Average Error: 0.0 → 0.0
Time: 2.3s
Precision: binary64
Cost: 6720
\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]
\[x \cdot x + y \cdot y \]
↓
\[\mathsf{fma}\left(y, y, x \cdot x\right) \]
(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

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded in x around 0 0.0

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

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, x \cdot x\right)} \]
    Proof
    (fma.f64 y y (*.f64 x x)): 0 points increase in error, 0 points decrease in error
    (fma.f64 y y (Rewrite<= unpow2_binary64 (pow.f64 x 2))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y y) (pow.f64 x 2))): 2 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2)) (pow.f64 x 2)): 0 points increase in error, 0 points decrease in error

  4. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost448
\[x \cdot x + y \cdot y \]
Alternative 2
Error7.9
Cost324
\[\begin{array}{l} \mathbf{if}\;y \leq 2.322735470880692 \cdot 10^{-91}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
Alternative 3
Error27.4
Cost192
\[y \cdot y \]

Error

Reproduce

herbie shell --seed 2022294 
(FPCore (x y)
  :name "Graphics.Rasterific.Linear:$cquadrance from Rasterific-0.6.1"
  :precision binary64
  (+ (* x x) (* y y)))