?

Average Accuracy: 54.3% → 100.0%
Time: 10.4s
Precision: binary64
Cost: 93128

?

\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 9000:\\ \;\;\;\;\frac{x + x \cdot \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 0.0424060604 \cdot {x}^{4}\right) + \mathsf{fma}\left({x}^{6}, 0.0072644182, {x}^{8} \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right)}{\mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{4}, 0.2909738639, {x}^{6} \cdot 0.0694555761\right)\right) + \left(\mathsf{fma}\left(x, x \cdot 0.7715471019, 1\right) + \mathsf{fma}\left({x}^{10}, 0.0008327945, 0.0003579942 \cdot {x}^{12}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{x \cdot \left(\frac{0.2514179000665374}{x \cdot x} + -0.5\right)}\\ \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (/
   (+
    (+
     (+
      (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x))))
      (* 0.0072644182 (* (* (* x x) (* x x)) (* x x))))
     (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x))))
    (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x))))
   (+
    (+
     (+
      (+
       (+
        (+ 1.0 (* 0.7715471019 (* x x)))
        (* 0.2909738639 (* (* x x) (* x x))))
       (* 0.0694555761 (* (* (* x x) (* x x)) (* x x))))
      (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x))))
     (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x))))
    (*
     (* 2.0 0.0001789971)
     (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x)))))
  x))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -5e+22)
   (/ 0.5 x)
   (if (<= x 9000.0)
     (/
      (+
       x
       (*
        x
        (+
         (fma 0.1049934947 (* x x) (* 0.0424060604 (pow x 4.0)))
         (fma
          (pow x 6.0)
          0.0072644182
          (* (pow x 8.0) (+ 0.0005064034 (* (* x x) 0.0001789971)))))))
      (+
       (fma
        (pow x 8.0)
        0.0140005442
        (fma (pow x 4.0) 0.2909738639 (* (pow x 6.0) 0.0694555761)))
       (+
        (fma x (* x 0.7715471019) 1.0)
        (fma (pow x 10.0) 0.0008327945 (* 0.0003579942 (pow x 12.0))))))
     (/ -0.25 (* x (+ (/ 0.2514179000665374 (* x x)) -0.5))))))
double code(double x) {
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * ((x * x) * (x * x)))) + (0.0072644182 * (((x * x) * (x * x)) * (x * x)))) + (0.0005064034 * ((((x * x) * (x * x)) * (x * x)) * (x * x)))) + (0.0001789971 * (((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)))) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * ((x * x) * (x * x)))) + (0.0694555761 * (((x * x) * (x * x)) * (x * x)))) + (0.0140005442 * ((((x * x) * (x * x)) * (x * x)) * (x * x)))) + (0.0008327945 * (((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)))) + ((2.0 * 0.0001789971) * ((((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)) * (x * x))))) * x;
}
↓
double code(double x) {
	double tmp;
	if (x <= -5e+22) {
		tmp = 0.5 / x;
	} else if (x <= 9000.0) {
		tmp = (x + (x * (fma(0.1049934947, (x * x), (0.0424060604 * pow(x, 4.0))) + fma(pow(x, 6.0), 0.0072644182, (pow(x, 8.0) * (0.0005064034 + ((x * x) * 0.0001789971))))))) / (fma(pow(x, 8.0), 0.0140005442, fma(pow(x, 4.0), 0.2909738639, (pow(x, 6.0) * 0.0694555761))) + (fma(x, (x * 0.7715471019), 1.0) + fma(pow(x, 10.0), 0.0008327945, (0.0003579942 * pow(x, 12.0)))));
	} else {
		tmp = -0.25 / (x * ((0.2514179000665374 / (x * x)) + -0.5));
	}
	return tmp;
}

