Average Error: 0.1 → 0.1
Time: 17.8s
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 r97264 = x;
        double r97265 = y;
        double r97266 = r97264 * r97265;
        double r97267 = z;
        double r97268 = r97266 + r97267;
        double r97269 = r97268 * r97265;
        double r97270 = t;
        double r97271 = r97269 + r97270;
        return r97271;
}
↓
double f(double x, double y, double z, double t) {
        double r97272 = x;
        double r97273 = y;
        double r97274 = z;
        double r97275 = fma(r97272, r97273, r97274);
        double r97276 = t;
        double r97277 = fma(r97275, r97273, r97276);
        return r97277;
}

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 2019212 +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))