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

Average Error: 31.5 → 13.4

Time: 1.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 6.7749600182820292 \cdot 10^{-301}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 7.32131004751992112 \cdot 10^{-188}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 8.75867643358271617 \cdot 10^{-28}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.4607953395609545 \cdot 10^{239}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

C

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)) <= 6.774960018282029e-301)) {
		VAR = 1.0;
	} else {
		double VAR_1;
		if ((((double) (((double) (y * 4.0)) * y)) <= 7.321310047519921e-188)) {
			VAR_1 = ((double) (((double) (((double) (x * x)) - ((double) (((double) (y * 4.0)) * y)))) / ((double) (((double) (x * x)) + ((double) (((double) (y * 4.0)) * y))))));
		} else {
			double VAR_2;
			if ((((double) (((double) (y * 4.0)) * y)) <= 8.758676433582716e-28)) {
				VAR_2 = 1.0;
			} else {
				double VAR_3;
				if ((((double) (((double) (y * 4.0)) * y)) <= 3.4607953395609545e+239)) {
					VAR_3 = ((double) (((double) (((double) (x * x)) - ((double) (((double) (y * 4.0)) * y)))) / ((double) (((double) (x * x)) + ((double) (((double) (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

Target

Original	31.5
Target	31.2
Herbie	13.4

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

Derivation

1. Split input into 3 regimes

2. if (* (* y 4.0) y) < 6.774960018282029e-301 or 7.321310047519921e-188 < (* (* y 4.0) y) < 8.758676433582716e-28

   1. Initial program 25.9

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

   2. Taylor expanded around inf 14.2

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

   if 6.774960018282029e-301 < (* (* y 4.0) y) < 7.321310047519921e-188 or 8.758676433582716e-28 < (* (* y 4.0) y) < 3.4607953395609545e+239

   1. Initial program 15.2

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

   if 3.4607953395609545e+239 < (* (* y 4.0) y)

   1. Initial program 55.8

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

   2. Taylor expanded around 0 10.6

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

4. Final simplification 13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 6.7749600182820292 \cdot 10^{-301}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 7.32131004751992112 \cdot 10^{-188}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 8.75867643358271617 \cdot 10^{-28}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.4607953395609545 \cdot 10^{239}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(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))))