Average Error: 0.4 → 0.2
Time: 12.2s
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 r157786 = x;
        double r157787 = y;
        double r157788 = r157787 - r157786;
        double r157789 = 6.0;
        double r157790 = r157788 * r157789;
        double r157791 = 2.0;
        double r157792 = 3.0;
        double r157793 = r157791 / r157792;
        double r157794 = z;
        double r157795 = r157793 - r157794;
        double r157796 = r157790 * r157795;
        double r157797 = r157786 + r157796;
        return r157797;
}
↓
double f(double x, double y, double z) {
        double r157798 = y;
        double r157799 = x;
        double r157800 = r157798 - r157799;
        double r157801 = 4.0;
        double r157802 = 6.0;
        double r157803 = z;
        double r157804 = r157802 * r157803;
        double r157805 = r157801 - r157804;
        double r157806 = fma(r157800, r157805, r157799);
        return r157806;
}

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 2019235 +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))))