function code(x)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(0.0424060604 * Float64(Float64(x * x) * Float64(x * x)))) + Float64(0.0072644182 * Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0005064034 * Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0001789971 * Float64(Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) / Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.7715471019 * Float64(x * x))) + Float64(0.2909738639 * Float64(Float64(x * x) * Float64(x * x)))) + Float64(0.0694555761 * Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0140005442 * Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0008327945 * Float64(Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) + Float64(Float64(2.0 * 0.0001789971) * Float64(Float64(Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x))))) * x)
end
↓
function code(x)
	tmp = 0.0
	if (x <= -5e+22)
		tmp = Float64(0.5 / x);
	elseif (x <= 9000.0)
		tmp = Float64(Float64(x + Float64(x * Float64(fma(0.1049934947, Float64(x * x), Float64(0.0424060604 * (x ^ 4.0))) + fma((x ^ 6.0), 0.0072644182, Float64((x ^ 8.0) * Float64(0.0005064034 + Float64(Float64(x * x) * 0.0001789971))))))) / Float64(fma((x ^ 8.0), 0.0140005442, fma((x ^ 4.0), 0.2909738639, Float64((x ^ 6.0) * 0.0694555761))) + Float64(fma(x, Float64(x * 0.7715471019), 1.0) + fma((x ^ 10.0), 0.0008327945, Float64(0.0003579942 * (x ^ 12.0))))));
	else
		tmp = Float64(-0.25 / Float64(x * Float64(Float64(0.2514179000665374 / Float64(x * x)) + -0.5)));
	end
	return tmp
end

code[x_] := N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(1.0 + N[(0.7715471019 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0694555761 * N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]
↓
code[x_] := If[LessEqual[x, -5e+22], N[(0.5 / x), $MachinePrecision], If[LessEqual[x, 9000.0], N[(N[(x + N[(x * N[(N[(0.1049934947 * N[(x * x), $MachinePrecision] + N[(0.0424060604 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 6.0], $MachinePrecision] * 0.0072644182 + N[(N[Power[x, 8.0], $MachinePrecision] * N[(0.0005064034 + N[(N[(x * x), $MachinePrecision] * 0.0001789971), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[x, 8.0], $MachinePrecision] * 0.0140005442 + N[(N[Power[x, 4.0], $MachinePrecision] * 0.2909738639 + N[(N[Power[x, 6.0], $MachinePrecision] * 0.0694555761), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(x * 0.7715471019), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[Power[x, 10.0], $MachinePrecision] * 0.0008327945 + N[(0.0003579942 * N[Power[x, 12.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 / N[(x * N[(N[(0.2514179000665374 / N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x
↓
\begin{array}{l}
\mathbf{if}\;x \leq -5 \cdot 10^{+22}:\\
\;\;\;\;\frac{0.5}{x}\\

\mathbf{elif}\;x \leq 9000:\\
\;\;\;\;\frac{x + x \cdot \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 0.0424060604 \cdot {x}^{4}\right) + \mathsf{fma}\left({x}^{6}, 0.0072644182, {x}^{8} \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right)}{\mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{4}, 0.2909738639, {x}^{6} \cdot 0.0694555761\right)\right) + \left(\mathsf{fma}\left(x, x \cdot 0.7715471019, 1\right) + \mathsf{fma}\left({x}^{10}, 0.0008327945, 0.0003579942 \cdot {x}^{12}\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{-0.25}{x \cdot \left(\frac{0.2514179000665374}{x \cdot x} + -0.5\right)}\\


\end{array}

Error?

Derivation?

  1. Split input into 3 regimes

  2. if x < -4.9999999999999996e22

    1. Initial program 1.1%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

    2. Simplified1.1%

      \[\leadsto \color{blue}{x \cdot \frac{\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(0.0005064034 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0001789971 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}{\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.2909738639 + \left(0.0694555761 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0140005442\right) + \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}} \]
      Proof

      [Start]1.1

      \[ \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      *-commutative [=>]1.1

      \[ \color{blue}{x \cdot \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}} \]

    3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]

    if -4.9999999999999996e22 < x < 9e3

    1. Initial program 99.9%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

    2. Simplified99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(0.0005064034 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0001789971 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}{\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.2909738639 + \left(0.0694555761 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0140005442\right) + \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}} \]
      Proof

      [Start]99.9

      \[ \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      *-commutative [=>]99.9

      \[ \color{blue}{x \cdot \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}} \]

