Average Error: 0.1 → 0.1
Time: 14.6s
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 r108909 = x;
        double r108910 = y;
        double r108911 = r108909 * r108910;
        double r108912 = z;
        double r108913 = r108911 + r108912;
        double r108914 = r108913 * r108910;
        double r108915 = t;
        double r108916 = r108914 + r108915;
        return r108916;
}
↓
double f(double x, double y, double z, double t) {
        double r108917 = x;
        double r108918 = y;
        double r108919 = z;
        double r108920 = fma(r108917, r108918, r108919);
        double r108921 = t;
        double r108922 = fma(r108920, r108918, r108921);
        return r108922;
}

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 2019325 +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))