Average Error: 0.1 → 0.1
Time: 14.1s
Precision: 64
\[\left(x \cdot y + z\right) \cdot y + t\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, t\right)\]
\left(x \cdot y + z\right) \cdot y + t
↓
\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, t\right)
double f(double x, double y, double z, double t) {
        double r78705 = x;
        double r78706 = y;
        double r78707 = r78705 * r78706;
        double r78708 = z;
        double r78709 = r78707 + r78708;
        double r78710 = r78709 * r78706;
        double r78711 = t;
        double r78712 = r78710 + r78711;
        return r78712;
}
↓
double f(double x, double y, double z, double t) {
        double r78713 = x;
        double r78714 = y;
        double r78715 = z;
        double r78716 = fma(r78713, r78714, r78715);
        double r78717 = t;
        double r78718 = fma(r78716, r78714, r78717);
        return r78718;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.1

    \[\left(x \cdot y + z\right) \cdot y + t\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, t\right)}\]

  3. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, t\right)\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z t)
  :name "Language.Haskell.HsColour.ColourHighlight:unbase from hscolour-1.23"
  :precision binary64
  (+ (* (+ (* x y) z) y) t))