Average Error: 0.0 → 0.0
Time: 995.0ms
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 r577702 = x;
        double r577703 = r577702 * r577702;
        double r577704 = 2.0;
        double r577705 = r577702 * r577704;
        double r577706 = y;
        double r577707 = r577705 * r577706;
        double r577708 = r577703 + r577707;
        double r577709 = r577706 * r577706;
        double r577710 = r577708 + r577709;
        return r577710;
}
↓
double f(double x, double y) {
        double r577711 = x;
        double r577712 = 2.0;
        double r577713 = r577711 * r577712;
        double r577714 = y;
        double r577715 = r577713 * r577714;
        double r577716 = fma(r577711, r577711, r577715);
        double r577717 = r577714 * r577714;
        double r577718 = r577716 + r577717;
        return r577718;
}

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 2020039 +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)))