Average Error: 0.4 → 0.2
Time: 3.6s
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 r217189 = x;
        double r217190 = y;
        double r217191 = r217190 - r217189;
        double r217192 = 6.0;
        double r217193 = r217191 * r217192;
        double r217194 = 2.0;
        double r217195 = 3.0;
        double r217196 = r217194 / r217195;
        double r217197 = z;
        double r217198 = r217196 - r217197;
        double r217199 = r217193 * r217198;
        double r217200 = r217189 + r217199;
        return r217200;
}
↓
double f(double x, double y, double z) {
        double r217201 = y;
        double r217202 = x;
        double r217203 = r217201 - r217202;
        double r217204 = 4.0;
        double r217205 = 6.0;
        double r217206 = z;
        double r217207 = r217205 * r217206;
        double r217208 = r217204 - r217207;
        double r217209 = fma(r217203, r217208, r217202);
        return r217209;
}

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 2020001 +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))))