Average Error: 0.1 → 0.1
Time: 4.3s
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 r160200 = x;
        double r160201 = y;
        double r160202 = r160200 * r160201;
        double r160203 = z;
        double r160204 = r160202 + r160203;
        double r160205 = r160204 * r160201;
        double r160206 = t;
        double r160207 = r160205 + r160206;
        return r160207;
}
↓
double f(double x, double y, double z, double t) {
        double r160208 = x;
        double r160209 = y;
        double r160210 = z;
        double r160211 = fma(r160208, r160209, r160210);
        double r160212 = t;
        double r160213 = fma(r160211, r160209, r160212);
        return r160213;
}

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 2020064 +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))