Average Error: 0.0 → 0.0
Time: 8.3s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(y, \mathsf{fma}\left(x, 2, y\right), x \cdot x\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(y, \mathsf{fma}\left(x, 2, y\right), x \cdot x\right)
double f(double x, double y) {
        double r355167 = x;
        double r355168 = y;
        double r355169 = r355167 + r355168;
        double r355170 = r355169 * r355169;
        return r355170;
}
↓
double f(double x, double y) {
        double r355171 = y;
        double r355172 = x;
        double r355173 = 2.0;
        double r355174 = fma(r355172, r355173, r355171);
        double r355175 = r355172 * r355172;
        double r355176 = fma(r355171, r355174, r355175);
        return r355176;
}

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

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 +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)))