Average Error: 0.1 → 0.1
Time: 3.7s
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 r149136 = x;
        double r149137 = y;
        double r149138 = r149136 * r149137;
        double r149139 = z;
        double r149140 = r149138 + r149139;
        double r149141 = r149140 * r149137;
        double r149142 = t;
        double r149143 = r149141 + r149142;
        return r149143;
}
↓
double f(double x, double y, double z, double t) {
        double r149144 = x;
        double r149145 = y;
        double r149146 = z;
        double r149147 = fma(r149144, r149145, r149146);
        double r149148 = t;
        double r149149 = fma(r149147, r149145, r149148);
        return r149149;
}

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 2020056 +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))