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

Average Error: 31.2 → 12.9

Time: 2.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot x \le 1.14334768981791205 \cdot 10^{-206}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 4.7279890716798893 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \frac{x}{x \cdot x + y \cdot \left(y \cdot 4\right)} - y \cdot \frac{y}{\frac{x \cdot x + y \cdot \left(y \cdot 4\right)}{4}}\\ \mathbf{elif}\;x \cdot x \le 6.7323680991419826 \cdot 10^{-26}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 2.2837601005071328 \cdot 10^{259}:\\ \;\;\;\;x \cdot \frac{x}{x \cdot x + y \cdot \left(y \cdot 4\right)} - y \cdot \frac{y}{\frac{x \cdot x + y \cdot \left(y \cdot 4\right)}{4}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

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) (x * x)) <= 1.143347689817912e-206)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((((double) (x * x)) <= 4.7279890716798893e-63)) {
			VAR_1 = ((double) (((double) (x * ((double) (x / ((double) (((double) (x * x)) + ((double) (y * ((double) (y * 4.0)))))))))) - ((double) (y * ((double) (y / ((double) (((double) (((double) (x * x)) + ((double) (y * ((double) (y * 4.0)))))) / 4.0))))))));
		} else {
			double VAR_2;
			if ((((double) (x * x)) <= 6.732368099141983e-26)) {
				VAR_2 = -1.0;
			} else {
				double VAR_3;
				if ((((double) (x * x)) <= 2.2837601005071328e+259)) {
					VAR_3 = ((double) (((double) (x * ((double) (x / ((double) (((double) (x * x)) + ((double) (y * ((double) (y * 4.0)))))))))) - ((double) (y * ((double) (y / ((double) (((double) (((double) (x * x)) + ((double) (y * ((double) (y * 4.0)))))) / 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

Target

Original	31.2
Target	30.9
Herbie	12.9

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

Derivation

1. Split input into 3 regimes

2. if (* x x) < 1.14334768981791205e-206 or 4.7279890716798893e-63 < (* x x) < 6.7323680991419826e-26

   1. Initial program 25.7

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

   2. Taylor expanded around 0 12.7

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

   if 1.14334768981791205e-206 < (* x x) < 4.7279890716798893e-63 or 6.7323680991419826e-26 < (* x x) < 2.2837601005071328e259

   1. Initial program 15.7

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

   3. Applied div-sub 15.7

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

   4. Simplified 15.9

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

   5. Simplified 15.4

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

   if 2.2837601005071328e259 < (* x x)

   1. Initial program 57.5

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

   2. Taylor expanded around inf 10.0

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

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \le 1.14334768981791205 \cdot 10^{-206}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 4.7279890716798893 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \frac{x}{x \cdot x + y \cdot \left(y \cdot 4\right)} - y \cdot \frac{y}{\frac{x \cdot x + y \cdot \left(y \cdot 4\right)}{4}}\\ \mathbf{elif}\;x \cdot x \le 6.7323680991419826 \cdot 10^{-26}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 2.2837601005071328 \cdot 10^{259}:\\ \;\;\;\;x \cdot \frac{x}{x \cdot x + y \cdot \left(y \cdot 4\right)} - y \cdot \frac{y}{\frac{x \cdot x + y \cdot \left(y \cdot 4\right)}{4}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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))))