Average Error: 0.0 → 0.0
Time: 1.7s
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 r589716 = x;
        double r589717 = r589716 * r589716;
        double r589718 = 2.0;
        double r589719 = r589716 * r589718;
        double r589720 = y;
        double r589721 = r589719 * r589720;
        double r589722 = r589717 + r589721;
        double r589723 = r589720 * r589720;
        double r589724 = r589722 + r589723;
        return r589724;
}
↓
double f(double x, double y) {
        double r589725 = x;
        double r589726 = 2.0;
        double r589727 = y;
        double r589728 = r589726 * r589727;
        double r589729 = r589728 + r589725;
        double r589730 = r589725 * r589729;
        double r589731 = r589727 * r589727;
        double r589732 = r589730 + r589731;
        return r589732;
}

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 2020064 
(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)))