Average Error: 0.0 → 0.0
Time: 1.5s
Precision: binary64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
↓
\[x \cdot \left(2 \cdot y\right) + \mathsf{fma}\left(y, y, x \cdot x\right) \]
(FPCore (x y) :precision binary64 (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))
↓
(FPCore (x y) :precision binary64 (+ (* x (* 2.0 y)) (fma y y (* x x))))
double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}
↓
double code(double x, double y) {
	return (x * (2.0 * y)) + fma(y, y, (x * x));
}

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

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

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

Error

Bits error versus x

Bits error versus y

Target

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

Derivation

  1. Initial program 0.0

    \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]

  2. Simplified0.0

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

  3. Applied fma-udef_binary640.0

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

  4. Applied fma-udef_binary640.0

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

  5. Applied distribute-lft-in_binary640.0

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

  6. Applied associate-+l+_binary640.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2022131 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

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

  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))