Average Error: 0.0 → 0
Time: 1.5s
Precision: binary64
Cost: 6592
\[x \cdot x + 1 \]
↓
\[\mathsf{fma}\left(x, x, 1\right) \]
(FPCore (x) :precision binary64 (+ (* x x) 1.0))
↓
(FPCore (x) :precision binary64 (fma x x 1.0))
double code(double x) {
	return (x * x) + 1.0;
}
↓
double code(double x) {
	return fma(x, x, 1.0);
}

function code(x)
	return Float64(Float64(x * x) + 1.0)
end
↓
function code(x)
	return fma(x, x, 1.0)
end

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

x \cdot x + 1
↓
\mathsf{fma}\left(x, x, 1\right)

Error

Derivation

  1. Initial program 0.0

    \[x \cdot x + 1 \]

  2. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 1\right)} \]
    Proof
    (fma.f64 x x 1): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x x) 1)): 1 points increase in error, 0 points decrease in error

  3. Final simplification0

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

Alternatives

Alternative 1
Error1.6
Cost452
\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.874764784896079 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
Alternative 2
Error0.0
Cost320
\[1 + x \cdot x \]
Alternative 3
Error20.7
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022308 
(FPCore (x)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, A"
  :precision binary64
  (+ (* x x) 1.0))