Average Error: 0.0 → 0.0
Time: 3.5s
Precision: binary64
\[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
↓
\[\frac{1}{\frac{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\right)}^{3}}}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}} - x \]
(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))
↓
(FPCore (x)
 :precision binary64
 (-
  (/
   1.0
   (/
    (cbrt (pow (fma x (fma x 0.04481 0.99229) 1.0) 3.0))
    (fma x 0.27061 2.30753)))
  x))
double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}
↓
double code(double x) {
	return (1.0 / (cbrt(pow(fma(x, fma(x, 0.04481, 0.99229), 1.0), 3.0)) / fma(x, 0.27061, 2.30753))) - x;
}

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

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

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

Error

Bits error versus x

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Applied add-cbrt-cube_binary640.0

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

  4. Simplified0.0

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

  5. Applied clear-num_binary640.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022129 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  :precision binary64
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))