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

Average Error: 31.6 → 4.2

Time: 2.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -7.29694135869560264 \cdot 10^{161}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -3.31275582958287272 \cdot 10^{-159}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(4, \frac{y}{\frac{x}{y}}, x\right)} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\ \mathbf{elif}\;y \le 2.88385576877549007 \cdot 10^{-163}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(4, \frac{y}{\frac{x}{y}}, x\right)} - 0\\ \mathbf{elif}\;y \le 2.5487421427540681 \cdot 10^{159}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(4, \frac{y}{\frac{x}{y}}, x\right)} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

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 VAR;
	if ((y <= -7.296941358695603e+161)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -3.3127558295828727e-159)) {
			VAR_1 = ((x / fma(4.0, (y / (x / y)), x)) - ((y * 4.0) / (fma(x, x, ((y * 4.0) * y)) / y)));
		} else {
			double VAR_2;
			if ((y <= 2.88385576877549e-163)) {
				VAR_2 = ((x / fma(4.0, (y / (x / y)), x)) - 0.0);
			} else {
				double VAR_3;
				if ((y <= 2.548742142754068e+159)) {
					VAR_3 = ((x / fma(4.0, (y / (x / y)), x)) - ((y * 4.0) / (fma(x, x, ((y * 4.0) * y)) / 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	31.6
Target	31.4
Herbie	4.2

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

Derivation

1. Split input into 3 regimes

2. if y < -7.296941358695603e+161 or 2.548742142754068e+159 < y

   1. Initial program 64.0

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

   2. Taylor expanded around 0 7.3

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

   if -7.296941358695603e+161 < y < -3.3127558295828727e-159 or 2.88385576877549e-163 < y < 2.548742142754068e+159

   1. Initial program 16.3

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

   3. Applied div-sub 16.3

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

   4. Simplified 15.9

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

   5. Simplified 16.0

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

   6. Taylor expanded around 0 1.4

      \[\leadsto \frac{x}{\color{blue}{4 \cdot \frac{{y}^{2}}{x} + x}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\]

   7. Simplified 1.4

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(4, \frac{{y}^{2}}{x}, x\right)}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\]
   8. Using strategy rm

   9. Applied unpow2 1.4

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

   10. Applied associate-/l* 1.0

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

   if -3.3127558295828727e-159 < y < 2.88385576877549e-163

   1. Initial program 30.7

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

   3. Applied div-sub 30.7

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

   4. Simplified 30.8

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

   5. Simplified 30.9

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

   6. Taylor expanded around 0 15.2

      \[\leadsto \frac{x}{\color{blue}{4 \cdot \frac{{y}^{2}}{x} + x}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\]

   7. Simplified 15.2

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(4, \frac{{y}^{2}}{x}, x\right)}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\]
   8. Using strategy rm

   9. Applied unpow2 15.2

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

   10. Applied associate-/l* 15.2

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

   11. Taylor expanded around 0 7.9

       \[\leadsto \frac{x}{\mathsf{fma}\left(4, \frac{y}{\frac{x}{y}}, x\right)} - \color{blue}{0}\]
3. Recombined 3 regimes into one program.

4. Final simplification 4.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.29694135869560264 \cdot 10^{161}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -3.31275582958287272 \cdot 10^{-159}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(4, \frac{y}{\frac{x}{y}}, x\right)} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\ \mathbf{elif}\;y \le 2.88385576877549007 \cdot 10^{-163}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(4, \frac{y}{\frac{x}{y}}, x\right)} - 0\\ \mathbf{elif}\;y \le 2.5487421427540681 \cdot 10^{159}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(4, \frac{y}{\frac{x}{y}}, x\right)} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +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))))