Average Error: 0.4 → 0.4
Time: 20.4s
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 r9508674 = x;
        double r9508675 = y;
        double r9508676 = r9508675 - r9508674;
        double r9508677 = 6.0;
        double r9508678 = r9508676 * r9508677;
        double r9508679 = 2.0;
        double r9508680 = 3.0;
        double r9508681 = r9508679 / r9508680;
        double r9508682 = z;
        double r9508683 = r9508681 - r9508682;
        double r9508684 = r9508678 * r9508683;
        double r9508685 = r9508674 + r9508684;
        return r9508685;
}
↓
double f(double x, double y, double z) {
        double r9508686 = x;
        double r9508687 = cbrt(r9508686);
        double r9508688 = r9508687 * r9508687;
        double r9508689 = 6.0;
        double r9508690 = 2.0;
        double r9508691 = 3.0;
        double r9508692 = r9508690 / r9508691;
        double r9508693 = z;
        double r9508694 = r9508692 - r9508693;
        double r9508695 = r9508689 * r9508694;
        double r9508696 = y;
        double r9508697 = r9508696 - r9508686;
        double r9508698 = r9508695 * r9508697;
        double r9508699 = fma(r9508688, r9508687, r9508698);
        return r9508699;
}

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-def0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019179 +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))))