Average Error: 0.4 → 0.4
Time: 4.8s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(\frac{2}{3} \cdot \left(y - x\right), 6, x\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \left(-z\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(\frac{2}{3} \cdot \left(y - x\right), 6, x\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \left(-z\right)
double f(double x, double y, double z) {
        double r269468 = x;
        double r269469 = y;
        double r269470 = r269469 - r269468;
        double r269471 = 6.0;
        double r269472 = r269470 * r269471;
        double r269473 = 2.0;
        double r269474 = 3.0;
        double r269475 = r269473 / r269474;
        double r269476 = z;
        double r269477 = r269475 - r269476;
        double r269478 = r269472 * r269477;
        double r269479 = r269468 + r269478;
        return r269479;
}
↓
double f(double x, double y, double z) {
        double r269480 = 2.0;
        double r269481 = 3.0;
        double r269482 = r269480 / r269481;
        double r269483 = y;
        double r269484 = x;
        double r269485 = r269483 - r269484;
        double r269486 = r269482 * r269485;
        double r269487 = 6.0;
        double r269488 = fma(r269486, r269487, r269484);
        double r269489 = r269485 * r269487;
        double r269490 = z;
        double r269491 = -r269490;
        double r269492 = r269489 * r269491;
        double r269493 = r269488 + r269492;
        return r269493;
}

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. Using strategy rm

  3. Applied sub-neg0.4

    \[\leadsto x + \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(-z\right)\right)}\]

  4. Applied distribute-lft-in0.4

    \[\leadsto x + \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(-z\right)\right)}\]

  5. Applied associate-+r+0.4

    \[\leadsto \color{blue}{\left(x + \left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3}\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \left(-z\right)}\]

  6. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} \cdot \left(y - x\right), 6, x\right)} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(-z\right)\]

  7. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\frac{2}{3} \cdot \left(y - x\right), 6, x\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \left(-z\right)\]

Reproduce

herbie shell --seed 2019353 +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))))