Average Error: 0.4 → 0.2
Time: 21.2s
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 r180305 = x;
        double r180306 = y;
        double r180307 = r180306 - r180305;
        double r180308 = 6.0;
        double r180309 = r180307 * r180308;
        double r180310 = 2.0;
        double r180311 = 3.0;
        double r180312 = r180310 / r180311;
        double r180313 = z;
        double r180314 = r180312 - r180313;
        double r180315 = r180309 * r180314;
        double r180316 = r180305 + r180315;
        return r180316;
}
↓
double f(double x, double y, double z) {
        double r180317 = y;
        double r180318 = x;
        double r180319 = r180317 - r180318;
        double r180320 = 4.0;
        double r180321 = 6.0;
        double r180322 = z;
        double r180323 = r180321 * r180322;
        double r180324 = r180320 - r180323;
        double r180325 = fma(r180319, r180324, r180318);
        return r180325;
}

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))))