Average Error: 0.4 → 0.2
Time: 10.4s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\frac{2.0}{3.0} - z\right)\]
↓
\[\mathsf{fma}\left(y - x, 4.0 - 6.0 \cdot z, x\right) + \left(6.0 \cdot \left(y - x\right)\right) \cdot \mathsf{fma}\left(-z, 1, z\right)\]
x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\frac{2.0}{3.0} - z\right)
↓
\mathsf{fma}\left(y - x, 4.0 - 6.0 \cdot z, x\right) + \left(6.0 \cdot \left(y - x\right)\right) \cdot \mathsf{fma}\left(-z, 1, z\right)
double f(double x, double y, double z) {
        double r4463922 = x;
        double r4463923 = y;
        double r4463924 = r4463923 - r4463922;
        double r4463925 = 6.0;
        double r4463926 = r4463924 * r4463925;
        double r4463927 = 2.0;
        double r4463928 = 3.0;
        double r4463929 = r4463927 / r4463928;
        double r4463930 = z;
        double r4463931 = r4463929 - r4463930;
        double r4463932 = r4463926 * r4463931;
        double r4463933 = r4463922 + r4463932;
        return r4463933;
}
↓
double f(double x, double y, double z) {
        double r4463934 = y;
        double r4463935 = x;
        double r4463936 = r4463934 - r4463935;
        double r4463937 = 4.0;
        double r4463938 = 6.0;
        double r4463939 = z;
        double r4463940 = r4463938 * r4463939;
        double r4463941 = r4463937 - r4463940;
        double r4463942 = fma(r4463936, r4463941, r4463935);
        double r4463943 = r4463938 * r4463936;
        double r4463944 = -r4463939;
        double r4463945 = 1.0;
        double r4463946 = fma(r4463944, r4463945, r4463939);
        double r4463947 = r4463943 * r4463946;
        double r4463948 = r4463942 + r4463947;
        return r4463948;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.4

    \[x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\frac{2.0}{3.0} - z\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.4

    \[\leadsto x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\frac{2.0}{3.0} - \color{blue}{1 \cdot z}\right)\]

  4. Applied add-sqr-sqrt0.4

    \[\leadsto x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\color{blue}{\sqrt{\frac{2.0}{3.0}} \cdot \sqrt{\frac{2.0}{3.0}}} - 1 \cdot z\right)\]

  5. Applied prod-diff0.4

    \[\leadsto x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{\frac{2.0}{3.0}}, \sqrt{\frac{2.0}{3.0}}, -z \cdot 1\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right)\right)}\]

  6. Applied distribute-lft-in0.4

    \[\leadsto x + \color{blue}{\left(\left(\left(y - x\right) \cdot 6.0\right) \cdot \mathsf{fma}\left(\sqrt{\frac{2.0}{3.0}}, \sqrt{\frac{2.0}{3.0}}, -z \cdot 1\right) + \left(\left(y - x\right) \cdot 6.0\right) \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)\right)}\]

  7. Applied associate-+r+0.4

    \[\leadsto \color{blue}{\left(x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \mathsf{fma}\left(\sqrt{\frac{2.0}{3.0}}, \sqrt{\frac{2.0}{3.0}}, -z \cdot 1\right)\right) + \left(\left(y - x\right) \cdot 6.0\right) \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)}\]

  8. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6.0 \cdot \left(\frac{2.0}{3.0} + \left(-z\right)\right), x\right)} + \left(\left(y - x\right) \cdot 6.0\right) \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)\]

  9. Taylor expanded around 0 0.2

    \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4.0 - 6.0 \cdot z}, x\right) + \left(\left(y - x\right) \cdot 6.0\right) \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)\]

  10. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(y - x, 4.0 - 6.0 \cdot z, x\right) + \left(6.0 \cdot \left(y - x\right)\right) \cdot \mathsf{fma}\left(-z, 1, z\right)\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))