Average Error: 31.2 → 12.6
Time: 2.9s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -8.0247309759737891 \cdot 10^{75}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.80022384883163463 \cdot 10^{-154}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;y \le 3.01738360492243472 \cdot 10^{-160}:\\ \;\;\;\;\log \left(e^{1}\right)\\ \mathbf{elif}\;y \le 1.61700023952303702 \cdot 10^{143}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
\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}\;y \le -8.0247309759737891 \cdot 10^{75}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.80022384883163463 \cdot 10^{-154}:\\
\;\;\;\;\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}\\

\mathbf{elif}\;y \le 3.01738360492243472 \cdot 10^{-160}:\\
\;\;\;\;\log \left(e^{1}\right)\\

\mathbf{elif}\;y \le 1.61700023952303702 \cdot 10^{143}:\\
\;\;\;\;\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}\\

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

\end{array}
double code(double x, double y) {
	return (((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y)));
}
↓
double code(double x, double y) {
	double VAR;
	if ((y <= -8.024730975973789e+75)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -1.8002238488316346e-154)) {
			VAR_1 = (((x * x) / ((x * x) + ((y * 4.0) * y))) - (((y * 4.0) * y) / ((x * x) + ((y * 4.0) * y))));
		} else {
			double VAR_2;
			if ((y <= 3.017383604922435e-160)) {
				VAR_2 = log(exp(1.0));
			} else {
				double VAR_3;
				if ((y <= 1.617000239523037e+143)) {
					VAR_3 = (((x * x) / ((x * x) + ((y * 4.0) * y))) - (((y * 4.0) * y) / ((x * x) + ((y * 4.0) * y))));
				} 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

Original31.2
Target30.9
Herbie12.6
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.974323384962678118:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -8.024730975973789e+75 or 1.617000239523037e+143 < y

    1. Initial program 53.2

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

    2. Taylor expanded around 0 11.3

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

    if -8.024730975973789e+75 < y < -1.8002238488316346e-154 or 3.017383604922435e-160 < y < 1.617000239523037e+143

    1. Initial program 16.0

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

    3. Applied div-sub16.0

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

    if -1.8002238488316346e-154 < y < 3.017383604922435e-160

    1. Initial program 29.9

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

    3. Applied add-log-exp29.9

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

    4. Taylor expanded around inf 8.4

      \[\leadsto \log \left(e^{\color{blue}{1}}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -8.0247309759737891 \cdot 10^{75}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.80022384883163463 \cdot 10^{-154}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;y \le 3.01738360492243472 \cdot 10^{-160}:\\ \;\;\;\;\log \left(e^{1}\right)\\ \mathbf{elif}\;y \le 1.61700023952303702 \cdot 10^{143}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4))) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4)))) 2) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))))

  (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))))