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

↓

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

(FPCore (x y) :precision binary64 (* 2.0 (- (* x x) (* x y))))

↓

(FPCore (x y) :precision binary64 (* 2.0 (fma y (- x) (* x x))))

double code(double x, double y) {
	return 2.0 * ((x * x) - (x * y));
}

↓

double code(double x, double y) {
	return 2.0 * fma(y, -x, (x * x));
}

function code(x, y)
	return Float64(2.0 * Float64(Float64(x * x) - Float64(x * y)))
end

↓

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

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

↓

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

2 \cdot \left(x \cdot x - x \cdot y\right)

↓

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

Original	0.0
Target	0.0
Herbie	0.0

Derivation

Initial program 0.0
\[2 \cdot \left(x \cdot x - x \cdot y\right) \]
Taylor expanded in x around 0 0.0
\[\leadsto 2 \cdot \color{blue}{\left({x}^{2} + -1 \cdot \left(y \cdot x\right)\right)} \]
Simplified0.0
\[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(y, -x, x \cdot x\right)} \]
Proof
(fma.f64 y (neg.f64 x) (*.f64 x x)): 0 points increase in error, 0 points decrease in error
(Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (neg.f64 x)) (*.f64 x x))): 1 points increase in error, 1 points decrease in error
(Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) (*.f64 y (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
(+.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) (*.f64 y (neg.f64 x))): 0 points increase in error, 1 points decrease in error
(+.f64 (pow.f64 x 2) (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 y x)))): 0 points increase in error, 0 points decrease in error
(+.f64 (pow.f64 x 2) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 y x)))): 0 points increase in error, 0 points decrease in error
Final simplification0.0
\[\leadsto 2 \cdot \mathsf{fma}\left(y, -x, x \cdot x\right) \]

Alternatives

Alternative 1
Error	8.2
Cost	584

\[\begin{array}{l} t_0 := y \cdot \left(x \cdot -2\right)\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{-62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-96}:\\ \;\;\;\;x \cdot \left(2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.0
Cost	576

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

Alternative 3
Error	0.0
Cost	448

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

Alternative 4
Error	32.3
Cost	320

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

Reproduce

herbie shell --seed 2022330 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, A"
  :precision binary64

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

  (* 2.0 (- (* x x) (* x y))))

Error

Target

Derivation

Alternatives

Error

Reproduce