Average Error: 0.1 → 0.1
Time: 10.3s
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 r131473 = x;
        double r131474 = y;
        double r131475 = r131473 * r131474;
        double r131476 = z;
        double r131477 = r131475 + r131476;
        double r131478 = r131477 * r131474;
        double r131479 = t;
        double r131480 = r131478 + r131479;
        return r131480;
}
↓
double f(double x, double y, double z, double t) {
        double r131481 = x;
        double r131482 = y;
        double r131483 = z;
        double r131484 = fma(r131481, r131482, r131483);
        double r131485 = t;
        double r131486 = fma(r131484, r131482, r131485);
        return r131486;
}

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 2020043 +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))