Average Error: 0.1 → 0.1
Time: 10.3s
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 r155195 = x;
        double r155196 = y;
        double r155197 = r155195 * r155196;
        double r155198 = z;
        double r155199 = r155197 + r155198;
        double r155200 = r155199 * r155196;
        double r155201 = t;
        double r155202 = r155200 + r155201;
        return r155202;
}
↓
double f(double x, double y, double z, double t) {
        double r155203 = x;
        double r155204 = y;
        double r155205 = z;
        double r155206 = fma(r155203, r155204, r155205);
        double r155207 = t;
        double r155208 = fma(r155206, r155204, r155207);
        return r155208;
}

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 2020045 +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))