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

Average Error: 33.3 → 0.4

Time: 3.0s

Precision: 64

Math

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

↓

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

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((x / y), (x / y), ((z / t) / (t / z)));
}

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	0.4

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

Derivation

1. Initial program 33.3

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

2. Simplified 19.1

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

3. Taylor expanded around 0 33.3

   \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}} + \frac{{x}^{2}}{{y}^{2}}}\]

4. Simplified 0.4

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

6. Applied clear-num 0.4

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

7. Applied un-div-inv 0.4

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

8. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020078 +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))))