Average Error: 0.1 → 0.1
Time: 3.5s
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 r153249 = x;
        double r153250 = y;
        double r153251 = r153249 * r153250;
        double r153252 = z;
        double r153253 = r153251 + r153252;
        double r153254 = r153253 * r153250;
        double r153255 = t;
        double r153256 = r153254 + r153255;
        return r153256;
}
↓
double f(double x, double y, double z, double t) {
        double r153257 = x;
        double r153258 = y;
        double r153259 = z;
        double r153260 = fma(r153257, r153258, r153259);
        double r153261 = t;
        double r153262 = fma(r153260, r153258, r153261);
        return r153262;
}

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 2020039 +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))