Average Error: 0.1 → 0.1
Time: 4.3s
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 r121361 = x;
        double r121362 = y;
        double r121363 = r121361 * r121362;
        double r121364 = z;
        double r121365 = r121363 + r121364;
        double r121366 = r121365 * r121362;
        double r121367 = t;
        double r121368 = r121366 + r121367;
        return r121368;
}
↓
double f(double x, double y, double z, double t) {
        double r121369 = x;
        double r121370 = y;
        double r121371 = z;
        double r121372 = fma(r121369, r121370, r121371);
        double r121373 = t;
        double r121374 = fma(r121372, r121370, r121373);
        return r121374;
}

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 2019353 +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))