Average Error: 0.1 → 0.1
Time: 12.6s
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 r100735 = x;
        double r100736 = y;
        double r100737 = r100735 * r100736;
        double r100738 = z;
        double r100739 = r100737 + r100738;
        double r100740 = r100739 * r100736;
        double r100741 = t;
        double r100742 = r100740 + r100741;
        return r100742;
}
↓
double f(double x, double y, double z, double t) {
        double r100743 = x;
        double r100744 = y;
        double r100745 = z;
        double r100746 = fma(r100743, r100744, r100745);
        double r100747 = t;
        double r100748 = fma(r100746, r100744, r100747);
        return r100748;
}

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 2019235 +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))