Average Error: 0.1 → 0.1
Time: 2.9s
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 r117145 = x;
        double r117146 = y;
        double r117147 = r117145 * r117146;
        double r117148 = z;
        double r117149 = r117147 + r117148;
        double r117150 = r117149 * r117146;
        double r117151 = t;
        double r117152 = r117150 + r117151;
        return r117152;
}
↓
double f(double x, double y, double z, double t) {
        double r117153 = x;
        double r117154 = y;
        double r117155 = z;
        double r117156 = fma(r117153, r117154, r117155);
        double r117157 = t;
        double r117158 = fma(r117156, r117154, r117157);
        return r117158;
}

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 2020065 +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))