Average Error: 0.0 → 0.0
Time: 7.6s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, x\right), y \cdot y\right)\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, x\right), y \cdot y\right)
double f(double x, double y) {
        double r370701 = x;
        double r370702 = r370701 * r370701;
        double r370703 = 2.0;
        double r370704 = r370701 * r370703;
        double r370705 = y;
        double r370706 = r370704 * r370705;
        double r370707 = r370702 + r370706;
        double r370708 = r370705 * r370705;
        double r370709 = r370707 + r370708;
        return r370709;
}
↓
double f(double x, double y) {
        double r370710 = x;
        double r370711 = y;
        double r370712 = 2.0;
        double r370713 = fma(r370711, r370712, r370710);
        double r370714 = r370711 * r370711;
        double r370715 = fma(r370710, r370713, r370714);
        return r370715;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 +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)))