Average Error: 0.4 → 0.3
Time: 23.0s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(y - x\right)\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(y - x\right)\right)
double f(double x, double y, double z) {
        double r10867876 = x;
        double r10867877 = y;
        double r10867878 = r10867877 - r10867876;
        double r10867879 = 6.0;
        double r10867880 = r10867878 * r10867879;
        double r10867881 = 2.0;
        double r10867882 = 3.0;
        double r10867883 = r10867881 / r10867882;
        double r10867884 = z;
        double r10867885 = r10867883 - r10867884;
        double r10867886 = r10867880 * r10867885;
        double r10867887 = r10867876 + r10867886;
        return r10867887;
}
↓
double f(double x, double y, double z) {
        double r10867888 = x;
        double r10867889 = cbrt(r10867888);
        double r10867890 = r10867889 * r10867889;
        double r10867891 = 6.0;
        double r10867892 = 2.0;
        double r10867893 = 3.0;
        double r10867894 = r10867892 / r10867893;
        double r10867895 = z;
        double r10867896 = r10867894 - r10867895;
        double r10867897 = r10867891 * r10867896;
        double r10867898 = y;
        double r10867899 = r10867898 - r10867888;
        double r10867900 = r10867897 * r10867899;
        double r10867901 = fma(r10867890, r10867889, r10867900);
        return r10867901;
}

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 associate-*l*0.2

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied fma-def0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +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))))