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

Average Error: 31.3 → 5.6

Time: 5.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -3.40649112388281499 \cdot 10^{151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 2.28590995318824201 \cdot 10^{150}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(y \cdot 4\right) \cdot y} + x}{{\left(\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}\right)}^{3}}}{\sqrt[3]{\mathsf{hypot}\left(x, \left|\sqrt[3]{y \cdot 4}\right| \cdot \sqrt{\sqrt[3]{y \cdot 4} \cdot y}\right)}} \cdot \frac{x - \sqrt{\left(y \cdot 4\right) \cdot y}}{\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

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 ((y <= -3.406491123882815e+151)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= 2.285909953188242e+150)) {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) sqrt(((double) (((double) (y * 4.0)) * y)))) + x)) / ((double) pow(((double) cbrt(((double) hypot(x, ((double) sqrt(((double) (((double) (y * 4.0)) * y)))))))), 3.0)))) / ((double) cbrt(((double) hypot(x, ((double) (((double) fabs(((double) cbrt(((double) (y * 4.0)))))) * ((double) sqrt(((double) (((double) cbrt(((double) (y * 4.0)))) * y)))))))))))) * ((double) (((double) (x - ((double) sqrt(((double) (((double) (y * 4.0)) * y)))))) / ((double) (((double) cbrt(((double) hypot(x, ((double) sqrt(((double) (((double) (y * 4.0)) * y)))))))) * ((double) cbrt(((double) hypot(x, ((double) sqrt(((double) (((double) (y * 4.0)) * y))))))))))))));
		} else {
			VAR_1 = -1.0;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	31.3
Target	31.1
Herbie	5.6

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

Derivation

1. Split input into 2 regimes

2. if y < -3.406491123882815e+151 or 2.285909953188242e+150 < y

   1. Initial program 63.3

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

   2. Taylor expanded around 0 8.4

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

   if -3.406491123882815e+151 < y < 2.285909953188242e+150

   1. Initial program 20.5

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

   3. Applied add-sqr-sqrt 20.5

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

   4. Simplified 20.5

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

   5. Simplified 20.5

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 21.4

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

   8. Applied add-cube-cbrt 21.8

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

   9. Applied swap-sqr 21.8

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

   10. Applied add-sqr-sqrt 21.8

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

   11. Applied difference-of-squares 21.8

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

   12. Applied times-frac 11.8

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

   13. Simplified 4.5

       \[\leadsto \color{blue}{\frac{\frac{\sqrt{\left(y \cdot 4\right) \cdot y} + x}{{\left(\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}\right)}^{3}}}{\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}} \cdot \frac{x - \sqrt{\left(y \cdot 4\right) \cdot y}}{\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}\]
   14. Using strategy rm

   15. Applied add-cube-cbrt 4.5

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

   16. Applied associate-*l* 4.5

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

   17. Applied sqrt-prod 4.6

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

   18. Simplified 4.6

       \[\leadsto \frac{\frac{\sqrt{\left(y \cdot 4\right) \cdot y} + x}{{\left(\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}\right)}^{3}}}{\sqrt[3]{\mathsf{hypot}\left(x, \color{blue}{\left|\sqrt[3]{y \cdot 4}\right|} \cdot \sqrt{\sqrt[3]{y \cdot 4} \cdot y}\right)}} \cdot \frac{x - \sqrt{\left(y \cdot 4\right) \cdot y}}{\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.40649112388281499 \cdot 10^{151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 2.28590995318824201 \cdot 10^{150}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(y \cdot 4\right) \cdot y} + x}{{\left(\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}\right)}^{3}}}{\sqrt[3]{\mathsf{hypot}\left(x, \left|\sqrt[3]{y \cdot 4}\right| \cdot \sqrt{\sqrt[3]{y \cdot 4} \cdot y}\right)}} \cdot \frac{x - \sqrt{\left(y \cdot 4\right) \cdot y}}{\sqrt[3]{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +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))))