Average Error: 0.0 → 0.0
Time: 1.9s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\left(x + y\right) \cdot \left(x + y\right) \]
↓
\[\mathsf{fma}\left(x, \mathsf{fma}\left(2, y, x\right), y \cdot y\right) \]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(x, \mathsf{fma}\left(2, y, x\right), y \cdot y\right)

(FPCore (x y) :precision binary64 (* (+ x y) (+ x y)))
↓
(FPCore (x y) :precision binary64 (fma x (fma 2.0 y x) (* y y)))
double code(double x, double y) {
	return (x + y) * (x + y);
}
↓
double code(double x, double y) {
	return fma(x, fma(2.0, y, x), (y * 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 add-cbrt-cube_binary6437.5

    \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

  3. Simplified37.5

    \[\leadsto \sqrt[3]{\color{blue}{{\left(y + x\right)}^{6}}} \]

  4. Taylor expanded in y around 0 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022077 
(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)))