Average Error: 0.0 → 0.0
Time: 3.9s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(y, 2, x\right), x, y \cdot y\right)\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(\mathsf{fma}\left(y, 2, x\right), x, y \cdot y\right)
double f(double x, double y) {
        double r672886 = x;
        double r672887 = r672886 * r672886;
        double r672888 = 2.0;
        double r672889 = r672886 * r672888;
        double r672890 = y;
        double r672891 = r672889 * r672890;
        double r672892 = r672887 + r672891;
        double r672893 = r672890 * r672890;
        double r672894 = r672892 + r672893;
        return r672894;
}
↓
double f(double x, double y) {
        double r672895 = y;
        double r672896 = 2.0;
        double r672897 = x;
        double r672898 = fma(r672895, r672896, r672897);
        double r672899 = r672895 * r672895;
        double r672900 = fma(r672898, r672897, r672899);
        return r672900;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right)\]

Derivation

  1. Initial program 0.0

    \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, 2, x\right), x, y \cdot y\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, 2, x\right), x, y \cdot y\right)\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :herbie-target
  (+ (* x x) (+ (* y y) (* (* x y) 2)))

  (+ (+ (* x x) (* (* x 2) y)) (* y y)))