Average Error: 0.0 → 0.0
Time: 1.7s
Precision: binary64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[x \cdot x + y \cdot \left(y + x \cdot 2\right)\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
x \cdot x + y \cdot \left(y + x \cdot 2\right)
(FPCore (x y) :precision binary64 (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))
↓
(FPCore (x y) :precision binary64 (+ (* x x) (* y (+ y (* x 2.0)))))
double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}
↓
double code(double x, double y) {
	return (x * x) + (y * (y + (x * 2.0)));
}

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. Simplified0.0

    \[\leadsto \color{blue}{x \cdot \left(x + 2 \cdot y\right) + y \cdot y}\]
  3. Using strategy rm

  4. Applied distribute-lft-in_binary640.0

    \[\leadsto \color{blue}{\left(x \cdot x + x \cdot \left(2 \cdot y\right)\right)} + y \cdot y\]

  5. Applied associate-+l+_binary640.0

    \[\leadsto \color{blue}{x \cdot x + \left(x \cdot \left(2 \cdot y\right) + y \cdot y\right)}\]

  6. Simplified0.0

    \[\leadsto x \cdot x + \color{blue}{y \cdot \left(y + x \cdot 2\right)}\]

  7. Final simplification0.0

    \[\leadsto x \cdot x + y \cdot \left(y + x \cdot 2\right)\]

Reproduce

herbie shell --seed 2020219 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :herbie-target
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))