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

Average Error: 0.0 → 0.0

Time: 1.5s

Precision: binary64

Cost: 6720

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 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, y \cdot y\right)} \]
   Proof
   (fma.f64 x x (*.f64 y y)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x x) (*.f64 y y))): 1 points increase in error, 0 points decrease in error

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	20.6
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-147}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-92}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-78}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 2
Error	0.0
Cost	448

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

Alternative 3
Error	27.0
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2022325 
(FPCore (x y)
  :name "Graphics.Rasterific.Linear:$cquadrance from Rasterific-0.6.1"
  :precision binary64
  (+ (* x x) (* y y)))