Average Error: 0.1 → 0.1
Time: 15.1s
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 r106996 = x;
        double r106997 = y;
        double r106998 = r106996 * r106997;
        double r106999 = z;
        double r107000 = r106998 + r106999;
        double r107001 = r107000 * r106997;
        double r107002 = t;
        double r107003 = r107001 + r107002;
        return r107003;
}
↓
double f(double x, double y, double z, double t) {
        double r107004 = x;
        double r107005 = y;
        double r107006 = z;
        double r107007 = fma(r107004, r107005, r107006);
        double r107008 = t;
        double r107009 = fma(r107007, r107005, r107008);
        return r107009;
}

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 2019304 +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))