Average Error: 0.0 → 0.1
Time: 6.4s
Precision: binary64
\[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\\ {t_0}^{-0.5} \cdot \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt{t_0}} - x \end{array} \]
(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (fma x 0.04481 0.99229) 1.0)))
   (- (* (pow t_0 -0.5) (/ (fma x 0.27061 2.30753) (sqrt t_0))) x)))
double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}
↓
double code(double x) {
	double t_0 = fma(x, fma(x, 0.04481, 0.99229), 1.0);
	return (pow(t_0, -0.5) * (fma(x, 0.27061, 2.30753) / sqrt(t_0))) - 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)
	t_0 = fma(x, fma(x, 0.04481, 0.99229), 1.0)
	return Float64(Float64((t_0 ^ -0.5) * Float64(fma(x, 0.27061, 2.30753) / sqrt(t_0))) - 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_] := Block[{t$95$0 = N[(x * N[(x * 0.04481 + 0.99229), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[(N[Power[t$95$0, -0.5], $MachinePrecision] * N[(N[(x * 0.27061 + 2.30753), $MachinePrecision] / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x
↓
\begin{array}{l}
t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\\
{t_0}^{-0.5} \cdot \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt{t_0}} - x
\end{array}

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 egg-rr0.1

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

  4. Applied egg-rr0.1

    \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\right)}^{-0.5}} \cdot \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} - x \]

  5. Final simplification0.1

    \[\leadsto {\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\right)}^{-0.5} \cdot \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} - x \]

Reproduce

herbie shell --seed 2022160 
(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))