Average Error: 0.0 → 0.0
Time: 1.1s
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 r670752 = x;
        double r670753 = r670752 * r670752;
        double r670754 = 2.0;
        double r670755 = r670752 * r670754;
        double r670756 = y;
        double r670757 = r670755 * r670756;
        double r670758 = r670753 + r670757;
        double r670759 = r670756 * r670756;
        double r670760 = r670758 + r670759;
        return r670760;
}
↓
double f(double x, double y) {
        double r670761 = x;
        double r670762 = 2.0;
        double r670763 = y;
        double r670764 = r670762 * r670763;
        double r670765 = r670764 + r670761;
        double r670766 = r670761 * r670765;
        double r670767 = r670763 * r670763;
        double r670768 = r670766 + r670767;
        return r670768;
}

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 2020039 
(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)))