Average Error: 0.1 → 0.1
Time: 8.6s
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 r137748 = x;
        double r137749 = y;
        double r137750 = r137748 * r137749;
        double r137751 = z;
        double r137752 = r137750 + r137751;
        double r137753 = r137752 * r137749;
        double r137754 = t;
        double r137755 = r137753 + r137754;
        return r137755;
}
↓
double f(double x, double y, double z, double t) {
        double r137756 = x;
        double r137757 = y;
        double r137758 = z;
        double r137759 = fma(r137756, r137757, r137758);
        double r137760 = t;
        double r137761 = fma(r137759, r137757, r137760);
        return r137761;
}

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 2020046 +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))