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

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 fma(x, x, fma(Float64(x * 2.0), y, Float64(y * y)))
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[(x * x + N[(N[(x * 2.0), $MachinePrecision] * y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Bits error versus y

Target

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

Derivation

  1. Initial program 0.0

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

  2. Applied egg-rr0.0

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

  3. Applied egg-rr0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022150 
(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)))