Average Error: 0.1 → 0.1
Time: 4.4s
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 r145400 = x;
        double r145401 = y;
        double r145402 = r145400 * r145401;
        double r145403 = z;
        double r145404 = r145402 + r145403;
        double r145405 = r145404 * r145401;
        double r145406 = t;
        double r145407 = r145405 + r145406;
        return r145407;
}
↓
double f(double x, double y, double z, double t) {
        double r145408 = x;
        double r145409 = y;
        double r145410 = z;
        double r145411 = fma(r145408, r145409, r145410);
        double r145412 = t;
        double r145413 = fma(r145411, r145409, r145412);
        return r145413;
}

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 2019353 +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))