Average Error: 0.4 → 0.3
Time: 12.9s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(y - x\right) \cdot \left(4 - 6 \cdot z\right)\right)\]
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(y - x\right) \cdot \left(4 - 6 \cdot z\right)\right)
double f(double x, double y, double z) {
        double r231897 = x;
        double r231898 = y;
        double r231899 = r231898 - r231897;
        double r231900 = 6.0;
        double r231901 = r231899 * r231900;
        double r231902 = 2.0;
        double r231903 = 3.0;
        double r231904 = r231902 / r231903;
        double r231905 = z;
        double r231906 = r231904 - r231905;
        double r231907 = r231901 * r231906;
        double r231908 = r231897 + r231907;
        return r231908;
}
↓
double f(double x, double y, double z) {
        double r231909 = x;
        double r231910 = cbrt(r231909);
        double r231911 = r231910 * r231910;
        double r231912 = y;
        double r231913 = r231912 - r231909;
        double r231914 = 4.0;
        double r231915 = 6.0;
        double r231916 = z;
        double r231917 = r231915 * r231916;
        double r231918 = r231914 - r231917;
        double r231919 = r231913 * r231918;
        double r231920 = fma(r231911, r231910, r231919);
        return r231920;
}

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. Simplified0.2

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

  5. Taylor expanded around 0 0.2

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

  7. Applied add-cube-cbrt0.3

    \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} + \left(y - x\right) \cdot \left(4 - 6 \cdot z\right)\]

  8. Applied fma-def0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(y - x\right) \cdot \left(4 - 6 \cdot z\right)\right)}\]

  9. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(y - x\right) \cdot \left(4 - 6 \cdot z\right)\right)\]

Reproduce

herbie shell --seed 2020045 +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))))