Average Error: 0.4 → 0.5
Time: 4.2s
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 r287535 = x;
        double r287536 = y;
        double r287537 = r287536 - r287535;
        double r287538 = 6.0;
        double r287539 = r287537 * r287538;
        double r287540 = 2.0;
        double r287541 = 3.0;
        double r287542 = r287540 / r287541;
        double r287543 = z;
        double r287544 = r287542 - r287543;
        double r287545 = r287539 * r287544;
        double r287546 = r287535 + r287545;
        return r287546;
}
↓
double f(double x, double y, double z) {
        double r287547 = x;
        double r287548 = cbrt(r287547);
        double r287549 = r287548 * r287548;
        double r287550 = y;
        double r287551 = r287550 - r287547;
        double r287552 = 6.0;
        double r287553 = r287551 * r287552;
        double r287554 = 2.0;
        double r287555 = 3.0;
        double r287556 = r287554 / r287555;
        double r287557 = z;
        double r287558 = r287556 - r287557;
        double r287559 = r287553 * r287558;
        double r287560 = fma(r287549, r287548, r287559);
        return r287560;
}

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))))