    3. Applied egg-rr99.9%

      \[\leadsto \color{blue}{\frac{x \cdot \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + 0.0424060604 \cdot \left(x \cdot {x}^{3}\right)\right)\right) + \left(0.0072644182 \cdot \left({x}^{3} \cdot {x}^{3}\right) + \mathsf{fma}\left(0.0005064034, {\left(x \cdot {x}^{3}\right)}^{2}, \left(0.0001789971 \cdot \left(x \cdot x\right)\right) \cdot {\left(x \cdot {x}^{3}\right)}^{2}\right)\right)\right)}{\left(\left(1 + x \cdot \left(x \cdot 0.7715471019\right)\right) + \left(\mathsf{fma}\left(x \cdot {x}^{3}, 0.2909738639, 0.0694555761 \cdot \left({x}^{3} \cdot {x}^{3}\right)\right) + {\left(x \cdot {x}^{3}\right)}^{2} \cdot 0.0140005442\right)\right) + \mathsf{fma}\left(x \cdot \left(x \cdot {\left(x \cdot {x}^{3}\right)}^{2}\right), 0.0008327945, 0.0003579942 \cdot {\left(x \cdot {x}^{3}\right)}^{3}\right)}} \]

    4. Simplified99.9%

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 + \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 0.0424060604 \cdot {x}^{4}\right) + \mathsf{fma}\left({x}^{6}, 0.0072644182, {x}^{8} \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{4}, 0.2909738639, {x}^{6} \cdot 0.0694555761\right)\right) + \left(\mathsf{fma}\left(x, x \cdot 0.7715471019, 1\right) + \mathsf{fma}\left({x}^{6} \cdot {x}^{4}, 0.0008327945, 0.0003579942 \cdot {x}^{12}\right)\right)}} \]
      Proof

      [Start]99.9

      \[ \frac{x \cdot \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + 0.0424060604 \cdot \left(x \cdot {x}^{3}\right)\right)\right) + \left(0.0072644182 \cdot \left({x}^{3} \cdot {x}^{3}\right) + \mathsf{fma}\left(0.0005064034, {\left(x \cdot {x}^{3}\right)}^{2}, \left(0.0001789971 \cdot \left(x \cdot x\right)\right) \cdot {\left(x \cdot {x}^{3}\right)}^{2}\right)\right)\right)}{\left(\left(1 + x \cdot \left(x \cdot 0.7715471019\right)\right) + \left(\mathsf{fma}\left(x \cdot {x}^{3}, 0.2909738639, 0.0694555761 \cdot \left({x}^{3} \cdot {x}^{3}\right)\right) + {\left(x \cdot {x}^{3}\right)}^{2} \cdot 0.0140005442\right)\right) + \mathsf{fma}\left(x \cdot \left(x \cdot {\left(x \cdot {x}^{3}\right)}^{2}\right), 0.0008327945, 0.0003579942 \cdot {\left(x \cdot {x}^{3}\right)}^{3}\right)} \]

      associate-*r/ [<=]99.9

      \[ \color{blue}{x \cdot \frac{\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + 0.0424060604 \cdot \left(x \cdot {x}^{3}\right)\right)\right) + \left(0.0072644182 \cdot \left({x}^{3} \cdot {x}^{3}\right) + \mathsf{fma}\left(0.0005064034, {\left(x \cdot {x}^{3}\right)}^{2}, \left(0.0001789971 \cdot \left(x \cdot x\right)\right) \cdot {\left(x \cdot {x}^{3}\right)}^{2}\right)\right)}{\left(\left(1 + x \cdot \left(x \cdot 0.7715471019\right)\right) + \left(\mathsf{fma}\left(x \cdot {x}^{3}, 0.2909738639, 0.0694555761 \cdot \left({x}^{3} \cdot {x}^{3}\right)\right) + {\left(x \cdot {x}^{3}\right)}^{2} \cdot 0.0140005442\right)\right) + \mathsf{fma}\left(x \cdot \left(x \cdot {\left(x \cdot {x}^{3}\right)}^{2}\right), 0.0008327945, 0.0003579942 \cdot {\left(x \cdot {x}^{3}\right)}^{3}\right)}} \]

