Average Error: 0.1 → 0.1
Time: 4.2s
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 r135309 = x;
        double r135310 = y;
        double r135311 = r135309 * r135310;
        double r135312 = z;
        double r135313 = r135311 + r135312;
        double r135314 = r135313 * r135310;
        double r135315 = t;
        double r135316 = r135314 + r135315;
        return r135316;
}
↓
double f(double x, double y, double z, double t) {
        double r135317 = x;
        double r135318 = y;
        double r135319 = z;
        double r135320 = fma(r135317, r135318, r135319);
        double r135321 = t;
        double r135322 = fma(r135320, r135318, r135321);
        return r135322;
}

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 2020035 +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))