Average Error: 0.1 → 0.1
Time: 4.4s
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 r123299 = x;
        double r123300 = y;
        double r123301 = r123299 * r123300;
        double r123302 = z;
        double r123303 = r123301 + r123302;
        double r123304 = r123303 * r123300;
        double r123305 = t;
        double r123306 = r123304 + r123305;
        return r123306;
}
↓
double f(double x, double y, double z, double t) {
        double r123307 = x;
        double r123308 = y;
        double r123309 = z;
        double r123310 = fma(r123307, r123308, r123309);
        double r123311 = t;
        double r123312 = fma(r123310, r123308, r123311);
        return r123312;
}

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 2020060 +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))