Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Average Accuracy: 100.0% → 100.0%

Time: 3.8s

Precision: binary64

Cost: 6912

Math

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

↓

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

FPCore

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

↓

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

C

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));
}

Julia

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(Float64(-y), x, Float64(x * x)))
end

Wolfram

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]

Target

Original	100.0%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 100.0%

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

2. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ 2 \cdot \left(x \cdot x - x \cdot y\right) \]
   sub-neg [=>] 100.0	\[ 2 \cdot \color{blue}{\left(x \cdot x + \left(-x \cdot y\right)\right)} \]
   +-commutative [=>] 100.0	\[ 2 \cdot \color{blue}{\left(\left(-x \cdot y\right) + x \cdot x\right)} \]
   *-commutative [=>] 100.0	\[ 2 \cdot \left(\left(-\color{blue}{y \cdot x}\right) + x \cdot x\right) \]
   distribute-lft-neg-in [=>] 100.0	\[ 2 \cdot \left(\color{blue}{\left(-y\right) \cdot x} + x \cdot x\right) \]
   fma-def [=>] 100.0	\[ 2 \cdot \color{blue}{\mathsf{fma}\left(-y, x, x \cdot x\right)} \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	87.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-87} \lor \neg \left(y \leq 3.5 \cdot 10^{-32}\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	576

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

Alternative 3
Accuracy	100.0%
Cost	448

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

Alternative 4
Accuracy	47.7%
Cost	320

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

Reproduce

herbie shell --seed 2023151 
(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))))