Average Error: 0.0 → 0.0
Time: 2.2s
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 r719614 = x;
        double r719615 = r719614 * r719614;
        double r719616 = 2.0;
        double r719617 = r719614 * r719616;
        double r719618 = y;
        double r719619 = r719617 * r719618;
        double r719620 = r719615 + r719619;
        double r719621 = r719618 * r719618;
        double r719622 = r719620 + r719621;
        return r719622;
}
↓
double f(double x, double y) {
        double r719623 = x;
        double r719624 = 2.0;
        double r719625 = y;
        double r719626 = r719624 * r719625;
        double r719627 = r719626 + r719623;
        double r719628 = r719623 * r719627;
        double r719629 = r719625 * r719625;
        double r719630 = r719628 + r719629;
        return r719630;
}

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 2019354 
(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)))