Average Error: 0.0 → 0.0
Time: 2.2s
Precision: binary64
\[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
↓
\[x + \sqrt[3]{{\left(\frac{\mathsf{fma}\left(x, -0.27061, -2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{3}} \]
(FPCore (x)
 :precision binary64
 (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))
↓
(FPCore (x)
 :precision binary64
 (+
  x
  (cbrt
   (pow
    (/ (fma x -0.27061 -2.30753) (fma x (fma x 0.04481 0.99229) 1.0))
    3.0))))
double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
}
↓
double code(double x) {
	return x + cbrt(pow((fma(x, -0.27061, -2.30753) / fma(x, fma(x, 0.04481, 0.99229), 1.0)), 3.0));
}

function code(x)
	return Float64(x - Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(Float64(0.99229 + Float64(x * 0.04481)) * x))))
end
↓
function code(x)
	return Float64(x + cbrt((Float64(fma(x, -0.27061, -2.30753) / fma(x, fma(x, 0.04481, 0.99229), 1.0)) ^ 3.0)))
end

code[x_] := N[(x - N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(x + N[Power[N[Power[N[(N[(x * -0.27061 + -2.30753), $MachinePrecision] / N[(x * N[(x * 0.04481 + 0.99229), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x}
↓
x + \sqrt[3]{{\left(\frac{\mathsf{fma}\left(x, -0.27061, -2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{3}}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

  2. Simplified0.0

    \[\leadsto \color{blue}{x + \frac{\mathsf{fma}\left(x, -0.27061, -2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} \]

  3. Applied egg-rr0.0

    \[\leadsto x + \color{blue}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(x, -0.27061, -2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{3}}} \]

  4. Final simplification0.0

    \[\leadsto x + \sqrt[3]{{\left(\frac{\mathsf{fma}\left(x, -0.27061, -2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{3}} \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))