Average Error: 0.0 → 0.0
Time: 6.6s
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 r411584 = x;
        double r411585 = r411584 * r411584;
        double r411586 = 2.0;
        double r411587 = r411584 * r411586;
        double r411588 = y;
        double r411589 = r411587 * r411588;
        double r411590 = r411585 + r411589;
        double r411591 = r411588 * r411588;
        double r411592 = r411590 + r411591;
        return r411592;
}
↓
double f(double x, double y) {
        double r411593 = x;
        double r411594 = y;
        double r411595 = 2.0;
        double r411596 = fma(r411594, r411595, r411593);
        double r411597 = r411594 * r411594;
        double r411598 = fma(r411593, r411596, r411597);
        return r411598;
}

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 2020043 +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)))