Average Error: 0.4 → 0.3
Time: 4.0s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(y - x, 6 \cdot \left(\frac{2}{3} - z\right), x\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right) \cdot \left(\left(y - x\right) \cdot 6\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(y - x, 6 \cdot \left(\frac{2}{3} - z\right), x\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right) \cdot \left(\left(y - x\right) \cdot 6\right)
double f(double x, double y, double z) {
        double r220980 = x;
        double r220981 = y;
        double r220982 = r220981 - r220980;
        double r220983 = 6.0;
        double r220984 = r220982 * r220983;
        double r220985 = 2.0;
        double r220986 = 3.0;
        double r220987 = r220985 / r220986;
        double r220988 = z;
        double r220989 = r220987 - r220988;
        double r220990 = r220984 * r220989;
        double r220991 = r220980 + r220990;
        return r220991;
}
↓
double f(double x, double y, double z) {
        double r220992 = y;
        double r220993 = x;
        double r220994 = r220992 - r220993;
        double r220995 = 6.0;
        double r220996 = 2.0;
        double r220997 = 3.0;
        double r220998 = r220996 / r220997;
        double r220999 = z;
        double r221000 = r220998 - r220999;
        double r221001 = r220995 * r221000;
        double r221002 = fma(r220994, r221001, r220993);
        double r221003 = -r220999;
        double r221004 = 1.0;
        double r221005 = r220999 * r221004;
        double r221006 = fma(r221003, r221004, r221005);
        double r221007 = r220994 * r220995;
        double r221008 = r221006 * r221007;
        double r221009 = r221002 + r221008;
        return r221009;
}

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 *-un-lft-identity0.4

    \[\leadsto x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right)\]

  4. Applied add-sqr-sqrt0.4

    \[\leadsto x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\color{blue}{\sqrt{\frac{2}{3}} \cdot \sqrt{\frac{2}{3}}} - 1 \cdot z\right)\]

  5. Applied prod-diff0.4

    \[\leadsto x + \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{\frac{2}{3}}, \sqrt{\frac{2}{3}}, -z \cdot 1\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right)\right)}\]

  6. Applied distribute-rgt-in0.4

    \[\leadsto x + \color{blue}{\left(\mathsf{fma}\left(\sqrt{\frac{2}{3}}, \sqrt{\frac{2}{3}}, -z \cdot 1\right) \cdot \left(\left(y - x\right) \cdot 6\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right) \cdot \left(\left(y - x\right) \cdot 6\right)\right)}\]

  7. Applied associate-+r+0.4

    \[\leadsto \color{blue}{\left(x + \mathsf{fma}\left(\sqrt{\frac{2}{3}}, \sqrt{\frac{2}{3}}, -z \cdot 1\right) \cdot \left(\left(y - x\right) \cdot 6\right)\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right) \cdot \left(\left(y - x\right) \cdot 6\right)}\]

  8. Simplified0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \left(\frac{2}{3} - z\right), x\right)} + \mathsf{fma}\left(-z, 1, z \cdot 1\right) \cdot \left(\left(y - x\right) \cdot 6\right)\]

  9. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \left(\frac{2}{3} - z\right), x\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right) \cdot \left(\left(y - x\right) \cdot 6\right)\]

Reproduce

herbie shell --seed 2020021 +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))))