Average Error: 0.0 → 0.0
Time: 2.6s
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 r783710 = x;
        double r783711 = r783710 * r783710;
        double r783712 = 2.0;
        double r783713 = r783710 * r783712;
        double r783714 = y;
        double r783715 = r783713 * r783714;
        double r783716 = r783711 + r783715;
        double r783717 = r783714 * r783714;
        double r783718 = r783716 + r783717;
        return r783718;
}
↓
double f(double x, double y) {
        double r783719 = y;
        double r783720 = 2.0;
        double r783721 = x;
        double r783722 = fma(r783719, r783720, r783721);
        double r783723 = r783719 * r783719;
        double r783724 = fma(r783722, r783721, r783723);
        return r783724;
}

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 2019354 +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)))