Average Error: 0.4 → 0.2
Time: 12.1s
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 r150173 = x;
        double r150174 = y;
        double r150175 = r150174 - r150173;
        double r150176 = 6.0;
        double r150177 = r150175 * r150176;
        double r150178 = 2.0;
        double r150179 = 3.0;
        double r150180 = r150178 / r150179;
        double r150181 = z;
        double r150182 = r150180 - r150181;
        double r150183 = r150177 * r150182;
        double r150184 = r150173 + r150183;
        return r150184;
}
↓
double f(double x, double y, double z) {
        double r150185 = y;
        double r150186 = x;
        double r150187 = r150185 - r150186;
        double r150188 = 4.0;
        double r150189 = 6.0;
        double r150190 = z;
        double r150191 = r150189 * r150190;
        double r150192 = r150188 - r150191;
        double r150193 = fma(r150187, r150192, r150186);
        return r150193;
}

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 2019212 +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))))