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 r130760 = x;
        double r130761 = y;
        double r130762 = r130760 * r130761;
        double r130763 = z;
        double r130764 = r130762 + r130763;
        double r130765 = r130764 * r130761;
        double r130766 = t;
        double r130767 = r130765 + r130766;
        return r130767;
}
↓
double f(double x, double y, double z, double t) {
        double r130768 = x;
        double r130769 = y;
        double r130770 = z;
        double r130771 = fma(r130768, r130769, r130770);
        double r130772 = t;
        double r130773 = fma(r130771, r130769, r130772);
        return r130773;
}

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 2020018 +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))