Average Error: 0.1 → 0.1
Time: 6.2s
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 r124833 = x;
        double r124834 = y;
        double r124835 = r124833 * r124834;
        double r124836 = z;
        double r124837 = r124835 + r124836;
        double r124838 = r124837 * r124834;
        double r124839 = t;
        double r124840 = r124838 + r124839;
        return r124840;
}
↓
double f(double x, double y, double z, double t) {
        double r124841 = x;
        double r124842 = y;
        double r124843 = z;
        double r124844 = fma(r124841, r124842, r124843);
        double r124845 = t;
        double r124846 = fma(r124844, r124842, r124845);
        return r124846;
}

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 2020047 +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))