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

Average Error: 33.3 → 1.4

Time: 6.7s

Precision: binary64

Math

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

↓

\[\frac{\sqrt[3]{x}}{y} \cdot \left(x \cdot \left(\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{y}\right)\right) + \frac{z}{t} \cdot \frac{z}{t}\]

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) (((double) cbrt(x)) / y)) * ((double) (x * ((double) (((double) cbrt(x)) * ((double) (((double) cbrt(x)) / y)))))))) + ((double) (((double) (z / t)) * ((double) (z / t))))));
}

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	1.4

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

Derivation

1. Initial program 33.3

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

2. Simplified 24.4

   \[\leadsto \color{blue}{x \cdot \frac{x}{y \cdot y} + z \cdot \frac{z}{t \cdot t}}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 43.2

   \[\leadsto x \cdot \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y \cdot y} + z \cdot \frac{z}{t \cdot t}\]

5. Applied times-frac 39.5

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

6. Applied add-sqr-sqrt 39.5

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

7. Applied unswap-sqr 37.8

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

8. Simplified 37.7

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

9. Simplified 13.5

   \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} + z \cdot \frac{z}{t \cdot t}\]
10. Using strategy rm

11. Applied add-sqr-sqrt 37.7

    \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + z \cdot \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{t \cdot t}\]

12. Applied times-frac 33.3

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

13. Applied add-sqr-sqrt 33.3

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

14. Applied unswap-sqr 31.4

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

15. Simplified 31.3

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

16. Simplified 0.4

    \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}}\]
17. Using strategy rm

18. Applied *-un-lft-identity 0.4

    \[\leadsto \frac{x}{y} \cdot \frac{x}{\color{blue}{1 \cdot y}} + \frac{z}{t} \cdot \frac{z}{t}\]

19. Applied add-cube-cbrt 0.8

    \[\leadsto \frac{x}{y} \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{1 \cdot y} + \frac{z}{t} \cdot \frac{z}{t}\]

20. Applied times-frac 0.8

    \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \frac{\sqrt[3]{x}}{y}\right)} + \frac{z}{t} \cdot \frac{z}{t}\]

21. Applied associate-*r* 1.4

    \[\leadsto \color{blue}{\left(\frac{x}{y} \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right) \cdot \frac{\sqrt[3]{x}}{y}} + \frac{z}{t} \cdot \frac{z}{t}\]

22. Simplified 1.4

    \[\leadsto \color{blue}{\left(x \cdot \left(\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{y}\right)\right)} \cdot \frac{\sqrt[3]{x}}{y} + \frac{z}{t} \cdot \frac{z}{t}\]

23. Final simplification 1.4

    \[\leadsto \frac{\sqrt[3]{x}}{y} \cdot \left(x \cdot \left(\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{y}\right)\right) + \frac{z}{t} \cdot \frac{z}{t}\]

Reproduce

herbie shell --seed 2020184 
(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) 2.0) (pow (/ z t) 2.0))

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