Average Error: 0.0 → 0.0
Time: 3.1s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot y\right) + y \cdot y
double f(double x, double y) {
        double r712571 = x;
        double r712572 = r712571 * r712571;
        double r712573 = 2.0;
        double r712574 = r712571 * r712573;
        double r712575 = y;
        double r712576 = r712574 * r712575;
        double r712577 = r712572 + r712576;
        double r712578 = r712575 * r712575;
        double r712579 = r712577 + r712578;
        return r712579;
}
↓
double f(double x, double y) {
        double r712580 = x;
        double r712581 = 2.0;
        double r712582 = r712580 * r712581;
        double r712583 = y;
        double r712584 = r712582 * r712583;
        double r712585 = fma(r712580, r712580, r712584);
        double r712586 = r712583 * r712583;
        double r712587 = r712585 + r712586;
        return r712587;
}

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 fma-def0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 +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)))