Average Error: 0.4 → 0.6
Time: 18.9s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \sqrt[3]{x}, \left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \sqrt[3]{x}, \left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)\right)
double f(double x, double y, double z) {
        double r9208034 = x;
        double r9208035 = y;
        double r9208036 = r9208035 - r9208034;
        double r9208037 = 6.0;
        double r9208038 = r9208036 * r9208037;
        double r9208039 = 2.0;
        double r9208040 = 3.0;
        double r9208041 = r9208039 / r9208040;
        double r9208042 = z;
        double r9208043 = r9208041 - r9208042;
        double r9208044 = r9208038 * r9208043;
        double r9208045 = r9208034 + r9208044;
        return r9208045;
}
↓
double f(double x, double y, double z) {
        double r9208046 = x;
        double r9208047 = cbrt(r9208046);
        double r9208048 = r9208047 * r9208047;
        double r9208049 = cbrt(r9208048);
        double r9208050 = r9208049 * r9208049;
        double r9208051 = r9208049 * r9208050;
        double r9208052 = 2.0;
        double r9208053 = 3.0;
        double r9208054 = r9208052 / r9208053;
        double r9208055 = z;
        double r9208056 = r9208054 - r9208055;
        double r9208057 = y;
        double r9208058 = r9208057 - r9208046;
        double r9208059 = 6.0;
        double r9208060 = r9208058 * r9208059;
        double r9208061 = r9208056 * r9208060;
        double r9208062 = fma(r9208051, r9208047, r9208061);
        return r9208062;
}

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. Using strategy rm

  6. Applied add-cube-cbrt0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019174 +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))))