Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1

Average Error: 0.01% → 0%

Time: 1.5s

Precision: binary64

Cost: 6656

Math

\[x - y \cdot y \]

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.01

   \[x - y \cdot y \]

2. Taylor expanded in x around 0 0.01

   \[\leadsto \color{blue}{-1 \cdot {y}^{2} + x} \]

3. Simplified 0

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

   [Start] 0.01	\[ -1 \cdot {y}^{2} + x \]
   mul-1-neg [=>] 0.01	\[ \color{blue}{\left(-{y}^{2}\right)} + x \]
   unpow2 [=>] 0.01	\[ \left(-\color{blue}{y \cdot y}\right) + x \]
   distribute-rgt-neg-out [<=] 0.01	\[ \color{blue}{y \cdot \left(-y\right)} + x \]
   fma-def [=>] 0	\[ \color{blue}{\mathsf{fma}\left(y, -y, x\right)} \]

4. Final simplification 0

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

Alternatives

Alternative 1
Error	16.88%
Cost	786

\[\begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-12} \lor \neg \left(y \leq 2.9 \cdot 10^{-29} \lor \neg \left(y \leq 1.75 \cdot 10^{-15}\right) \land y \leq 7.5 \cdot 10^{+40}\right):\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.01%
Cost	320

\[x - y \cdot y \]

Alternative 3
Error	32.72%
Cost	64

\[x \]

Reproduce

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