Average Error: 0.4 → 0.5
Time: 13.8s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\right)\]
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(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\right)
double f(double x, double y, double z) {
        double r278600 = x;
        double r278601 = y;
        double r278602 = r278601 - r278600;
        double r278603 = 6.0;
        double r278604 = r278602 * r278603;
        double r278605 = 2.0;
        double r278606 = 3.0;
        double r278607 = r278605 / r278606;
        double r278608 = z;
        double r278609 = r278607 - r278608;
        double r278610 = r278604 * r278609;
        double r278611 = r278600 + r278610;
        return r278611;
}
↓
double f(double x, double y, double z) {
        double r278612 = x;
        double r278613 = cbrt(r278612);
        double r278614 = r278613 * r278613;
        double r278615 = y;
        double r278616 = r278615 - r278612;
        double r278617 = 6.0;
        double r278618 = r278616 * r278617;
        double r278619 = 2.0;
        double r278620 = 3.0;
        double r278621 = r278619 / r278620;
        double r278622 = z;
        double r278623 = r278621 - r278622;
        double r278624 = r278618 * r278623;
        double r278625 = fma(r278614, r278613, r278624);
        return r278625;
}

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 add-cube-cbrt0.5

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

  4. Applied fma-def0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2020047 +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))))