Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Average Error: 0.0 → 0.0

Time: 1.6s

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	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

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

2. Taylor expanded in x around 0 0.0

   \[\leadsto 2 \cdot \color{blue}{\left({x}^{2} + -1 \cdot \left(y \cdot x\right)\right)} \]

3. Simplified 0.0

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(x, x, \left(-y\right) \cdot x\right)} \]
   Proof
   (fma.f64 x x (*.f64 (neg.f64 y) x)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x x (Rewrite=> distribute-lft-neg-out_binary64 (neg.f64 (*.f64 y x)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x x (Rewrite<= distribute-rgt-neg-out_binary64 (*.f64 y (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 x x) (*.f64 y (neg.f64 x)))): 1 points increase in error, 1 points decrease in error
   (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) (*.f64 y (neg.f64 x))): 0 points increase in error, 0 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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	8.4
Cost	584

\[\begin{array}{l} t_0 := x \cdot \left(y \cdot -2\right)\\ \mathbf{if}\;y \leq -6 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \left(2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.0
Cost	448

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

Alternative 3
Error	32.9
Cost	320

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

Reproduce

herbie shell --seed 2022325 
(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))))