Average Error: 0.4 → 0.2
Time: 13.4s
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 r157661 = x;
        double r157662 = y;
        double r157663 = r157662 - r157661;
        double r157664 = 6.0;
        double r157665 = r157663 * r157664;
        double r157666 = 2.0;
        double r157667 = 3.0;
        double r157668 = r157666 / r157667;
        double r157669 = z;
        double r157670 = r157668 - r157669;
        double r157671 = r157665 * r157670;
        double r157672 = r157661 + r157671;
        return r157672;
}
↓
double f(double x, double y, double z) {
        double r157673 = y;
        double r157674 = x;
        double r157675 = r157673 - r157674;
        double r157676 = 4.0;
        double r157677 = 6.0;
        double r157678 = z;
        double r157679 = r157677 * r157678;
        double r157680 = r157676 - r157679;
        double r157681 = fma(r157675, r157680, r157674);
        return r157681;
}

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 2019209 +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))))