Average Error: 31.4 → 12.7
Time: 1.9s
Precision: binary64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 3.03505332871859726 \cdot 10^{-286}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.20597489333461231 \cdot 10^{-11}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 1.02905667657082406 \cdot 10^{31}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 1.5901440680608424 \cdot 10^{273}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
\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}\;\left(y \cdot 4\right) \cdot y \le 3.03505332871859726 \cdot 10^{-286}:\\
\;\;\;\;1\\

\mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.20597489333461231 \cdot 10^{-11}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}\\

\mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 1.02905667657082406 \cdot 10^{31}:\\
\;\;\;\;1\\

\mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 1.5901440680608424 \cdot 10^{273}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}\\

\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 ((((double) (((double) (y * 4.0)) * y)) <= 3.0350533287185973e-286)) {
		VAR = 1.0;
	} else {
		double VAR_1;
		if ((((double) (((double) (y * 4.0)) * y)) <= 3.205974893334612e-11)) {
			VAR_1 = ((double) cbrt(((double) pow(((double) (((double) (((double) (x * x)) - ((double) (((double) (y * 4.0)) * y)))) / ((double) (((double) (x * x)) + ((double) (((double) (y * 4.0)) * y)))))), 3.0))));
		} else {
			double VAR_2;
			if ((((double) (((double) (y * 4.0)) * y)) <= 1.029056676570824e+31)) {
				VAR_2 = 1.0;
			} else {
				double VAR_3;
				if ((((double) (((double) (y * 4.0)) * y)) <= 1.5901440680608424e+273)) {
					VAR_3 = ((double) cbrt(((double) pow(((double) (((double) (((double) (x * x)) - ((double) (((double) (y * 4.0)) * y)))) / ((double) (((double) (x * x)) + ((double) (((double) (y * 4.0)) * y)))))), 3.0))));
				} 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

Original31.4
Target31.1
Herbie12.7
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.974323384962678118:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (* y 4.0) y) < 3.03505332871859726e-286 or 3.20597489333461231e-11 < (* (* y 4.0) y) < 1.02905667657082406e31

    1. Initial program 26.7

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

    2. Taylor expanded around inf 10.9

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

    if 3.03505332871859726e-286 < (* (* y 4.0) y) < 3.20597489333461231e-11 or 1.02905667657082406e31 < (* (* y 4.0) y) < 1.5901440680608424e273

    1. Initial program 16.4

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

    3. Applied add-cbrt-cube48.1

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}}\]

    4. Applied add-cbrt-cube48.6

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)}}}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}\]

    5. Applied cbrt-undiv48.6

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)}{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}}\]

    6. Simplified16.5

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}}\]

    if 1.5901440680608424e273 < (* (* y 4.0) y)

    1. Initial program 59.1

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

    2. Taylor expanded around 0 8.8

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

  4. Final simplification12.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 3.03505332871859726 \cdot 10^{-286}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.20597489333461231 \cdot 10^{-11}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 1.02905667657082406 \cdot 10^{31}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 1.5901440680608424 \cdot 10^{273}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2020149 
(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))))