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

Average Error: 31.8 → 14.1

Time: 1.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -3.4362841072323272 \cdot 10^{150}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -3.8396732637812812 \cdot 10^{-74}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;y \le 1.41284087263746274 \cdot 10^{-110}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 4.54642378775940041 \cdot 10^{-26}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;y \le 2.97781259123946345 \cdot 10^{57}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

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 temp;
	if ((y <= -3.436284107232327e+150)) {
		temp = -1.0;
	} else {
		double temp_1;
		if ((y <= -3.839673263781281e-74)) {
			temp_1 = (((x * x) - ((y * 4.0) * y)) / (sqrt(((x * x) + ((y * 4.0) * y))) * sqrt(((x * x) + ((y * 4.0) * y)))));
		} else {
			double temp_2;
			if ((y <= 1.4128408726374627e-110)) {
				temp_2 = 1.0;
			} else {
				double temp_3;
				if ((y <= 4.5464237877594004e-26)) {
					temp_3 = (((x * x) - ((y * 4.0) * y)) / (sqrt(((x * x) + ((y * 4.0) * y))) * sqrt(((x * x) + ((y * 4.0) * y)))));
				} else {
					double temp_4;
					if ((y <= 2.9778125912394635e+57)) {
						temp_4 = 1.0;
					} else {
						temp_4 = -1.0;
					}
					temp_3 = temp_4;
				}
				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	14.1

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

Derivation

1. Split input into 3 regimes

2. if y < -3.436284107232327e+150 or 2.9778125912394635e+57 < y

   1. Initial program 51.9

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

   2. Taylor expanded around 0 12.1

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

   if -3.436284107232327e+150 < y < -3.839673263781281e-74 or 1.4128408726374627e-110 < y < 4.5464237877594004e-26

   1. Initial program 15.5

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

   3. Applied add-sqr-sqrt 15.5

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

   if -3.839673263781281e-74 < y < 1.4128408726374627e-110 or 4.5464237877594004e-26 < y < 2.9778125912394635e+57

   1. Initial program 24.8

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

   2. Taylor expanded around inf 14.9

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

4. Final simplification 14.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.4362841072323272 \cdot 10^{150}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -3.8396732637812812 \cdot 10^{-74}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;y \le 1.41284087263746274 \cdot 10^{-110}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 4.54642378775940041 \cdot 10^{-26}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;y \le 2.97781259123946345 \cdot 10^{57}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))))