Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D

Average Error: 0.4 → 0.2

Time: 2.0s

Precision: 64

Math

\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]

↓

\[\left(4 \cdot y - 3 \cdot x\right) + \left(-z\right) \cdot \left(\left(y - x\right) \cdot 6\right)\]

C

double code(double x, double y, double z) {
	return (x + (((y - x) * 6.0) * ((2.0 / 3.0) - z)));
}

↓

double code(double x, double y, double z) {
	return (((4.0 * y) - (3.0 * x)) + (-z * ((y - x) * 6.0)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

1. Initial program 0.4

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
2. Using strategy rm

3. Applied sub-neg 0.4

   \[\leadsto x + \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(-z\right)\right)}\]

4. Applied distribute-rgt-in 0.4

   \[\leadsto x + \color{blue}{\left(\frac{2}{3} \cdot \left(\left(y - x\right) \cdot 6\right) + \left(-z\right) \cdot \left(\left(y - x\right) \cdot 6\right)\right)}\]

5. Applied associate-+r+ 0.4

   \[\leadsto \color{blue}{\left(x + \frac{2}{3} \cdot \left(\left(y - x\right) \cdot 6\right)\right) + \left(-z\right) \cdot \left(\left(y - x\right) \cdot 6\right)}\]

6. Simplified 0.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} \cdot \left(y - x\right), 6, x\right)} + \left(-z\right) \cdot \left(\left(y - x\right) \cdot 6\right)\]

7. Taylor expanded around 0 0.2

   \[\leadsto \color{blue}{\left(4 \cdot y - 3 \cdot x\right)} + \left(-z\right) \cdot \left(\left(y - x\right) \cdot 6\right)\]

8. Final simplification 0.2

   \[\leadsto \left(4 \cdot y - 3 \cdot x\right) + \left(-z\right) \cdot \left(\left(y - x\right) \cdot 6\right)\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  :precision binary64
  (+ x (* (* (- y x) 6) (- (/ 2 3) z))))