Average Error: 0.0 → 0.0
Time: 13.6s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, x \cdot y + x \cdot y\right)\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, x \cdot y + x \cdot y\right)\right)
double f(double x, double y) {
        double r166269447 = x;
        double r166269448 = y;
        double r166269449 = r166269447 + r166269448;
        double r166269450 = r166269449 * r166269449;
        return r166269450;
}
↓
double f(double x, double y) {
        double r166269451 = y;
        double r166269452 = x;
        double r166269453 = r166269452 * r166269451;
        double r166269454 = r166269453 + r166269453;
        double r166269455 = fma(r166269452, r166269452, r166269454);
        double r166269456 = fma(r166269451, r166269451, r166269455);
        return r166269456;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[x \cdot x + \left(y \cdot y + 2 \cdot \left(y \cdot x\right)\right)\]

Derivation

  1. Initial program 0.0

    \[\left(x + y\right) \cdot \left(x + y\right)\]

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Taylor expanded around 0 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"

  :herbie-target
  (+ (* x x) (+ (* y y) (* 2.0 (* y x))))

  (* (+ x y) (+ x y)))