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

Average Error: 0.0 → 0

Time: 1.0s

Precision: binary64

Cost: 6592

Math

\[x \cdot x + 1 \]

↓

\[\mathsf{fma}\left(x, x, 1\right) \]

FPCore

(FPCore (x) :precision binary64 (+ (* x x) 1.0))

↓

(FPCore (x) :precision binary64 (fma x x 1.0))

C

double code(double x) {
	return (x * x) + 1.0;
}

↓

double code(double x) {
	return fma(x, x, 1.0);
}

Julia

function code(x)
	return Float64(Float64(x * x) + 1.0)
end

↓

function code(x)
	return fma(x, x, 1.0)
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]

↓

code[x_] := N[(x * x + 1.0), $MachinePrecision]

Derivation

1. Initial program 0.0

   \[x \cdot x + 1 \]

2. Simplified 0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 1\right)} \]
   Proof

   [Start] 0.0	\[ x \cdot x + 1 \]
   fma-def [=>] 0	\[ \color{blue}{\mathsf{fma}\left(x, x, 1\right)} \]

3. Final simplification 0

   \[\leadsto \mathsf{fma}\left(x, x, 1\right) \]

Alternatives

Alternative 1
Error	0.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	0.0
Cost	320

\[1 + x \cdot x \]

Alternative 3
Error	20.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023073 
(FPCore (x)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, A"
  :precision binary64
  (+ (* x x) 1.0))