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 r201944 = x;
        double r201945 = y;
        double r201946 = r201944 * r201945;
        double r201947 = z;
        double r201948 = r201946 + r201947;
        double r201949 = r201948 * r201945;
        double r201950 = t;
        double r201951 = r201949 + r201950;
        return r201951;
}
↓
double f(double x, double y, double z, double t) {
        double r201952 = x;
        double r201953 = y;
        double r201954 = z;
        double r201955 = fma(r201952, r201953, r201954);
        double r201956 = t;
        double r201957 = fma(r201955, r201953, r201956);
        return r201957;
}

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 2019362 +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))