Average Error: 0.4 → 0.2
Time: 4.2s
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 r294500 = x;
        double r294501 = y;
        double r294502 = r294501 - r294500;
        double r294503 = 6.0;
        double r294504 = r294502 * r294503;
        double r294505 = 2.0;
        double r294506 = 3.0;
        double r294507 = r294505 / r294506;
        double r294508 = z;
        double r294509 = r294507 - r294508;
        double r294510 = r294504 * r294509;
        double r294511 = r294500 + r294510;
        return r294511;
}
↓
double f(double x, double y, double z) {
        double r294512 = y;
        double r294513 = x;
        double r294514 = r294512 - r294513;
        double r294515 = 4.0;
        double r294516 = 6.0;
        double r294517 = z;
        double r294518 = r294516 * r294517;
        double r294519 = r294515 - r294518;
        double r294520 = fma(r294514, r294519, r294513);
        return r294520;
}

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 2019362 +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))))