Average Error: 0.1 → 0.1
Time: 4.7s
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 r137491 = x;
        double r137492 = y;
        double r137493 = r137491 * r137492;
        double r137494 = z;
        double r137495 = r137493 + r137494;
        double r137496 = r137495 * r137492;
        double r137497 = t;
        double r137498 = r137496 + r137497;
        return r137498;
}
↓
double f(double x, double y, double z, double t) {
        double r137499 = x;
        double r137500 = y;
        double r137501 = z;
        double r137502 = fma(r137499, r137500, r137501);
        double r137503 = t;
        double r137504 = fma(r137502, r137500, r137503);
        return r137504;
}

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 2020001 +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))