Average Error: 0.4 → 0.2
Time: 3.7s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(y - x, 4 - 6 \cdot z, x\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(y - x, 4 - 6 \cdot z, x\right)
double f(double x, double y, double z) {
        double r256007 = x;
        double r256008 = y;
        double r256009 = r256008 - r256007;
        double r256010 = 6.0;
        double r256011 = r256009 * r256010;
        double r256012 = 2.0;
        double r256013 = 3.0;
        double r256014 = r256012 / r256013;
        double r256015 = z;
        double r256016 = r256014 - r256015;
        double r256017 = r256011 * r256016;
        double r256018 = r256007 + r256017;
        return r256018;
}
↓
double f(double x, double y, double z) {
        double r256019 = y;
        double r256020 = x;
        double r256021 = r256019 - r256020;
        double r256022 = 4.0;
        double r256023 = 6.0;
        double r256024 = z;
        double r256025 = r256023 * r256024;
        double r256026 = r256022 - r256025;
        double r256027 = fma(r256021, r256026, r256020);
        return r256027;
}

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. Simplified0.2

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

  3. Taylor expanded around 0 0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020062 +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))))