Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Average Error: 0.03% → 0.02%

Time: 2.5s

Precision: binary64

Cost: 6912

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

Target

Original	0.03%
Target	0.01%
Herbie	0.02%

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

Derivation

1. Initial program 0.03

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

2. Applied egg-rr 0.02

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

3. Final simplification 0.02

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

Alternatives

Alternative 1
Error	12.77%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-28} \lor \neg \left(y \leq 1.25 \cdot 10^{-5}\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot x\right)\\ \end{array} \]

Alternative 2
Error	0.01%
Cost	448

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

Alternative 3
Error	50.52%
Cost	320

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

Reproduce

herbie shell --seed 2023121 
(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))))