Average Error: 0.4 → 0.3
Time: 20.3s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(y - x\right) \cdot \left(\left(\frac{2}{3} - z\right) \cdot 6\right)\right)\]
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(y - x\right) \cdot \left(\left(\frac{2}{3} - z\right) \cdot 6\right)\right)
double f(double x, double y, double z) {
        double r307492 = x;
        double r307493 = y;
        double r307494 = r307493 - r307492;
        double r307495 = 6.0;
        double r307496 = r307494 * r307495;
        double r307497 = 2.0;
        double r307498 = 3.0;
        double r307499 = r307497 / r307498;
        double r307500 = z;
        double r307501 = r307499 - r307500;
        double r307502 = r307496 * r307501;
        double r307503 = r307492 + r307502;
        return r307503;
}
↓
double f(double x, double y, double z) {
        double r307504 = x;
        double r307505 = cbrt(r307504);
        double r307506 = r307505 * r307505;
        double r307507 = y;
        double r307508 = r307507 - r307504;
        double r307509 = 2.0;
        double r307510 = 3.0;
        double r307511 = r307509 / r307510;
        double r307512 = z;
        double r307513 = r307511 - r307512;
        double r307514 = 6.0;
        double r307515 = r307513 * r307514;
        double r307516 = r307508 * r307515;
        double r307517 = fma(r307506, r307505, r307516);
        return r307517;
}

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. Simplified0.2

    \[\leadsto x + \left(y - x\right) \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right)}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.3

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

  7. Applied fma-def0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020043 +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))))