Average Error: 0.1 → 0.1
Time: 16.5s
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 r121187 = x;
        double r121188 = y;
        double r121189 = r121187 * r121188;
        double r121190 = z;
        double r121191 = r121189 + r121190;
        double r121192 = r121191 * r121188;
        double r121193 = t;
        double r121194 = r121192 + r121193;
        return r121194;
}
↓
double f(double x, double y, double z, double t) {
        double r121195 = x;
        double r121196 = y;
        double r121197 = z;
        double r121198 = fma(r121195, r121196, r121197);
        double r121199 = t;
        double r121200 = fma(r121198, r121196, r121199);
        return r121200;
}

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 2019303 +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))