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

Average Error: 34.1 → 1.1

Time: 5.2s

Precision: 64

Math

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

↓

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

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 fma((z / t), (z / t), ((((x / (cbrt(y) * cbrt(y))) / cbrt(y)) / cbrt(y)) * (x / (cbrt(y) * cbrt(y)))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	34.1
Target	0.4
Herbie	1.1

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

Derivation

1. Initial program 34.1

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

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 times-frac 0.4

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

6. Applied add-cube-cbrt 0.8

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

7. Applied add-sqr-sqrt 32.4

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

8. Applied times-frac 32.4

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

9. Applied add-cube-cbrt 32.5

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

10. Applied add-sqr-sqrt 32.6

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

11. Applied times-frac 32.6

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

12. Applied swap-sqr 32.6

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

13. Simplified 32.6

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

14. Simplified 1.1

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

15. Final simplification 1.1

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

Reproduce

herbie shell --seed 2020058 +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))))