Graphics.Rasterific.Linear:$cquadrance from Rasterific-0.6.1

Average Error: 0.01% → 0.01%

Time: 1.3s

Precision: binary64

Cost: 6720

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

\[x \cdot x + y \cdot y \]

↓

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

FPCore

(FPCore (x y) :precision binary64 (+ (* x x) (* y y)))

↓

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

C

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

↓

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

Julia

function code(x, y)
	return Float64(Float64(x * x) + Float64(y * y))
end

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.01

   \[x \cdot x + y \cdot y \]

2. Simplified 0.01

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

   [Start] 0.01	\[ x \cdot x + y \cdot y \]
   fma-def [=>] 0.01	\[ \color{blue}{\mathsf{fma}\left(x, x, y \cdot y\right)} \]

3. Final simplification 0.01

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

Alternatives

Alternative 1
Error	14.09%
Cost	589

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-23} \lor \neg \left(x \leq -3.05 \cdot 10^{-50}\right) \land x \leq -2.3 \cdot 10^{-143}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 2
Error	0.01%
Cost	448

\[y \cdot y + x \cdot x \]

Alternative 3
Error	43.16%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023115 
(FPCore (x y)
  :name "Graphics.Rasterific.Linear:$cquadrance from Rasterific-0.6.1"
  :precision binary64
  (+ (* x x) (* y y)))