Average Error: 0.0 → 0.0
Time: 2.2s
Precision: binary64
Cost: 704
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[x \cdot x + y \cdot \left(y + x \cdot 2\right)\]
\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
\[x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right)\]

Alternatives

Alternative 1
Error14.6
Cost2060
\[\begin{array}{l} \mathbf{if}\;y \leq -1.1211958490574929 \cdot 10^{-24}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \mathbf{elif}\;y \leq -1.867192135062791 \cdot 10^{-94}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{elif}\;y \leq -1.7891248847175876 \cdot 10^{-140}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \mathbf{elif}\;y \leq 4.9185382277459305 \cdot 10^{-85}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \leq 1.3544328070757095 \cdot 10^{-24} \lor \neg \left(y \leq 5.905816456577433 \cdot 10^{+33}\right):\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \end{array}\]
Alternative 2
Error14.7
Cost2060
\[\begin{array}{l} \mathbf{if}\;y \leq -8.021633434638617 \cdot 10^{-25}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq -1.2797953691330216 \cdot 10^{-95}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{elif}\;y \leq -2.0241671006664829 \cdot 10^{-140}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq 2.7679225815299694 \cdot 10^{-84}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \leq 4.1282807451357015 \cdot 10^{-25} \lor \neg \left(y \leq 5.905816456577433 \cdot 10^{+33}\right):\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \end{array}\]
Alternative 3
Error13.4
Cost785
\[\begin{array}{l} \mathbf{if}\;x \leq -9.997791734835163 \cdot 10^{-26} \lor \neg \left(x \leq 2.1877696116171055 \cdot 10^{-115} \lor \neg \left(x \leq 7.5544806308271365 \cdot 10^{-90}\right) \land x \leq 7.150922772501069 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array}\]
Alternative 4
Error28.3
Cost192
\[x \cdot x\]
Alternative 5
Error55.2
Cost64
\[0\]
Alternative 6
Error60.7
Cost64
\[1\]

Error

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-rgt-in_binary64_184420.0

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

  5. Applied associate-+l+_binary64_184250.0

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

  6. Simplified0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2021044 
(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)))