Average Error: 0.0 → 0.0
Time: 1.4s
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 r638892 = x;
        double r638893 = r638892 * r638892;
        double r638894 = 2.0;
        double r638895 = r638892 * r638894;
        double r638896 = y;
        double r638897 = r638895 * r638896;
        double r638898 = r638893 + r638897;
        double r638899 = r638896 * r638896;
        double r638900 = r638898 + r638899;
        return r638900;
}
↓
double f(double x, double y) {
        double r638901 = y;
        double r638902 = 2.0;
        double r638903 = x;
        double r638904 = fma(r638901, r638902, r638903);
        double r638905 = r638901 * r638901;
        double r638906 = fma(r638904, r638903, r638905);
        return r638906;
}

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 2020001 +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)))