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

Average Error: 31.8 → 13.4

Time: 2.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot x \le 4.62860521086858958 \cdot 10^{-182}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 2.08191304081905237 \cdot 10^{-41}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;x \cdot x \le 1.23303926919671843 \cdot 10^{-11}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 5.4237368993854683 \cdot 10^{116}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

C

double code(double x, double y) {
	return (((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y)));
}

↓

double code(double x, double y) {
	double temp;
	if (((x * x) <= 4.6286052108685896e-182)) {
		temp = -1.0;
	} else {
		double temp_1;
		if (((x * x) <= 2.0819130408190524e-41)) {
			temp_1 = (((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y)));
		} else {
			double temp_2;
			if (((x * x) <= 1.2330392691967184e-11)) {
				temp_2 = -1.0;
			} else {
				double temp_3;
				if (((x * x) <= 5.423736899385468e+116)) {
					temp_3 = ((cbrt((((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y)))) * cbrt((((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y))))) * cbrt((((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y)))));
				} else {
					temp_3 = 1.0;
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	31.8
Target	31.5
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 4 regimes

2. if (* x x) < 4.6286052108685896e-182 or 2.0819130408190524e-41 < (* x x) < 1.2330392691967184e-11

   1. Initial program 25.1

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

   2. Taylor expanded around 0 12.6

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

   if 4.6286052108685896e-182 < (* x x) < 2.0819130408190524e-41

   1. Initial program 15.8

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

   if 1.2330392691967184e-11 < (* x x) < 5.423736899385468e+116

   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 add-cube-cbrt 15.7

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}}\]

   if 5.423736899385468e+116 < (* x x)

   1. Initial program 46.0

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

   2. Taylor expanded around inf 13.0

      \[\leadsto \color{blue}{1}\]
3. Recombined 4 regimes into one program.

4. Final simplification 13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \le 4.62860521086858958 \cdot 10^{-182}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 2.08191304081905237 \cdot 10^{-41}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;x \cdot x \le 1.23303926919671843 \cdot 10^{-11}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 5.4237368993854683 \cdot 10^{116}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 
(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))))