Average Error: 0.4 → 0.2
Time: 20.9s
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 r177964 = x;
        double r177965 = y;
        double r177966 = r177965 - r177964;
        double r177967 = 6.0;
        double r177968 = r177966 * r177967;
        double r177969 = 2.0;
        double r177970 = 3.0;
        double r177971 = r177969 / r177970;
        double r177972 = z;
        double r177973 = r177971 - r177972;
        double r177974 = r177968 * r177973;
        double r177975 = r177964 + r177974;
        return r177975;
}
↓
double f(double x, double y, double z) {
        double r177976 = y;
        double r177977 = x;
        double r177978 = r177976 - r177977;
        double r177979 = 4.0;
        double r177980 = 6.0;
        double r177981 = z;
        double r177982 = r177980 * r177981;
        double r177983 = r177979 - r177982;
        double r177984 = fma(r177978, r177983, r177977);
        return r177984;
}

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 2019323 +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))))