Average Error: 0.4 → 0.4
Time: 5.1s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(\frac{2}{3} \cdot \left(y - x\right), 6, x\right) + \left(-z\right) \cdot \left(\left(y - x\right) \cdot 6\right)\]
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(-z\right) \cdot \left(\left(y - x\right) \cdot 6\right)
double f(double x, double y, double z) {
        double r215747 = x;
        double r215748 = y;
        double r215749 = r215748 - r215747;
        double r215750 = 6.0;
        double r215751 = r215749 * r215750;
        double r215752 = 2.0;
        double r215753 = 3.0;
        double r215754 = r215752 / r215753;
        double r215755 = z;
        double r215756 = r215754 - r215755;
        double r215757 = r215751 * r215756;
        double r215758 = r215747 + r215757;
        return r215758;
}
↓
double f(double x, double y, double z) {
        double r215759 = 2.0;
        double r215760 = 3.0;
        double r215761 = r215759 / r215760;
        double r215762 = y;
        double r215763 = x;
        double r215764 = r215762 - r215763;
        double r215765 = r215761 * r215764;
        double r215766 = 6.0;
        double r215767 = fma(r215765, r215766, r215763);
        double r215768 = z;
        double r215769 = -r215768;
        double r215770 = r215764 * r215766;
        double r215771 = r215769 * r215770;
        double r215772 = r215767 + r215771;
        return r215772;
}

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-rgt-in0.4

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

  5. Applied associate-+r+0.4

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

  6. Simplified0.4

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

  7. Final simplification0.4

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

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))))