Average Error: 0.0 → 0
Time: 721.0ms
Precision: binary64
\[\left(x \cdot x\right) \cdot 2 - 1 \]
↓
\[\mathsf{fma}\left(x, x \cdot 2, -1\right) \]
(FPCore (x) :precision binary64 (- (* (* x x) 2.0) 1.0))
↓
(FPCore (x) :precision binary64 (fma x (* x 2.0) -1.0))
double code(double x) {
	return ((x * x) * 2.0) - 1.0;
}
↓
double code(double x) {
	return fma(x, (x * 2.0), -1.0);
}

function code(x)
	return Float64(Float64(Float64(x * x) * 2.0) - 1.0)
end
↓
function code(x)
	return fma(x, Float64(x * 2.0), -1.0)
end

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision] - 1.0), $MachinePrecision]
↓
code[x_] := N[(x * N[(x * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]

\left(x \cdot x\right) \cdot 2 - 1
↓
\mathsf{fma}\left(x, x \cdot 2, -1\right)

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\left(x \cdot x\right) \cdot 2 - 1 \]

  2. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 2, -1\right)} \]

  3. Final simplification0

    \[\leadsto \mathsf{fma}\left(x, x \cdot 2, -1\right) \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x)
  :name "Numeric.SpecFunctions:logGammaCorrection from math-functions-0.1.5.2"
  :precision binary64
  (- (* (* x x) 2.0) 1.0))