 10^{154}:\\
\;\;\;\;x \cdot \frac{x}{x \cdot x + y \cdot \left(y \cdot 4\right)} - y \cdot \left(4 \cdot \frac{y}{x \cdot x + y \cdot \left(y \cdot 4\right)}\right)\\

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

\end{array}
double code(double x, double y) {
	return ((double) (((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 <= -1.3391086476024713e+154)) {
		VAR = 1.0;
	} else {
		double VAR_1;
		if ((x <= -1.0517208507748325e-149)) {
			VAR_1 = ((double) (((double) (x * ((double) (x / ((double) (((double) (x * x)) + ((double) (y * ((double) (y * 4.0)))))))))) - ((double) (y * ((double) (4.0 * ((double) (y / ((double) (((double) (x * x)) + ((double) (y * ((double) (y * 4.0))))))))))))));
		} else {
			double VAR_2;
			if ((x <= 4.865934429524617e-79)) {
				VAR_2 = -1.0;
			} else {
				double VAR_3;
				if ((x <= 1.1102102412278923e+154)) {
					VAR_3 = ((double) (((double) (x * ((double) (x / ((double) (((double) (x * x)) + ((double) (y * ((double) (y * 4.0)))))))))) - ((double) (y * ((double) (4.0 * ((double) (y / ((double) (((double) (x * x)) + ((double) (y * ((double) (y * 4.0))))))))))))));
				} else {
					VAR_3 = 1.0;
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		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

Original30.1
Target30.2
Herbie12.3
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.974323384962678118:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3391086476024713e154 or 1.110210241227892e154 < x

    1. Initial program 62.2

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

    2. Taylor expanded around inf 8.9

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

    if -1.3391086476024713e154 < x < -1.05172085077483252e-149 or 4.86593442952461677e-79 < x < 1.110210241227892e154

    1. Initial program 15.0

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
    2. Using strategy rm

    3. Applied div-sub15.0

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

    4. Simplified15.2

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

    5. Simplified14.9

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

    if -1.05172085077483252e-149 < x < 4.86593442952461677e-79

    1. Initial program 26.4

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

    2. Taylor expanded around 0 11.3

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

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3391086476024713 \cdot 10^{154}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -1.05172085077483252 \cdot 10^{-149}:\\ \;\;\;\;x \cdot \frac{x}{x \cdot x + y \cdot \left(y \cdot 4\right)} - y \cdot \left(4 \cdot \frac{y}{x \cdot x + y \cdot \left(y \cdot 4\right)}\right)\\ \mathbf{elif}\;x \le 4.86593442952461677 \cdot 10^{-79}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 1.110210241227892 \cdot 10^{154}:\\ \;\;\;\;x \cdot \frac{x}{x \cdot x + y \cdot \left(y \cdot 4\right)} - y \cdot \left(4 \cdot \frac{y}{x \cdot x + y \cdot \left(y \cdot 4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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))))