Average Error: 0.0 → 0.0
Time: 2.5s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(x, x, \mathsf{fma}\left(2 \cdot x, y, {y}^{2}\right)\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(x, x, \mathsf{fma}\left(2 \cdot x, y, {y}^{2}\right)\right)
double f(double x, double y) {
        double r668590 = x;
        double r668591 = y;
        double r668592 = r668590 + r668591;
        double r668593 = r668592 * r668592;
        return r668593;
}
↓
double f(double x, double y) {
        double r668594 = x;
        double r668595 = 2.0;
        double r668596 = r668595 * r668594;
        double r668597 = y;
        double r668598 = pow(r668597, r668595);
        double r668599 = fma(r668596, r668597, r668598);
        double r668600 = fma(r668594, r668594, r668599);
        return r668600;
}

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. Using strategy rm

  3. Applied flip-+0.0

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

  4. Applied associate-*r/18.6

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

  5. Simplified18.6

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

  6. Taylor expanded around 0 0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"
  :precision binary64

  :herbie-target
  (+ (* x x) (+ (* y y) (* 2 (* y x))))

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