Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3

Average Error: 30.6 → 16.4

Time: 1.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.60391653418792414 \cdot 10^{59}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -4.352408394580921 \cdot 10^{21}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le -1.2849439179793821 \cdot 10^{-40}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le 1.70625720008210268 \cdot 10^{-27}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

C

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 ((x <= -2.603916534187924e+59)) {
		VAR = 1.0;
	} else {
		double VAR_1;
		if ((x <= -4.352408394580921e+21)) {
			VAR_1 = -1.0;
		} else {
			double VAR_2;
			if ((x <= -1.284943917979382e-40)) {
				VAR_2 = 1.0;
			} else {
				double VAR_3;
				if ((x <= 1.7062572000821027e-27)) {
					VAR_3 = -1.0;
				} 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

Target

Original	30.6
Target	30.3
Herbie	16.4

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

Derivation

1. Split input into 2 regimes

2. if x < -2.603916534187924e+59 or -4.352408394580921e+21 < x < -1.284943917979382e-40 or 1.7062572000821027e-27 < x

   1. Initial program 38.4

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

   2. Taylor expanded around inf 17.2

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

   if -2.603916534187924e+59 < x < -4.352408394580921e+21 or -1.284943917979382e-40 < x < 1.7062572000821027e-27

   1. Initial program 22.1

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

   2. Taylor expanded around 0 15.5

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

4. Final simplification 16.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.60391653418792414 \cdot 10^{59}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -4.352408394580921 \cdot 10^{21}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le -1.2849439179793821 \cdot 10^{-40}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le 1.70625720008210268 \cdot 10^{-27}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 +o rules:numerics
(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))))