Average Error: 0.1 → 0.1
Time: 4.5s
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 r138532 = x;
        double r138533 = y;
        double r138534 = r138532 * r138533;
        double r138535 = z;
        double r138536 = r138534 + r138535;
        double r138537 = r138536 * r138533;
        double r138538 = t;
        double r138539 = r138537 + r138538;
        return r138539;
}
↓
double f(double x, double y, double z, double t) {
        double r138540 = x;
        double r138541 = y;
        double r138542 = z;
        double r138543 = fma(r138540, r138541, r138542);
        double r138544 = t;
        double r138545 = fma(r138543, r138541, r138544);
        return r138545;
}

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 2020024 +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))