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

Average Error: 0.4 → 0.2

Time: 18.4s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(-x, 3, y \cdot 4\right) + \left(-z\right) \cdot \left(\left(y - x\right) \cdot 6\right)\]

C

double f(double x, double y, double z) {
        double r202006 = x;
        double r202007 = y;
        double r202008 = r202007 - r202006;
        double r202009 = 6.0;
        double r202010 = r202008 * r202009;
        double r202011 = 2.0;
        double r202012 = 3.0;
        double r202013 = r202011 / r202012;
        double r202014 = z;
        double r202015 = r202013 - r202014;
        double r202016 = r202010 * r202015;
        double r202017 = r202006 + r202016;
        return r202017;
}

↓

double f(double x, double y, double z) {
        double r202018 = x;
        double r202019 = -r202018;
        double r202020 = 3.0;
        double r202021 = y;
        double r202022 = 4.0;
        double r202023 = r202021 * r202022;
        double r202024 = fma(r202019, r202020, r202023);
        double r202025 = z;
        double r202026 = -r202025;
        double r202027 = r202021 - r202018;
        double r202028 = 6.0;
        double r202029 = r202027 * r202028;
        double r202030 = r202026 * r202029;
        double r202031 = r202024 + r202030;
        return r202031;
}

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 0.2

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

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. Simplified 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 3, y \cdot 4\right)} + \left(-z\right) \cdot \left(\left(y - x\right) \cdot 6\right)\]
9. Using strategy rm

10. Applied *-un-lft-identity 0.2

    \[\leadsto \mathsf{fma}\left(-x, 3, y \cdot 4\right) + \color{blue}{\left(1 \cdot \left(-z\right)\right)} \cdot \left(\left(y - x\right) \cdot 6\right)\]

11. Applied associate-*l* 0.2

    \[\leadsto \mathsf{fma}\left(-x, 3, y \cdot 4\right) + \color{blue}{1 \cdot \left(\left(-z\right) \cdot \left(\left(y - x\right) \cdot 6\right)\right)}\]

12. Simplified 0.2

    \[\leadsto \mathsf{fma}\left(-x, 3, y \cdot 4\right) + 1 \cdot \color{blue}{\left(\left(\left(y - x\right) \cdot z\right) \cdot \left(-6\right)\right)}\]
13. Using strategy rm

14. Applied distribute-rgt-neg-out 0.2

    \[\leadsto \mathsf{fma}\left(-x, 3, y \cdot 4\right) + 1 \cdot \color{blue}{\left(-\left(\left(y - x\right) \cdot z\right) \cdot 6\right)}\]

15. Simplified 0.2

    \[\leadsto \mathsf{fma}\left(-x, 3, y \cdot 4\right) + 1 \cdot \left(-\color{blue}{\left(6 \cdot \left(y - x\right)\right) \cdot z}\right)\]

16. Final simplification 0.2

    \[\leadsto \mathsf{fma}\left(-x, 3, y \cdot 4\right) + \left(-z\right) \cdot \left(\left(y - x\right) \cdot 6\right)\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))