Average Error: 0.4 → 0.4
Time: 51.7s
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 r13763567 = x;
        double r13763568 = y;
        double r13763569 = r13763568 - r13763567;
        double r13763570 = 6.0;
        double r13763571 = r13763569 * r13763570;
        double r13763572 = 2.0;
        double r13763573 = 3.0;
        double r13763574 = r13763572 / r13763573;
        double r13763575 = z;
        double r13763576 = r13763574 - r13763575;
        double r13763577 = r13763571 * r13763576;
        double r13763578 = r13763567 + r13763577;
        return r13763578;
}
↓
double f(double x, double y, double z) {
        double r13763579 = x;
        double r13763580 = cbrt(r13763579);
        double r13763581 = r13763580 * r13763580;
        double r13763582 = 6.0;
        double r13763583 = 2.0;
        double r13763584 = 3.0;
        double r13763585 = r13763583 / r13763584;
        double r13763586 = z;
        double r13763587 = r13763585 - r13763586;
        double r13763588 = r13763582 * r13763587;
        double r13763589 = y;
        double r13763590 = r13763589 - r13763579;
        double r13763591 = r13763588 * r13763590;
        double r13763592 = fma(r13763581, r13763580, r13763591);
        return r13763592;
}

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-def0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019168 +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))))