Average Error: 0.0 → 0.0
Time: 1.6s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, y\right) \cdot y\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, y\right) \cdot y\right)
double f(double x, double y) {
        double r631624 = x;
        double r631625 = y;
        double r631626 = r631624 + r631625;
        double r631627 = r631626 * r631626;
        return r631627;
}
↓
double f(double x, double y) {
        double r631628 = x;
        double r631629 = 2.0;
        double r631630 = y;
        double r631631 = fma(r631628, r631629, r631630);
        double r631632 = r631631 * r631630;
        double r631633 = fma(r631628, r631628, r631632);
        return r631633;
}

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 distribute-lft-in0.0

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

  4. Simplified0.0

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

  5. Simplified0.0

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

  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. Taylor expanded around 0 0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 +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)))