Average Error: 0.0 → 0.0
Time: 1.5s
Precision: binary64
\[\left(x + y\right) \cdot \left(x + y\right) \]
↓
\[\mathsf{fma}\left(x, x, y \cdot \mathsf{fma}\left(2, x, y\right)\right) \]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(x, x, y \cdot \mathsf{fma}\left(2, x, y\right)\right)

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

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[x \cdot x + \left(y \cdot y + 2 \cdot \left(y \cdot x\right)\right) \]

Derivation

  1. Initial program 0.0

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

  2. Applied distribute-rgt-in_binary640.0

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

  3. Applied distribute-rgt-in_binary640.0

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

  4. Applied associate-+l+_binary640.0

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

  5. Simplified0.0

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

  6. Taylor expanded in x around 0 0.0

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

  7. Simplified0.0

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

  8. Applied fma-def_binary640.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2021275 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"
  :precision binary64

  :herbie-target
  (+ (* x x) (+ (* y y) (* 2.0 (* y x))))

  (* (+ x y) (+ x y)))