Average Error: 0.1 → 0.1
Time: 4.6s
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 r114728 = x;
        double r114729 = y;
        double r114730 = r114728 * r114729;
        double r114731 = z;
        double r114732 = r114730 + r114731;
        double r114733 = r114732 * r114729;
        double r114734 = t;
        double r114735 = r114733 + r114734;
        return r114735;
}
↓
double f(double x, double y, double z, double t) {
        double r114736 = x;
        double r114737 = y;
        double r114738 = z;
        double r114739 = fma(r114736, r114737, r114738);
        double r114740 = t;
        double r114741 = fma(r114739, r114737, r114740);
        return r114741;
}

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 2019346 +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))