Average Error: 0.0 → 0.0
Time: 1.6s
Precision: binary64
\[\left(x + y\right) \cdot \left(x + y\right) \]
↓
\[\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, x\right), y \cdot y\right) \]
(FPCore (x y) :precision binary64 (* (+ x y) (+ x y)))
↓
(FPCore (x y) :precision binary64 (fma x (fma y 2.0 x) (* y y)))
double code(double x, double y) {
	return (x + y) * (x + y);
}
↓
double code(double x, double y) {
	return fma(x, fma(y, 2.0, x), (y * y));
}

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

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

\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, x\right), y \cdot y\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. 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. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022170 
(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)))