Average Error: 0.0 → 0.0
Time: 3.4s
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 r640980 = x;
        double r640981 = r640980 * r640980;
        double r640982 = 2.0;
        double r640983 = r640980 * r640982;
        double r640984 = y;
        double r640985 = r640983 * r640984;
        double r640986 = r640981 + r640985;
        double r640987 = r640984 * r640984;
        double r640988 = r640986 + r640987;
        return r640988;
}
↓
double f(double x, double y) {
        double r640989 = y;
        double r640990 = 2.0;
        double r640991 = x;
        double r640992 = fma(r640989, r640990, r640991);
        double r640993 = r640989 * r640989;
        double r640994 = fma(r640992, r640991, r640993);
        return r640994;
}

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 2020089 +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)))