Average Error: 0.0 → 0.0
Time: 1.8s
Precision: binary64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[x \cdot \left(x + y \cdot 2\right) + y \cdot y\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
x \cdot \left(x + y \cdot 2\right) + y \cdot y
(FPCore (x y) :precision binary64 (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))
↓
(FPCore (x y) :precision binary64 (+ (* x (+ x (* y 2.0))) (* y y)))
double code(double x, double y) {
	return ((double) (((double) (((double) (x * x)) + ((double) (((double) (x * 2.0)) * y)))) + ((double) (y * y))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (x * ((double) (x + ((double) (y * 2.0)))))) + ((double) (y * y))));
}

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 Error: 0.0 bits

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

  2. SimplifiedError: 0.0 bits

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

  4. Applied distribute-lft-inError: 0.0 bits

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

  5. Applied associate-+r+Error: 0.0 bits

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

  6. SimplifiedError: 0.0 bits

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

  7. Final simplificationError: 0.0 bits

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

Reproduce

herbie shell --seed 2020204 
(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)))