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

Average Error: 33.7 → 0.6

Time: 4.5s

Precision: 64

Math

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

↓

\[\sqrt{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\frac{3}{2}}} \cdot \left(\sqrt{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\frac{3}{2}}} \cdot \sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}\right)\]

C

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

↓

double code(double x, double y, double z, double t) {
	return (sqrt(pow(hypot((z / t), (x / y)), 1.5)) * (sqrt(pow(hypot((z / t), (x / y)), 1.5)) * sqrt(hypot((z / t), (x / y)))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	33.7
Target	0.4
Herbie	0.6

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

Derivation

1. Initial program 33.7

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

2. Simplified 19.1

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

4. Applied add-sqr-sqrt 19.1

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

5. Simplified 19.1

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

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. Applied associate-*r* 0.6

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

10. Simplified 0.8

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

12. Applied pow1/2 0.8

    \[\leadsto {\color{blue}{\left({\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\frac{1}{2}}\right)}}^{3} \cdot \sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}\]

13. Applied pow-pow 0.5

    \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\left(\frac{1}{2} \cdot 3\right)}} \cdot \sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}\]

14. Simplified 0.5

    \[\leadsto {\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\color{blue}{\frac{3}{2}}} \cdot \sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}\]
15. Using strategy rm

16. Applied add-sqr-sqrt 0.6

    \[\leadsto \color{blue}{\left(\sqrt{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\frac{3}{2}}} \cdot \sqrt{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\frac{3}{2}}}\right)} \cdot \sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}\]

17. Applied associate-*l* 0.6

    \[\leadsto \color{blue}{\sqrt{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\frac{3}{2}}} \cdot \left(\sqrt{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\frac{3}{2}}} \cdot \sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}\right)}\]

18. Final simplification 0.6

    \[\leadsto \sqrt{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\frac{3}{2}}} \cdot \left(\sqrt{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\frac{3}{2}}} \cdot \sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}\right)\]

Reproduce

herbie shell --seed 2020100 +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))))