Average Error: 0.0 → 0.0
Time: 4.4s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(y, \mathsf{fma}\left(2, x, y\right), x \cdot x\right)\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(y, \mathsf{fma}\left(2, x, y\right), x \cdot x\right)
double f(double x, double y) {
        double r164902823 = x;
        double r164902824 = r164902823 * r164902823;
        double r164902825 = 2.0;
        double r164902826 = r164902823 * r164902825;
        double r164902827 = y;
        double r164902828 = r164902826 * r164902827;
        double r164902829 = r164902824 + r164902828;
        double r164902830 = r164902827 * r164902827;
        double r164902831 = r164902829 + r164902830;
        return r164902831;
}
↓
double f(double x, double y) {
        double r164902832 = y;
        double r164902833 = 2.0;
        double r164902834 = x;
        double r164902835 = fma(r164902833, r164902834, r164902832);
        double r164902836 = r164902834 * r164902834;
        double r164902837 = fma(r164902832, r164902835, r164902836);
        return r164902837;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right)\]

Derivation

  1. Initial program 0.0

    \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"

  :herbie-target
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))