Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1

Average Error: 33.3 → 0.5

Time: 5.1s

Precision: 64

Math

\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]

↓

\[\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right) \cdot \sqrt{\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}}\right) \cdot {\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\frac{3}{4}}\]

C

double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (x * x)) / ((double) (y * y)))) + ((double) (((double) (z * z)) / ((double) (t * t))))));
}

↓

double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) hypot(((double) (z / t)), ((double) (x / y)))) * ((double) sqrt(((double) sqrt(((double) hypot(((double) (z / t)), ((double) (x / y)))))))))) * ((double) pow(((double) hypot(((double) (z / t)), ((double) (x / y)))), 0.75))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	33.3
Target	0.4
Herbie	0.5

\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

1. Initial program 33.3

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]

2. Simplified 18.7

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 18.8

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)}}\]

5. Simplified 18.7

   \[\leadsto \color{blue}{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)}\]

6. Simplified 0.4

   \[\leadsto \mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right) \cdot \color{blue}{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}\]
7. Using strategy rm

8. Applied add-sqr-sqrt 0.6

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

10. Applied add-sqr-sqrt 0.6

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

11. Applied sqrt-prod 0.7

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

12. Applied associate-*l* 0.7

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

13. Applied associate-*r* 0.7

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

15. Applied pow1/2 0.7

    \[\leadsto \left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right) \cdot \sqrt{\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}}\right) \cdot \left(\sqrt{\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}} \cdot \color{blue}{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\frac{1}{2}}}\right)\]

16. Applied pow1/2 0.7

    \[\leadsto \left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right) \cdot \sqrt{\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}}\right) \cdot \left(\sqrt{\color{blue}{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\frac{1}{2}}}} \cdot {\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\frac{1}{2}}\right)\]

17. Applied sqrt-pow1 0.6

    \[\leadsto \left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right) \cdot \sqrt{\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}}\right) \cdot \left(\color{blue}{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot {\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\frac{1}{2}}\right)\]

18. Applied pow-prod-up 0.5

    \[\leadsto \left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right) \cdot \sqrt{\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}}\right) \cdot \color{blue}{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{1}{2}\right)}}\]

19. Simplified 0.5

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

20. Final simplification 0.5

    \[\leadsto \left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right) \cdot \sqrt{\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}}\right) \cdot {\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\frac{3}{4}}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

  :herbie-target
  (+ (pow (/ x y) 2) (pow (/ z t) 2))

  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))