Average Error: 0.0 → 0.0
Time: 9.6s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, x\right), y \cdot y\right)\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, x\right), y \cdot y\right)
double f(double x, double y) {
        double r481902 = x;
        double r481903 = r481902 * r481902;
        double r481904 = 2.0;
        double r481905 = r481902 * r481904;
        double r481906 = y;
        double r481907 = r481905 * r481906;
        double r481908 = r481903 + r481907;
        double r481909 = r481906 * r481906;
        double r481910 = r481908 + r481909;
        return r481910;
}
↓
double f(double x, double y) {
        double r481911 = x;
        double r481912 = y;
        double r481913 = 2.0;
        double r481914 = fma(r481912, r481913, r481911);
        double r481915 = r481912 * r481912;
        double r481916 = fma(r481911, r481914, r481915);
        return r481916;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

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

  (+ (+ (* x x) (* (* x 2) y)) (* y y)))