Average Error: 0.0 → 0.0
Time: 6.1s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\sqrt{y \cdot \left(y + 2 \cdot x\right) + x \cdot x} \cdot \sqrt{y \cdot \left(y + 2 \cdot x\right) + x \cdot x}\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\sqrt{y \cdot \left(y + 2 \cdot x\right) + x \cdot x} \cdot \sqrt{y \cdot \left(y + 2 \cdot x\right) + x \cdot x}
(FPCore (x y) :precision binary64 (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))
↓
(FPCore (x y)
 :precision binary64
 (*
  (sqrt (+ (* y (+ y (* 2.0 x))) (* x x)))
  (sqrt (+ (* y (+ y (* 2.0 x))) (* x x)))))
double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}
↓
double code(double x, double y) {
	return sqrt((y * (y + (2.0 * x))) + (x * x)) * sqrt((y * (y + (2.0 * x))) + (x * x));
}

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 add-sqr-sqrt_binary64_178320.0

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

  5. Simplified0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2021093 
(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)))