Average Error: 0.1 → 0.1
Time: 14.3s
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 r112073 = x;
        double r112074 = y;
        double r112075 = r112073 * r112074;
        double r112076 = z;
        double r112077 = r112075 + r112076;
        double r112078 = r112077 * r112074;
        double r112079 = t;
        double r112080 = r112078 + r112079;
        return r112080;
}
↓
double f(double x, double y, double z, double t) {
        double r112081 = x;
        double r112082 = y;
        double r112083 = z;
        double r112084 = fma(r112081, r112082, r112083);
        double r112085 = t;
        double r112086 = fma(r112084, r112082, r112085);
        return r112086;
}

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 2019326 +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))