\[x - y \cdot y \]

↓

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

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

↓

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

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

↓

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

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

↓

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

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

↓

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

x - y \cdot y

↓

\mathsf{fma}\left(y, -y, x\right)

Derivation

Initial program 0.0
\[x - y \cdot y \]
Taylor expanded in x around 0 0.0
\[\leadsto \color{blue}{-1 \cdot {y}^{2} + x} \]
Simplified0
\[\leadsto \color{blue}{\mathsf{fma}\left(y, -y, x\right)} \]
Proof
(fma.f64 y (neg.f64 y) x): 0 points increase in error, 0 points decrease in error
(Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (neg.f64 y)) x)): 0 points increase in error, 0 points decrease in error
(+.f64 (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 y y))) x): 0 points increase in error, 0 points decrease in error
(+.f64 (neg.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2))) x): 0 points increase in error, 0 points decrease in error
(+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 y 2))) x): 0 points increase in error, 0 points decrease in error
Final simplification0
\[\leadsto \mathsf{fma}\left(y, -y, x\right) \]

Alternatives

Alternative 1
Error	10.6
Cost	521

\[\begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-20} \lor \neg \left(y \leq 1.7 \cdot 10^{-32}\right):\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.0
Cost	320

\[x - y \cdot y \]

Alternative 3
Error	22.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022340 
(FPCore (x y)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1"
  :precision binary64
  (- x (* y y)))

Error

Derivation

Alternatives

Error

Reproduce