Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Average Error: 0.02% → 0.02%

Time: 3.1s

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	0.02%
Target	0.02%
Herbie	0.02%

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

Derivation

1. Initial program 0.02

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

2. Applied egg-rr 0.02

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

3. Taylor expanded in x around 0 0.05

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

4. Simplified 0.02

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

   [Start] 0.05	\[ 2 \cdot {x}^{2} + 2 \cdot \left(\left(-2 \cdot y + y\right) \cdot x\right) \]
   distribute-lft-out [=>] 0.05	\[ \color{blue}{2 \cdot \left({x}^{2} + \left(-2 \cdot y + y\right) \cdot x\right)} \]
   +-commutative [<=] 0.05	\[ 2 \cdot \color{blue}{\left(\left(-2 \cdot y + y\right) \cdot x + {x}^{2}\right)} \]
   distribute-lft1-in [=>] 0.03	\[ 2 \cdot \left(\color{blue}{\left(\left(-2 + 1\right) \cdot y\right)} \cdot x + {x}^{2}\right) \]
   metadata-eval [=>] 0.03	\[ 2 \cdot \left(\left(\color{blue}{-1} \cdot y\right) \cdot x + {x}^{2}\right) \]
   fma-def [=>] 0.02	\[ 2 \cdot \color{blue}{\mathsf{fma}\left(-1 \cdot y, x, {x}^{2}\right)} \]
   mul-1-neg [=>] 0.02	\[ 2 \cdot \mathsf{fma}\left(\color{blue}{-y}, x, {x}^{2}\right) \]
   unpow2 [=>] 0.02	\[ 2 \cdot \mathsf{fma}\left(-y, x, \color{blue}{x \cdot x}\right) \]

5. Final simplification 0.02

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

Alternatives

Alternative 1
Error	13.25%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -310000 \lor \neg \left(y \leq 4.8 \cdot 10^{-44}\right):\\ \;\;\;\;x \cdot \left(y \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot x\right)\\ \end{array} \]

Alternative 2
Error	0.02%
Cost	448

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

Alternative 3
Error	34.35%
Cost	320

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

Reproduce

herbie shell --seed 2023090 
(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))))