Average Error: 0.0 → 0.0
Time: 927.0ms
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 r646375 = x;
        double r646376 = r646375 * r646375;
        double r646377 = 2.0;
        double r646378 = r646375 * r646377;
        double r646379 = y;
        double r646380 = r646378 * r646379;
        double r646381 = r646376 + r646380;
        double r646382 = r646379 * r646379;
        double r646383 = r646381 + r646382;
        return r646383;
}
↓
double f(double x, double y) {
        double r646384 = y;
        double r646385 = 2.0;
        double r646386 = x;
        double r646387 = fma(r646384, r646385, r646386);
        double r646388 = r646384 * r646384;
        double r646389 = fma(r646387, r646386, r646388);
        return r646389;
}

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 2020062 +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)))