Average Error: 0.0 → 0.0
Time: 1.2s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[x \cdot x + \mathsf{fma}\left(2, x, y\right) \cdot y\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
x \cdot x + \mathsf{fma}\left(2, x, y\right) \cdot y
double f(double x, double y) {
        double r625733 = x;
        double r625734 = r625733 * r625733;
        double r625735 = 2.0;
        double r625736 = r625733 * r625735;
        double r625737 = y;
        double r625738 = r625736 * r625737;
        double r625739 = r625734 + r625738;
        double r625740 = r625737 * r625737;
        double r625741 = r625739 + r625740;
        return r625741;
}
↓
double f(double x, double y) {
        double r625742 = x;
        double r625743 = r625742 * r625742;
        double r625744 = 2.0;
        double r625745 = y;
        double r625746 = fma(r625744, r625742, r625745);
        double r625747 = r625746 * r625745;
        double r625748 = r625743 + r625747;
        return r625748;
}

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. Using strategy rm

  3. Applied associate-+l+0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020065 +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)))