Average Error: 0.0 → 0.0
Time: 4.4s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(2, y \cdot x, \mathsf{fma}\left(y, y, x \cdot x\right)\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(2, y \cdot x, \mathsf{fma}\left(y, y, x \cdot x\right)\right)
double f(double x, double y) {
        double r28506767 = x;
        double r28506768 = y;
        double r28506769 = r28506767 + r28506768;
        double r28506770 = r28506769 * r28506769;
        return r28506770;
}
↓
double f(double x, double y) {
        double r28506771 = 2.0;
        double r28506772 = y;
        double r28506773 = x;
        double r28506774 = r28506772 * r28506773;
        double r28506775 = r28506773 * r28506773;
        double r28506776 = fma(r28506772, r28506772, r28506775);
        double r28506777 = fma(r28506771, r28506774, r28506776);
        return r28506777;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"

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

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