Average Error: 0.0 → 0.0
Time: 15.2s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, x\right), y \cdot y\right)\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, x\right), y \cdot y\right)
double f(double x, double y) {
        double r418058 = x;
        double r418059 = r418058 * r418058;
        double r418060 = 2.0;
        double r418061 = r418058 * r418060;
        double r418062 = y;
        double r418063 = r418061 * r418062;
        double r418064 = r418059 + r418063;
        double r418065 = r418062 * r418062;
        double r418066 = r418064 + r418065;
        return r418066;
}
↓
double f(double x, double y) {
        double r418067 = x;
        double r418068 = y;
        double r418069 = 2.0;
        double r418070 = fma(r418068, r418069, r418067);
        double r418071 = r418068 * r418068;
        double r418072 = fma(r418067, r418070, r418071);
        return r418072;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019199 +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)))