Average Error: 0.1 → 0.1
Time: 4.6s
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 r133625 = x;
        double r133626 = y;
        double r133627 = r133625 * r133626;
        double r133628 = z;
        double r133629 = r133627 + r133628;
        double r133630 = r133629 * r133626;
        double r133631 = t;
        double r133632 = r133630 + r133631;
        return r133632;
}
↓
double f(double x, double y, double z, double t) {
        double r133633 = x;
        double r133634 = y;
        double r133635 = z;
        double r133636 = fma(r133633, r133634, r133635);
        double r133637 = t;
        double r133638 = fma(r133636, r133634, r133637);
        return r133638;
}

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 2020057 +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))