Average Error: 0.0 → 0.0
Time: 1.2s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[x \cdot x + y \cdot y \]
↓
\[\mathsf{fma}\left(y, y, x \cdot x\right) \]
(FPCore (x y) :precision binary64 (+ (* x x) (* y y)))
↓
(FPCore (x y) :precision binary64 (fma y y (* x x)))
double code(double x, double y) {
	return (x * x) + (y * y);
}
↓
double code(double x, double y) {
	return fma(y, y, (x * x));
}

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

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

x \cdot x + y \cdot y
↓
\mathsf{fma}\left(y, y, x \cdot x\right)

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Taylor expanded in x around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022134 
(FPCore (x y)
  :name "Graphics.Rasterific.Linear:$cquadrance from Rasterific-0.6.1"
  :precision binary64
  (+ (* x x) (* y y)))