Average Error: 0.1 → 0.1
Time: 7.0s
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 r147462 = x;
        double r147463 = y;
        double r147464 = r147462 * r147463;
        double r147465 = z;
        double r147466 = r147464 + r147465;
        double r147467 = r147466 * r147463;
        double r147468 = t;
        double r147469 = r147467 + r147468;
        return r147469;
}
↓
double f(double x, double y, double z, double t) {
        double r147470 = x;
        double r147471 = y;
        double r147472 = z;
        double r147473 = fma(r147470, r147471, r147472);
        double r147474 = t;
        double r147475 = fma(r147473, r147471, r147474);
        return r147475;
}

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 2020047 +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))