Average Error: 0.0 → 0.0
Time: 1.8s
Precision: binary64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
↓
\[y \cdot \mathsf{fma}\left(2, x, y\right) + x \cdot x \]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
y \cdot \mathsf{fma}\left(2, x, y\right) + x \cdot x

(FPCore (x y) :precision binary64 (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))
↓
(FPCore (x y) :precision binary64 (+ (* y (fma 2.0 x y)) (* x x)))
double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}
↓
double code(double x, double y) {
	return (y * fma(2.0, x, y)) + (x * x);
}

Error

Bits error versus x

Bits error versus y

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

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

  3. Taylor expanded in x around 0 0.0

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

  4. Simplified0.0

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

  5. Applied fma-udef_binary640.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022125 
(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)))