Average Error: 0.4 → 0.2
Time: 4.4s
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 r337894 = x;
        double r337895 = y;
        double r337896 = r337895 - r337894;
        double r337897 = 6.0;
        double r337898 = r337896 * r337897;
        double r337899 = 2.0;
        double r337900 = 3.0;
        double r337901 = r337899 / r337900;
        double r337902 = z;
        double r337903 = r337901 - r337902;
        double r337904 = r337898 * r337903;
        double r337905 = r337894 + r337904;
        return r337905;
}
↓
double f(double x, double y, double z) {
        double r337906 = y;
        double r337907 = x;
        double r337908 = r337906 - r337907;
        double r337909 = 4.0;
        double r337910 = 6.0;
        double r337911 = z;
        double r337912 = r337910 * r337911;
        double r337913 = r337909 - r337912;
        double r337914 = fma(r337908, r337913, r337907);
        return r337914;
}

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 2020081 +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))))