Average Error: 0.4 → 0.3
Time: 19.4s
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 r10874168 = x;
        double r10874169 = y;
        double r10874170 = r10874169 - r10874168;
        double r10874171 = 6.0;
        double r10874172 = r10874170 * r10874171;
        double r10874173 = 2.0;
        double r10874174 = 3.0;
        double r10874175 = r10874173 / r10874174;
        double r10874176 = z;
        double r10874177 = r10874175 - r10874176;
        double r10874178 = r10874172 * r10874177;
        double r10874179 = r10874168 + r10874178;
        return r10874179;
}
↓
double f(double x, double y, double z) {
        double r10874180 = x;
        double r10874181 = cbrt(r10874180);
        double r10874182 = r10874181 * r10874181;
        double r10874183 = 6.0;
        double r10874184 = 2.0;
        double r10874185 = 3.0;
        double r10874186 = r10874184 / r10874185;
        double r10874187 = z;
        double r10874188 = r10874186 - r10874187;
        double r10874189 = r10874183 * r10874188;
        double r10874190 = y;
        double r10874191 = r10874190 - r10874180;
        double r10874192 = r10874189 * r10874191;
        double r10874193 = fma(r10874182, r10874181, r10874192);
        return r10874193;
}

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))))