Average Error: 0.1 → 0.1
Time: 17.1s
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 r97329 = x;
        double r97330 = y;
        double r97331 = r97329 * r97330;
        double r97332 = z;
        double r97333 = r97331 + r97332;
        double r97334 = r97333 * r97330;
        double r97335 = t;
        double r97336 = r97334 + r97335;
        return r97336;
}
↓
double f(double x, double y, double z, double t) {
        double r97337 = x;
        double r97338 = y;
        double r97339 = z;
        double r97340 = fma(r97337, r97338, r97339);
        double r97341 = t;
        double r97342 = fma(r97340, r97338, r97341);
        return r97342;
}

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 2019322 +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))