Average Error: 0.0 → 0.0
Time: 2.5s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot y\right) + y \cdot y
double f(double x, double y) {
        double r628768 = x;
        double r628769 = r628768 * r628768;
        double r628770 = 2.0;
        double r628771 = r628768 * r628770;
        double r628772 = y;
        double r628773 = r628771 * r628772;
        double r628774 = r628769 + r628773;
        double r628775 = r628772 * r628772;
        double r628776 = r628774 + r628775;
        return r628776;
}
↓
double f(double x, double y) {
        double r628777 = x;
        double r628778 = 2.0;
        double r628779 = r628777 * r628778;
        double r628780 = y;
        double r628781 = r628779 * r628780;
        double r628782 = fma(r628777, r628777, r628781);
        double r628783 = r628780 * r628780;
        double r628784 = r628782 + r628783;
        return r628784;
}

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. Using strategy rm

  3. Applied fma-def0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 +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)))