Average Error: 0.0 → 0.0
Time: 2.3s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(\sqrt{x \cdot x}, \sqrt{x \cdot x}, \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(\sqrt{x \cdot x}, \sqrt{x \cdot x}, \left(x \cdot 2\right) \cdot y\right) + y \cdot 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) fma(((double) sqrt(((double) (x * x)))), ((double) sqrt(((double) (x * x)))), ((double) (((double) (x * 2.0)) * y)))) + ((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 0.0

    \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.0

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

  4. Applied fma-def0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(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)))