Average Error: 0.1 → 0.1
Time: 4.3s
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 r126049 = x;
        double r126050 = y;
        double r126051 = r126049 * r126050;
        double r126052 = z;
        double r126053 = r126051 + r126052;
        double r126054 = r126053 * r126050;
        double r126055 = t;
        double r126056 = r126054 + r126055;
        return r126056;
}
↓
double f(double x, double y, double z, double t) {
        double r126057 = x;
        double r126058 = y;
        double r126059 = z;
        double r126060 = fma(r126057, r126058, r126059);
        double r126061 = t;
        double r126062 = fma(r126060, r126058, r126061);
        return r126062;
}

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 2020020 +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))