Average Error: 0.1 → 0.1
Time: 9.0s
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 r154917 = x;
        double r154918 = y;
        double r154919 = r154917 * r154918;
        double r154920 = z;
        double r154921 = r154919 + r154920;
        double r154922 = r154921 * r154918;
        double r154923 = t;
        double r154924 = r154922 + r154923;
        return r154924;
}
↓
double f(double x, double y, double z, double t) {
        double r154925 = x;
        double r154926 = y;
        double r154927 = z;
        double r154928 = fma(r154925, r154926, r154927);
        double r154929 = t;
        double r154930 = fma(r154928, r154926, r154929);
        return r154930;
}

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 2019208 +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))