Average Error: 0.0 → 0.0
Time: 1.2s
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 r685562 = x;
        double r685563 = r685562 * r685562;
        double r685564 = 2.0;
        double r685565 = r685562 * r685564;
        double r685566 = y;
        double r685567 = r685565 * r685566;
        double r685568 = r685563 + r685567;
        double r685569 = r685566 * r685566;
        double r685570 = r685568 + r685569;
        return r685570;
}
↓
double f(double x, double y) {
        double r685571 = x;
        double r685572 = 2.0;
        double r685573 = y;
        double r685574 = r685572 * r685573;
        double r685575 = r685574 + r685571;
        double r685576 = r685571 * r685575;
        double r685577 = r685573 * r685573;
        double r685578 = r685576 + r685577;
        return r685578;
}

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 2019362 
(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)))