Average Error: 0.4 → 0.2
Time: 4.1s
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 r210773 = x;
        double r210774 = y;
        double r210775 = r210774 - r210773;
        double r210776 = 6.0;
        double r210777 = r210775 * r210776;
        double r210778 = 2.0;
        double r210779 = 3.0;
        double r210780 = r210778 / r210779;
        double r210781 = z;
        double r210782 = r210780 - r210781;
        double r210783 = r210777 * r210782;
        double r210784 = r210773 + r210783;
        return r210784;
}
↓
double f(double x, double y, double z) {
        double r210785 = y;
        double r210786 = x;
        double r210787 = r210785 - r210786;
        double r210788 = 4.0;
        double r210789 = 6.0;
        double r210790 = z;
        double r210791 = r210789 * r210790;
        double r210792 = r210788 - r210791;
        double r210793 = fma(r210787, r210792, r210786);
        return r210793;
}

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 2020034 +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))))