Average Error: 0.1 → 0.1
Time: 16.7s
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 r86614 = x;
        double r86615 = y;
        double r86616 = r86614 * r86615;
        double r86617 = z;
        double r86618 = r86616 + r86617;
        double r86619 = r86618 * r86615;
        double r86620 = t;
        double r86621 = r86619 + r86620;
        return r86621;
}
↓
double f(double x, double y, double z, double t) {
        double r86622 = x;
        double r86623 = y;
        double r86624 = z;
        double r86625 = fma(r86622, r86623, r86624);
        double r86626 = t;
        double r86627 = fma(r86625, r86623, r86626);
        return r86627;
}

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 2019306 +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))