Average Error: 0.1 → 0.1
Time: 870.0ms
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 r150741 = x;
        double r150742 = y;
        double r150743 = r150741 * r150742;
        double r150744 = z;
        double r150745 = r150743 + r150744;
        double r150746 = r150745 * r150742;
        double r150747 = t;
        double r150748 = r150746 + r150747;
        return r150748;
}
↓
double f(double x, double y, double z, double t) {
        double r150749 = x;
        double r150750 = y;
        double r150751 = z;
        double r150752 = fma(r150749, r150750, r150751);
        double r150753 = t;
        double r150754 = fma(r150752, r150750, r150753);
        return r150754;
}

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 2020036 +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))