Average Error: 0.0 → 0.0
Time: 2.9s
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 r523423 = x;
        double r523424 = r523423 * r523423;
        double r523425 = 2.0;
        double r523426 = r523423 * r523425;
        double r523427 = y;
        double r523428 = r523426 * r523427;
        double r523429 = r523424 + r523428;
        double r523430 = r523427 * r523427;
        double r523431 = r523429 + r523430;
        return r523431;
}
↓
double f(double x, double y) {
        double r523432 = x;
        double r523433 = 2.0;
        double r523434 = y;
        double r523435 = r523433 * r523434;
        double r523436 = r523435 + r523432;
        double r523437 = r523432 * r523436;
        double r523438 = r523434 * r523434;
        double r523439 = r523437 + r523438;
        return r523439;
}

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 1978988140 
(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)))