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

Average Error: 0.01% → 0%

Time: 1.8s

Precision: binary64

Cost: 6720

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.01

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

2. Taylor expanded in x around 0 0.01

   \[\leadsto \color{blue}{{y}^{2} + {x}^{2}} \]

3. Simplified 0

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

   [Start] 0.01	\[ {y}^{2} + {x}^{2} \]
   unpow2 [=>] 0.01	\[ \color{blue}{y \cdot y} + {x}^{2} \]
   unpow2 [=>] 0.01	\[ y \cdot y + \color{blue}{x \cdot x} \]
   fma-udef [<=] 0	\[ \color{blue}{\mathsf{fma}\left(y, y, x \cdot x\right)} \]

4. Final simplification 0

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

Alternatives

Alternative 1
Error	12.74%
Cost	589

\[\begin{array}{l} \mathbf{if}\;y \leq 1.08 \cdot 10^{-160} \lor \neg \left(y \leq 5.2 \cdot 10^{-71}\right) \land y \leq 2.25 \cdot 10^{-57}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 2
Error	0.01%
Cost	448

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

Alternative 3
Error	43.05%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y)
  :name "Graphics.Rasterific.Linear:$cquadrance from Rasterific-0.6.1"
  :precision binary64
  (+ (* x x) (* y y)))