      *-lft-identity [<=]99.9

      \[ \color{blue}{\left(1 \cdot x\right)} \cdot \frac{\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + 0.0424060604 \cdot \left(x \cdot {x}^{3}\right)\right)\right) + \left(0.0072644182 \cdot \left({x}^{3} \cdot {x}^{3}\right) + \mathsf{fma}\left(0.0005064034, {\left(x \cdot {x}^{3}\right)}^{2}, \left(0.0001789971 \cdot \left(x \cdot x\right)\right) \cdot {\left(x \cdot {x}^{3}\right)}^{2}\right)\right)}{\left(\left(1 + x \cdot \left(x \cdot 0.7715471019\right)\right) + \left(\mathsf{fma}\left(x \cdot {x}^{3}, 0.2909738639, 0.0694555761 \cdot \left({x}^{3} \cdot {x}^{3}\right)\right) + {\left(x \cdot {x}^{3}\right)}^{2} \cdot 0.0140005442\right)\right) + \mathsf{fma}\left(x \cdot \left(x \cdot {\left(x \cdot {x}^{3}\right)}^{2}\right), 0.0008327945, 0.0003579942 \cdot {\left(x \cdot {x}^{3}\right)}^{3}\right)} \]

    5. Applied egg-rr99.9%

      \[\leadsto \color{blue}{\left(x + \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 0.0424060604 \cdot {x}^{4}\right) + \mathsf{fma}\left({x}^{6}, 0.0072644182, {x}^{8} \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right) \cdot x\right) \cdot \frac{1}{\mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{4}, 0.2909738639, {x}^{6} \cdot 0.0694555761\right)\right) + \left(\mathsf{fma}\left(x, x \cdot 0.7715471019, 1\right) + \mathsf{fma}\left({x}^{10}, 0.0008327945, 0.0003579942 \cdot {x}^{12}\right)\right)}} \]

    6. Applied egg-rr99.9%

      \[\leadsto \color{blue}{\frac{x + x \cdot \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 0.0424060604 \cdot {x}^{4}\right) + \mathsf{fma}\left({x}^{6}, 0.0072644182, {x}^{8} \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{4}, 0.2909738639, {x}^{6} \cdot 0.0694555761\right)\right) + \left(\mathsf{fma}\left(x, x \cdot 0.7715471019, 1\right) + \mathsf{fma}\left({x}^{10}, 0.0008327945, 0.0003579942 \cdot {x}^{12}\right)\right)}} \]

    7. Simplified99.9%

      \[\leadsto \color{blue}{\frac{x + x \cdot \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 0.0424060604 \cdot {x}^{4}\right) + \mathsf{fma}\left({x}^{6}, 0.0072644182, {x}^{8} \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right)}{\mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{4}, 0.2909738639, {x}^{6} \cdot 0.0694555761\right)\right) + \left(\mathsf{fma}\left(x, x \cdot 0.7715471019, 1\right) + \mathsf{fma}\left({x}^{10}, 0.0008327945, 0.0003579942 \cdot {x}^{12}\right)\right)}} \]
      Proof

      [Start]99.9

      \[ \frac{x + x \cdot \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 0.0424060604 \cdot {x}^{4}\right) + \mathsf{fma}\left({x}^{6}, 0.0072644182, {x}^{8} \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{4}, 0.2909738639, {x}^{6} \cdot 0.0694555761\right)\right) + \left(\mathsf{fma}\left(x, x \cdot 0.7715471019, 1\right) + \mathsf{fma}\left({x}^{10}, 0.0008327945, 0.0003579942 \cdot {x}^{12}\right)\right)} \]

      *-rgt-identity [<=]99.9

      \[ \frac{\color{blue}{\left(x + x \cdot \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 0.0424060604 \cdot {x}^{4}\right) + \mathsf{fma}\left({x}^{6}, 0.0072644182, {x}^{8} \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)\right) \cdot 1}}{\mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{4}, 0.2909738639, {x}^{6} \cdot 0.0694555761\right)\right) + \left(\mathsf{fma}\left(x, x \cdot 0.7715471019, 1\right) + \mathsf{fma}\left({x}^{10}, 0.0008327945, 0.0003579942 \cdot {x}^{12}\right)\right)} \]

