Average Error: 0.0 → 0.0
Time: 4.2s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(y, \mathsf{fma}\left(2, x, y\right), x \cdot x\right)\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(y, \mathsf{fma}\left(2, x, y\right), x \cdot x\right)
double f(double x, double y) {
        double r537613 = x;
        double r537614 = r537613 * r537613;
        double r537615 = 2.0;
        double r537616 = r537613 * r537615;
        double r537617 = y;
        double r537618 = r537616 * r537617;
        double r537619 = r537614 + r537618;
        double r537620 = r537617 * r537617;
        double r537621 = r537619 + r537620;
        return r537621;
}
↓
double f(double x, double y) {
        double r537622 = y;
        double r537623 = 2.0;
        double r537624 = x;
        double r537625 = fma(r537623, r537624, r537622);
        double r537626 = r537624 * r537624;
        double r537627 = fma(r537622, r537625, r537626);
        return r537627;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"

  :herbie-target
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))