Average Error: 0.4 → 0.5
Time: 14.5s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\right)\]
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 r316528 = x;
        double r316529 = y;
        double r316530 = r316529 - r316528;
        double r316531 = 6.0;
        double r316532 = r316530 * r316531;
        double r316533 = 2.0;
        double r316534 = 3.0;
        double r316535 = r316533 / r316534;
        double r316536 = z;
        double r316537 = r316535 - r316536;
        double r316538 = r316532 * r316537;
        double r316539 = r316528 + r316538;
        return r316539;
}
↓
double f(double x, double y, double z) {
        double r316540 = x;
        double r316541 = cbrt(r316540);
        double r316542 = r316541 * r316541;
        double r316543 = y;
        double r316544 = r316543 - r316540;
        double r316545 = 6.0;
        double r316546 = r316544 * r316545;
        double r316547 = 2.0;
        double r316548 = 3.0;
        double r316549 = r316547 / r316548;
        double r316550 = z;
        double r316551 = r316549 - r316550;
        double r316552 = r316546 * r316551;
        double r316553 = fma(r316542, r316541, r316552);
        return r316553;
}

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))))