Jmat.Real.dawson

Average Error: 45.7% → 0.02%

Time: 11.2s

Precision: binary64

Cost: 87368

Math

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

↓

\[\begin{array}{l} t_0 := {\left(x \cdot x\right)}^{4}\\ t_1 := {\left(x \cdot x\right)}^{5}\\ \mathbf{if}\;x \leq -7000:\\ \;\;\;\;\frac{1}{x \cdot x} \cdot \frac{0.2514179000665374}{x} + \frac{0.5}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;\frac{x}{\frac{1 + \left(\mathsf{fma}\left({x}^{6}, 0.0694555761, t_0 \cdot 0.0140005442\right) + \left(t_1 \cdot \left(0.0008327945 + x \cdot \left(x \cdot 0.0003579942\right)\right) + \mathsf{fma}\left(x \cdot x, 0.7715471019, 0.2909738639 \cdot {x}^{4}\right)\right)\right)}{1 + \left(\mathsf{fma}\left(0.0005064034, t_0, t_1 \cdot 0.0001789971\right) + \mathsf{fma}\left(x, x \cdot 0.1049934947, \mathsf{fma}\left(0.0424060604, {x}^{4}, {x}^{6} \cdot 0.0072644182\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

FPCore

(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
 (let* ((t_0 (pow (* x x) 4.0)) (t_1 (pow (* x x) 5.0)))
   (if (<= x -7000.0)
     (+ (* (/ 1.0 (* x x)) (/ 0.2514179000665374 x)) (/ 0.5 x))
     (if (<= x 50000000.0)
       (/
        x
        (/
         (+
          1.0
          (+
           (fma (pow x 6.0) 0.0694555761 (* t_0 0.0140005442))
           (+
            (* t_1 (+ 0.0008327945 (* x (* x 0.0003579942))))
            (fma (* x x) 0.7715471019 (* 0.2909738639 (pow x 4.0))))))
         (+
          1.0
          (+
           (fma 0.0005064034 t_0 (* t_1 0.0001789971))
           (fma
            x
            (* x 0.1049934947)
            (fma 0.0424060604 (pow x 4.0) (* (pow x 6.0) 0.0072644182)))))))
       (/ 0.5 x)))))

C

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 t_0 = pow((x * x), 4.0);
	double t_1 = pow((x * x), 5.0);
	double tmp;
	if (x <= -7000.0) {
		tmp = ((1.0 / (x * x)) * (0.2514179000665374 / x)) + (0.5 / x);
	} else if (x <= 50000000.0) {
		tmp = x / ((1.0 + (fma(pow(x, 6.0), 0.0694555761, (t_0 * 0.0140005442)) + ((t_1 * (0.0008327945 + (x * (x * 0.0003579942)))) + fma((x * x), 0.7715471019, (0.2909738639 * pow(x, 4.0)))))) / (1.0 + (fma(0.0005064034, t_0, (t_1 * 0.0001789971)) + fma(x, (x * 0.1049934947), fma(0.0424060604, pow(x, 4.0), (pow(x, 6.0) * 0.0072644182))))));
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Julia

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)
	t_0 = Float64(x * x) ^ 4.0
	t_1 = Float64(x * x) ^ 5.0
	tmp = 0.0
	if (x <= -7000.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(x * x)) * Float64(0.2514179000665374 / x)) + Float64(0.5 / x));
	elseif (x <= 50000000.0)
		tmp = Float64(x / Float64(Float64(1.0 + Float64(fma((x ^ 6.0), 0.0694555761, Float64(t_0 * 0.0140005442)) + Float64(Float64(t_1 * Float64(0.0008327945 + Float64(x * Float64(x * 0.0003579942)))) + fma(Float64(x * x), 0.7715471019, Float64(0.2909738639 * (x ^ 4.0)))))) / Float64(1.0 + Float64(fma(0.0005064034, t_0, Float64(t_1 * 0.0001789971)) + fma(x, Float64(x * 0.1049934947), fma(0.0424060604, (x ^ 4.0), Float64((x ^ 6.0) * 0.0072644182)))))));
	else
		tmp = Float64(0.5 / x);
	end
	return tmp
end

