Average Error: 0.1 → 0.1
Time: 16.3s
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 r134934 = x;
        double r134935 = y;
        double r134936 = r134934 * r134935;
        double r134937 = z;
        double r134938 = r134936 + r134937;
        double r134939 = r134938 * r134935;
        double r134940 = t;
        double r134941 = r134939 + r134940;
        return r134941;
}
↓
double f(double x, double y, double z, double t) {
        double r134942 = x;
        double r134943 = y;
        double r134944 = z;
        double r134945 = fma(r134942, r134943, r134944);
        double r134946 = t;
        double r134947 = fma(r134945, r134943, r134946);
        return r134947;
}

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 2019303 +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))