Average Error: 0.4 → 0.5
Time: 20.1s
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 r226728 = x;
        double r226729 = y;
        double r226730 = r226729 - r226728;
        double r226731 = 6.0;
        double r226732 = r226730 * r226731;
        double r226733 = 2.0;
        double r226734 = 3.0;
        double r226735 = r226733 / r226734;
        double r226736 = z;
        double r226737 = r226735 - r226736;
        double r226738 = r226732 * r226737;
        double r226739 = r226728 + r226738;
        return r226739;
}
↓
double f(double x, double y, double z) {
        double r226740 = x;
        double r226741 = cbrt(r226740);
        double r226742 = r226741 * r226741;
        double r226743 = y;
        double r226744 = r226743 - r226740;
        double r226745 = 6.0;
        double r226746 = r226744 * r226745;
        double r226747 = 2.0;
        double r226748 = 3.0;
        double r226749 = r226747 / r226748;
        double r226750 = z;
        double r226751 = r226749 - r226750;
        double r226752 = r226746 * r226751;
        double r226753 = fma(r226742, r226741, r226752);
        return r226753;
}

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))))