Average Error: 0.0 → 0.0
Time: 2.8s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[x \cdot \left(2 \cdot y + x\right) + y \cdot y\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
x \cdot \left(2 \cdot y + x\right) + y \cdot y
double f(double x, double y) {
        double r720841 = x;
        double r720842 = r720841 * r720841;
        double r720843 = 2.0;
        double r720844 = r720841 * r720843;
        double r720845 = y;
        double r720846 = r720844 * r720845;
        double r720847 = r720842 + r720846;
        double r720848 = r720845 * r720845;
        double r720849 = r720847 + r720848;
        return r720849;
}
↓
double f(double x, double y) {
        double r720850 = x;
        double r720851 = 2.0;
        double r720852 = y;
        double r720853 = r720851 * r720852;
        double r720854 = r720853 + r720850;
        double r720855 = r720850 * r720854;
        double r720856 = r720852 * r720852;
        double r720857 = r720855 + r720856;
        return r720857;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020035 
(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)))