Average Error: 0.4 → 0.3
Time: 24.5s
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 r11198250 = x;
        double r11198251 = y;
        double r11198252 = r11198251 - r11198250;
        double r11198253 = 6.0;
        double r11198254 = r11198252 * r11198253;
        double r11198255 = 2.0;
        double r11198256 = 3.0;
        double r11198257 = r11198255 / r11198256;
        double r11198258 = z;
        double r11198259 = r11198257 - r11198258;
        double r11198260 = r11198254 * r11198259;
        double r11198261 = r11198250 + r11198260;
        return r11198261;
}
↓
double f(double x, double y, double z) {
        double r11198262 = x;
        double r11198263 = cbrt(r11198262);
        double r11198264 = r11198263 * r11198263;
        double r11198265 = 6.0;
        double r11198266 = 2.0;
        double r11198267 = 3.0;
        double r11198268 = r11198266 / r11198267;
        double r11198269 = z;
        double r11198270 = r11198268 - r11198269;
        double r11198271 = r11198265 * r11198270;
        double r11198272 = y;
        double r11198273 = r11198272 - r11198262;
        double r11198274 = r11198271 * r11198273;
        double r11198275 = fma(r11198264, r11198263, r11198274);
        return r11198275;
}

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))))