Average Error: 0.1 → 0.1
Time: 4.2s
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 r134164 = x;
        double r134165 = y;
        double r134166 = r134164 * r134165;
        double r134167 = z;
        double r134168 = r134166 + r134167;
        double r134169 = r134168 * r134165;
        double r134170 = t;
        double r134171 = r134169 + r134170;
        return r134171;
}
↓
double f(double x, double y, double z, double t) {
        double r134172 = x;
        double r134173 = y;
        double r134174 = z;
        double r134175 = fma(r134172, r134173, r134174);
        double r134176 = t;
        double r134177 = fma(r134175, r134173, r134176);
        return r134177;
}

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))