Average Error: 0.2 → 0.2
Time: 1.7s
Precision: binary64
\[\frac{x \cdot x - 3}{6} \]
↓
\[\frac{\mathsf{fma}\left(x, x, -3\right)}{6} \]
(FPCore (x) :precision binary64 (/ (- (* x x) 3.0) 6.0))
↓
(FPCore (x) :precision binary64 (/ (fma x x -3.0) 6.0))
double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}
↓
double code(double x) {
	return fma(x, x, -3.0) / 6.0;
}

function code(x)
	return Float64(Float64(Float64(x * x) - 3.0) / 6.0)
end
↓
function code(x)
	return Float64(fma(x, x, -3.0) / 6.0)
end

code[x_] := N[(N[(N[(x * x), $MachinePrecision] - 3.0), $MachinePrecision] / 6.0), $MachinePrecision]
↓
code[x_] := N[(N[(x * x + -3.0), $MachinePrecision] / 6.0), $MachinePrecision]

\frac{x \cdot x - 3}{6}
↓
\frac{\mathsf{fma}\left(x, x, -3\right)}{6}

Error

Bits error versus x

Derivation

  1. Initial program 0.2

    \[\frac{x \cdot x - 3}{6} \]

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, -3\right)}{6}} \]

  3. Final simplification0.2

    \[\leadsto \frac{\mathsf{fma}\left(x, x, -3\right)}{6} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))