Average Error: 0.0 → 0.0
Time: 3.7s
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 r722876 = x;
        double r722877 = r722876 * r722876;
        double r722878 = 2.0;
        double r722879 = r722876 * r722878;
        double r722880 = y;
        double r722881 = r722879 * r722880;
        double r722882 = r722877 + r722881;
        double r722883 = r722880 * r722880;
        double r722884 = r722882 + r722883;
        return r722884;
}
↓
double f(double x, double y) {
        double r722885 = x;
        double r722886 = 2.0;
        double r722887 = y;
        double r722888 = r722886 * r722887;
        double r722889 = r722888 + r722885;
        double r722890 = r722885 * r722889;
        double r722891 = r722887 * r722887;
        double r722892 = r722890 + r722891;
        return r722892;
}

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 2019322 
(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)))