      *-commutative [=>]99.9

      \[ \frac{\color{blue}{1 \cdot \left(x + x \cdot \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 0.0424060604 \cdot {x}^{4}\right) + \mathsf{fma}\left({x}^{6}, 0.0072644182, {x}^{8} \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)\right)}}{\mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{4}, 0.2909738639, {x}^{6} \cdot 0.0694555761\right)\right) + \left(\mathsf{fma}\left(x, x \cdot 0.7715471019, 1\right) + \mathsf{fma}\left({x}^{10}, 0.0008327945, 0.0003579942 \cdot {x}^{12}\right)\right)} \]

    if 9e3 < x

    1. Initial program 6.9%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

    2. Simplified6.9%

      \[\leadsto \color{blue}{x \cdot \frac{\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(0.0005064034 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0001789971 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}{\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.2909738639 + \left(0.0694555761 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0140005442\right) + \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}} \]
      Proof

      [Start]6.9

      \[ \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      *-commutative [=>]6.9

      \[ \color{blue}{x \cdot \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}} \]

    3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}} \]

    4. Simplified100.0%

      \[\leadsto \color{blue}{\frac{0.2514179000665374}{{x}^{3}} + \frac{0.5}{x}} \]
      Proof

      [Start]100.0

      \[ 0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x} \]

      associate-*r/ [=>]100.0

      \[ \color{blue}{\frac{0.2514179000665374 \cdot 1}{{x}^{3}}} + 0.5 \cdot \frac{1}{x} \]

      metadata-eval [=>]100.0

      \[ \frac{\color{blue}{0.2514179000665374}}{{x}^{3}} + 0.5 \cdot \frac{1}{x} \]

      associate-*r/ [=>]100.0

      \[ \frac{0.2514179000665374}{{x}^{3}} + \color{blue}{\frac{0.5 \cdot 1}{x}} \]

      metadata-eval [=>]100.0

      \[ \frac{0.2514179000665374}{{x}^{3}} + \frac{\color{blue}{0.5}}{x} \]

    5. Applied egg-rr100.0%

      \[\leadsto \frac{0.2514179000665374}{\color{blue}{\left(x \cdot x\right) \cdot x}} + \frac{0.5}{x} \]

    6. Applied egg-rr33.5%

      \[\leadsto \color{blue}{\frac{\frac{0.06321096047386739}{{x}^{6}} - \frac{\frac{0.25}{x}}{x}}{x \cdot \frac{\frac{0.2514179000665374}{x}}{x} - x \cdot 0.5} \cdot \left(x \cdot x\right)} \]

    7. Simplified49.0%

      \[\leadsto \color{blue}{\frac{\left(\frac{0.06321096047386739}{{x}^{6}} - \frac{0.25}{x \cdot x}\right) \cdot \left(x \cdot x\right)}{x \cdot \left(\frac{0.2514179000665374}{x \cdot x} + -0.5\right)}} \]
      Proof

      [Start]33.5

      \[ \frac{\frac{0.06321096047386739}{{x}^{6}} - \frac{\frac{0.25}{x}}{x}}{x \cdot \frac{\frac{0.2514179000665374}{x}}{x} - x \cdot 0.5} \cdot \left(x \cdot x\right) \]

      associate-*l/ [=>]48.9

      \[ \color{blue}{\frac{\left(\frac{0.06321096047386739}{{x}^{6}} - \frac{\frac{0.25}{x}}{x}\right) \cdot \left(x \cdot x\right)}{x \cdot \frac{\frac{0.2514179000665374}{x}}{x} - x \cdot 0.5}} \]

      associate-/l/ [=>]49.0

      \[ \frac{\left(\frac{0.06321096047386739}{{x}^{6}} - \color{blue}{\frac{0.25}{x \cdot x}}\right) \cdot \left(x \cdot x\right)}{x \cdot \frac{\frac{0.2514179000665374}{x}}{x} - x \cdot 0.5} \]

      distribute-lft-out-- [=>]49.0

      \[ \frac{\left(\frac{0.06321096047386739}{{x}^{6}} - \frac{0.25}{x \cdot x}\right) \cdot \left(x \cdot x\right)}{\color{blue}{x \cdot \left(\frac{\frac{0.2514179000665374}{x}}{x} - 0.5\right)}} \]

      sub-neg [=>]49.0

      \[ \frac{\left(\frac{0.06321096047386739}{{x}^{6}} - \frac{0.25}{x \cdot x}\right) \cdot \left(x \cdot x\right)}{x \cdot \color{blue}{\left(\frac{\frac{0.2514179000665374}{x}}{x} + \left(-0.5\right)\right)}} \]

