Average Error: 0.0 → 0.0
Time: 1.6s
Precision: binary64
\[\left(x + y\right) \cdot \left(x + y\right) \]
↓
\[{x}^{2} + y \cdot \left(y + x \cdot 2\right) \]
\left(x + y\right) \cdot \left(x + y\right)
↓
{x}^{2} + y \cdot \left(y + x \cdot 2\right)

(FPCore (x y) :precision binary64 (* (+ x y) (+ x y)))
↓
(FPCore (x y) :precision binary64 (+ (pow x 2.0) (* y (+ y (* x 2.0)))))
double code(double x, double y) {
	return (x + y) * (x + y);
}
↓
double code(double x, double y) {
	return pow(x, 2.0) + (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
\[x \cdot x + \left(y \cdot y + 2 \cdot \left(y \cdot x\right)\right) \]

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded in x around 0 0.0

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

  3. Applied unpow2_binary640.0

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

  4. Applied fma-def_binary640.0

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

  5. Applied fma-udef_binary640.0

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

  6. Applied associate-+r+_binary640.0

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

  7. Simplified0.0

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

  8. Applied +-commutative_binary640.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2022121 
(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)))