Average Error: 31.8 → 12.7
Time: 1.8s
Precision: binary64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.7956671044621403 \cdot 10^{140}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -2.9881672222301808 \cdot 10^{-46}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot \left(y \cdot 4\right)}{x \cdot x - y \cdot \left(y \cdot 4\right)}}\\ \mathbf{elif}\;x \le 6.59557436800266193 \cdot 10^{-86}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 1.97234418195485838 \cdot 10^{95}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot \left(y \cdot 4\right)}{x \cdot x - y \cdot \left(y \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\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 -5.7956671044621403 \cdot 10^{140}:\\
\;\;\;\;1\\

\mathbf{elif}\;x \le -2.9881672222301808 \cdot 10^{-46}:\\
\;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot \left(y \cdot 4\right)}{x \cdot x - y \cdot \left(y \cdot 4\right)}}\\

\mathbf{elif}\;x \le 6.59557436800266193 \cdot 10^{-86}:\\
\;\;\;\;-1\\

\mathbf{elif}\;x \le 1.97234418195485838 \cdot 10^{95}:\\
\;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot \left(y \cdot 4\right)}{x \cdot x - y \cdot \left(y \cdot 4\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 <= -5.79566710446214e+140)) {
		VAR = 1.0;
	} else {
		double VAR_1;
		if ((x <= -2.9881672222301808e-46)) {
			VAR_1 = ((double) (1.0 / ((double) (((double) (((double) (x * x)) + ((double) (y * ((double) (y * 4.0)))))) / ((double) (((double) (x * x)) - ((double) (y * ((double) (y * 4.0))))))))));
		} else {
			double VAR_2;
			if ((x <= 6.595574368002662e-86)) {
				VAR_2 = -1.0;
			} else {
				double VAR_3;
				if ((x <= 1.9723441819548584e+95)) {
					VAR_3 = ((double) (1.0 / ((double) (((double) (((double) (x * x)) + ((double) (y * ((double) (y * 4.0)))))) / ((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

Original31.8
Target31.5
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 x < -5.7956671044621403e140 or 1.97234418195485838e95 < x

    1. Initial program 54.9

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

    2. Taylor expanded around inf 9.8

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

    if -5.7956671044621403e140 < x < -2.9881672222301808e-46 or 6.59557436800266193e-86 < x < 1.97234418195485838e95

    1. Initial program 15.6

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

    3. Applied clear-num15.6

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x - \left(y \cdot 4\right) \cdot y}}}\]

    4. Simplified15.6

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x + y \cdot \left(y \cdot 4\right)}{x \cdot x - y \cdot \left(y \cdot 4\right)}}}\]

    if -2.9881672222301808e-46 < x < 6.59557436800266193e-86

    1. Initial program 25.6

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

    2. Taylor expanded around 0 12.8

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

  4. Final simplification12.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.7956671044621403 \cdot 10^{140}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -2.9881672222301808 \cdot 10^{-46}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot \left(y \cdot 4\right)}{x \cdot x - y \cdot \left(y \cdot 4\right)}}\\ \mathbf{elif}\;x \le 6.59557436800266193 \cdot 10^{-86}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 1.97234418195485838 \cdot 10^{95}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot \left(y \cdot 4\right)}{x \cdot x - y \cdot \left(y \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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))))