Average Error: 0.1 → 0.1
Time: 9.9s
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 r147072 = x;
        double r147073 = y;
        double r147074 = r147072 * r147073;
        double r147075 = z;
        double r147076 = r147074 + r147075;
        double r147077 = r147076 * r147073;
        double r147078 = t;
        double r147079 = r147077 + r147078;
        return r147079;
}
↓
double f(double x, double y, double z, double t) {
        double r147080 = x;
        double r147081 = y;
        double r147082 = z;
        double r147083 = fma(r147080, r147081, r147082);
        double r147084 = t;
        double r147085 = fma(r147083, r147081, r147084);
        return r147085;
}

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 2020042 +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))