Average Error: 0.0 → 0.0
Time: 3.3s
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 r647520 = x;
        double r647521 = r647520 * r647520;
        double r647522 = 2.0;
        double r647523 = r647520 * r647522;
        double r647524 = y;
        double r647525 = r647523 * r647524;
        double r647526 = r647521 + r647525;
        double r647527 = r647524 * r647524;
        double r647528 = r647526 + r647527;
        return r647528;
}
↓
double f(double x, double y) {
        double r647529 = y;
        double r647530 = 2.0;
        double r647531 = x;
        double r647532 = fma(r647529, r647530, r647531);
        double r647533 = r647529 * r647529;
        double r647534 = fma(r647532, r647531, r647533);
        return r647534;
}

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 2020035 +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)))