Average Error: 0.1 → 0.1
Time: 14.2s
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 r174195 = x;
        double r174196 = y;
        double r174197 = r174195 * r174196;
        double r174198 = z;
        double r174199 = r174197 + r174198;
        double r174200 = r174199 * r174196;
        double r174201 = t;
        double r174202 = r174200 + r174201;
        return r174202;
}
↓
double f(double x, double y, double z, double t) {
        double r174203 = x;
        double r174204 = y;
        double r174205 = z;
        double r174206 = fma(r174203, r174204, r174205);
        double r174207 = t;
        double r174208 = fma(r174206, r174204, r174207);
        return r174208;
}

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))