Average Error: 0.4 → 0.2
Time: 23.3s
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 r39742 = x;
        double r39743 = y;
        double r39744 = r39743 - r39742;
        double r39745 = 6.0;
        double r39746 = r39744 * r39745;
        double r39747 = 2.0;
        double r39748 = 3.0;
        double r39749 = r39747 / r39748;
        double r39750 = z;
        double r39751 = r39749 - r39750;
        double r39752 = r39746 * r39751;
        double r39753 = r39742 + r39752;
        return r39753;
}
↓
double f(double x, double y, double z) {
        double r39754 = y;
        double r39755 = x;
        double r39756 = r39754 - r39755;
        double r39757 = 4.0;
        double r39758 = 6.0;
        double r39759 = z;
        double r39760 = r39758 * r39759;
        double r39761 = r39757 - r39760;
        double r39762 = fma(r39756, r39761, r39755);
        return r39762;
}

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 2019310 +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))))