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 r124613 = x;
        double r124614 = y;
        double r124615 = r124613 * r124614;
        double r124616 = z;
        double r124617 = r124615 + r124616;
        double r124618 = r124617 * r124614;
        double r124619 = t;
        double r124620 = r124618 + r124619;
        return r124620;
}
↓
double f(double x, double y, double z, double t) {
        double r124621 = x;
        double r124622 = y;
        double r124623 = z;
        double r124624 = fma(r124621, r124622, r124623);
        double r124625 = t;
        double r124626 = fma(r124624, r124622, r124625);
        return r124626;
}

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 2019352 +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))