Average Error: 0.1 → 0.1
Time: 4.2s
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 r151830 = x;
        double r151831 = y;
        double r151832 = r151830 * r151831;
        double r151833 = z;
        double r151834 = r151832 + r151833;
        double r151835 = r151834 * r151831;
        double r151836 = t;
        double r151837 = r151835 + r151836;
        return r151837;
}
↓
double f(double x, double y, double z, double t) {
        double r151838 = x;
        double r151839 = y;
        double r151840 = z;
        double r151841 = fma(r151838, r151839, r151840);
        double r151842 = t;
        double r151843 = fma(r151841, r151839, r151842);
        return r151843;
}

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 2020046 +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))