Average Error: 0.1 → 0.1
Time: 4.6s
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 r121731 = x;
        double r121732 = y;
        double r121733 = r121731 * r121732;
        double r121734 = z;
        double r121735 = r121733 + r121734;
        double r121736 = r121735 * r121732;
        double r121737 = t;
        double r121738 = r121736 + r121737;
        return r121738;
}
↓
double f(double x, double y, double z, double t) {
        double r121739 = x;
        double r121740 = y;
        double r121741 = z;
        double r121742 = fma(r121739, r121740, r121741);
        double r121743 = t;
        double r121744 = fma(r121742, r121740, r121743);
        return r121744;
}

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 2020001 +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))