Average Error: 0.4 → 0.2
Time: 4.2s
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 r195733 = x;
        double r195734 = y;
        double r195735 = r195734 - r195733;
        double r195736 = 6.0;
        double r195737 = r195735 * r195736;
        double r195738 = 2.0;
        double r195739 = 3.0;
        double r195740 = r195738 / r195739;
        double r195741 = z;
        double r195742 = r195740 - r195741;
        double r195743 = r195737 * r195742;
        double r195744 = r195733 + r195743;
        return r195744;
}
↓
double f(double x, double y, double z) {
        double r195745 = y;
        double r195746 = x;
        double r195747 = r195745 - r195746;
        double r195748 = 4.0;
        double r195749 = 6.0;
        double r195750 = z;
        double r195751 = r195749 * r195750;
        double r195752 = r195748 - r195751;
        double r195753 = fma(r195747, r195752, r195746);
        return r195753;
}

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 2020020 +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))))