Average Error: 0.1 → 0.1
Time: 3.5s
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 r143255 = x;
        double r143256 = y;
        double r143257 = r143255 * r143256;
        double r143258 = z;
        double r143259 = r143257 + r143258;
        double r143260 = r143259 * r143256;
        double r143261 = t;
        double r143262 = r143260 + r143261;
        return r143262;
}
↓
double f(double x, double y, double z, double t) {
        double r143263 = x;
        double r143264 = y;
        double r143265 = z;
        double r143266 = fma(r143263, r143264, r143265);
        double r143267 = t;
        double r143268 = fma(r143266, r143264, r143267);
        return r143268;
}

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 2020034 +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))