Average Error: 0.0 → 0.0
Time: 11.4s
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 r661372 = x;
        double r661373 = r661372 * r661372;
        double r661374 = 2.0;
        double r661375 = r661372 * r661374;
        double r661376 = y;
        double r661377 = r661375 * r661376;
        double r661378 = r661373 + r661377;
        double r661379 = r661376 * r661376;
        double r661380 = r661378 + r661379;
        return r661380;
}
↓
double f(double x, double y) {
        double r661381 = x;
        double r661382 = y;
        double r661383 = 2.0;
        double r661384 = fma(r661382, r661383, r661381);
        double r661385 = r661382 * r661382;
        double r661386 = fma(r661381, r661384, r661385);
        return r661386;
}

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 2020045 +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)))