      associate-/l/ [=>]49.0

      \[ \frac{\left(\frac{0.06321096047386739}{{x}^{6}} - \frac{0.25}{x \cdot x}\right) \cdot \left(x \cdot x\right)}{x \cdot \left(\color{blue}{\frac{0.2514179000665374}{x \cdot x}} + \left(-0.5\right)\right)} \]

      metadata-eval [=>]49.0

      \[ \frac{\left(\frac{0.06321096047386739}{{x}^{6}} - \frac{0.25}{x \cdot x}\right) \cdot \left(x \cdot x\right)}{x \cdot \left(\frac{0.2514179000665374}{x \cdot x} + \color{blue}{-0.5}\right)} \]

    8. Taylor expanded in x around inf 100.0%

      \[\leadsto \frac{\color{blue}{-0.25}}{x \cdot \left(\frac{0.2514179000665374}{x \cdot x} + -0.5\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification100.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 9000:\\ \;\;\;\;\frac{x + x \cdot \left(\mathsf{fma}\left(0.1049934947, x \cdot x, 0.0424060604 \cdot {x}^{4}\right) + \mathsf{fma}\left({x}^{6}, 0.0072644182, {x}^{8} \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right)}{\mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{4}, 0.2909738639, {x}^{6} \cdot 0.0694555761\right)\right) + \left(\mathsf{fma}\left(x, x \cdot 0.7715471019, 1\right) + \mathsf{fma}\left({x}^{10}, 0.0008327945, 0.0003579942 \cdot {x}^{12}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{x \cdot \left(\frac{0.2514179000665374}{x \cdot x} + -0.5\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy100.0%
Cost11208
\[\begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := x \cdot t_0\\ t_2 := t_1 \cdot t_1\\ t_3 := \left(x \cdot x\right) \cdot t_2\\ \mathbf{if}\;x \leq -12000:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{t_0}\\ \mathbf{elif}\;x \leq 2000:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t_1\right) + 0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot t_1\right)\right) + \left(0.0005064034 \cdot t_2 + 0.0001789971 \cdot t_3\right)}{\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + \left(0.2909738639 \cdot t_1 + \left(x \cdot x\right) \cdot \left(0.0694555761 \cdot t_1\right)\right)\right) + 0.0140005442 \cdot t_2\right) + \left(0.0008327945 \cdot t_3 + 0.0003579942 \cdot \left(t_1 \cdot t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{x \cdot \left(\frac{0.2514179000665374}{x \cdot x} + -0.5\right)}\\ \end{array} \]
Alternative 2
Accuracy99.6%
Cost969
\[\begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -0.95 \lor \neg \left(x \leq 0.95\right):\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{t_0}\\ \mathbf{else}:\\ \;\;\;\;x + t_0 \cdot -0.6665536072\\ \end{array} \]
Alternative 3
Accuracy99.6%
Cost968
\[\begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{t_0}\\ \mathbf{elif}\;x \leq 1.02:\\ \;\;\;\;x + t_0 \cdot -0.6665536072\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{x \cdot \left(\frac{0.2514179000665374}{x \cdot x} + -0.5\right)}\\ \end{array} \]
Alternative 4
Accuracy99.4%
Cost841
\[\begin{array}{l} \mathbf{if}\;x \leq -0.78 \lor \neg \left(x \leq 0.8\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.6665536072\right)\\ \end{array} \]
Alternative 5
Accuracy99.4%
Cost840
\[\begin{array}{l} \mathbf{if}\;x \leq -0.78:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;x + \left(x \cdot \left(x \cdot x\right)\right) \cdot -0.6665536072\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
Alternative 6
Accuracy99.1%
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -0.7:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
Alternative 7
Accuracy52.0%
Cost64
\[x \]

Error

Reproduce?

herbie shell --seed 2023122 
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
  (* (/ (+ (+ (+ (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x)))) (* 0.0072644182 (* (* (* x x) (* x x)) (* x x)))) (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (+ (+ (+ (+ (+ (+ 1.0 (* 0.7715471019 (* x x))) (* 0.2909738639 (* (* x x) (* x x)))) (* 0.0694555761 (* (* (* x x) (* x x)) (* x x)))) (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (* (* 2.0 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))