Average Error: 0.0 → 0.0
Time: 10.3s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(y, y, x \cdot \left(x + y \cdot 2\right)\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(y, y, x \cdot \left(x + y \cdot 2\right)\right)
double f(double x, double y) {
        double r24573689 = x;
        double r24573690 = y;
        double r24573691 = r24573689 + r24573690;
        double r24573692 = r24573691 * r24573691;
        return r24573692;
}
↓
double f(double x, double y) {
        double r24573693 = y;
        double r24573694 = x;
        double r24573695 = 2.0;
        double r24573696 = r24573693 * r24573695;
        double r24573697 = r24573694 + r24573696;
        double r24573698 = r24573694 * r24573697;
        double r24573699 = fma(r24573693, r24573693, r24573698);
        return r24573699;
}

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

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

  4. Taylor expanded around 0 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +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)))