Average Error: 0.0 → 0.0
Time: 1.7s
Precision: binary64
\[\left(x + y\right) \cdot \left(x + y\right) \]
↓
\[\mathsf{fma}\left(x, x, y \cdot \mathsf{fma}\left(x, 2, y\right)\right) \]
(FPCore (x y) :precision binary64 (* (+ x y) (+ x y)))
↓
(FPCore (x y) :precision binary64 (fma x x (* y (fma x 2.0 y))))
double code(double x, double y) {
	return (x + y) * (x + y);
}
↓
double code(double x, double y) {
	return fma(x, x, (y * fma(x, 2.0, y)));
}

function code(x, y)
	return Float64(Float64(x + y) * Float64(x + y))
end
↓
function code(x, y)
	return fma(x, x, Float64(y * fma(x, 2.0, y)))
end

code[x_, y_] := N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[(x * x + N[(y * N[(x * 2.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(x, x, y \cdot \mathsf{fma}\left(x, 2, y\right)\right)

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 egg-rr0.0

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

  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. Taylor expanded in y around 0 0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2022166 
(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)))