Average Error: 0.4 → 0.2
Time: 4.4s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(y - x, 4 - 6 \cdot z, x\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(y - x, 4 - 6 \cdot z, x\right)
double f(double x, double y, double z) {
        double r194756 = x;
        double r194757 = y;
        double r194758 = r194757 - r194756;
        double r194759 = 6.0;
        double r194760 = r194758 * r194759;
        double r194761 = 2.0;
        double r194762 = 3.0;
        double r194763 = r194761 / r194762;
        double r194764 = z;
        double r194765 = r194763 - r194764;
        double r194766 = r194760 * r194765;
        double r194767 = r194756 + r194766;
        return r194767;
}
↓
double f(double x, double y, double z) {
        double r194768 = y;
        double r194769 = x;
        double r194770 = r194768 - r194769;
        double r194771 = 4.0;
        double r194772 = 6.0;
        double r194773 = z;
        double r194774 = r194772 * r194773;
        double r194775 = r194771 - r194774;
        double r194776 = fma(r194770, r194775, r194769);
        return r194776;
}

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. Simplified0.2

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

  3. Taylor expanded around 0 0.2

    \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4 - 6 \cdot z}, x\right)\]

  4. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(y - x, 4 - 6 \cdot z, x\right)\]

Reproduce

herbie shell --seed 2020027 +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))))