Wolfram

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_] := Block[{t$95$0 = N[Power[N[(x * x), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x * x), $MachinePrecision], 5.0], $MachinePrecision]}, If[LessEqual[x, -7000.0], N[(N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.2514179000665374 / x), $MachinePrecision]), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 50000000.0], N[(x / N[(N[(1.0 + N[(N[(N[Power[x, 6.0], $MachinePrecision] * 0.0694555761 + N[(t$95$0 * 0.0140005442), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * N[(0.0008327945 + N[(x * N[(x * 0.0003579942), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.7715471019 + N[(0.2909738639 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.0005064034 * t$95$0 + N[(t$95$1 * 0.0001789971), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * 0.1049934947), $MachinePrecision] + N[(0.0424060604 * N[Power[x, 4.0], $MachinePrecision] + N[(N[Power[x, 6.0], $MachinePrecision] * 0.0072644182), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7e3

   1. Initial program 93.17

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

   2. Simplified 93.18

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

      [Start] 93.17

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

      *-commutative [=>] 93.17

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

   3. Taylor expanded in x around inf 0

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

   4. Simplified 0

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

      [Start] 0	\[ 0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x} \]
      associate-*r/ [=>] 0	\[ \color{blue}{\frac{0.2514179000665374 \cdot 1}{{x}^{3}}} + 0.5 \cdot \frac{1}{x} \]
      metadata-eval [=>] 0	\[ \frac{\color{blue}{0.2514179000665374}}{{x}^{3}} + 0.5 \cdot \frac{1}{x} \]
      associate-*r/ [=>] 0	\[ \frac{0.2514179000665374}{{x}^{3}} + \color{blue}{\frac{0.5 \cdot 1}{x}} \]
      metadata-eval [=>] 0	\[ \frac{0.2514179000665374}{{x}^{3}} + \frac{\color{blue}{0.5}}{x} \]

   5. Applied egg-rr 0

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

   if -7e3 < x < 5e7

   1. Initial program 0.04

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

   2. Applied egg-rr 0.04

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

   3. Applied egg-rr 0.04

      \[\leadsto \color{blue}{1 \cdot \frac{x \cdot \left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \mathsf{fma}\left(0.0424060604, {\left(x \cdot x\right)}^{2}, 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + \mathsf{fma}\left(0.0005064034, x \cdot \left(x \cdot {\left(x \cdot x\right)}^{3}\right), 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)\right)}{\left(1 + \mathsf{fma}\left(x \cdot x, 0.7715471019, {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{3}, 0.0694555761, \left(x \cdot x\right) \cdot \left({\left(x \cdot x\right)}^{3} \cdot 0.0140005442\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{5}\right)\right)\right)}} \]

   4. Simplified 0.04

      \[\leadsto \color{blue}{\frac{x \cdot \left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(\mathsf{fma}\left(0.0424060604, x \cdot {x}^{3}, 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right) + \mathsf{fma}\left(0.0005064034, \left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}, 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)\right)\right)}{1 + \left(\mathsf{fma}\left(x \cdot x, 0.7715471019, \left(x \cdot {x}^{3}\right) \cdot 0.2909738639\right) + \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{3}, 0.0694555761, \left(x \cdot x\right) \cdot \left({\left(x \cdot x\right)}^{3} \cdot 0.0140005442\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \left(0.0003579942 \cdot \left(x \cdot x\right)\right) \cdot {\left(x \cdot x\right)}^{5}\right)\right)\right)}} \]
      Proof

      [Start] 0.04

      \[ 1 \cdot \frac{x \cdot \left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \mathsf{fma}\left(0.0424060604, {\left(x \cdot x\right)}^{2}, 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + \mathsf{fma}\left(0.0005064034, x \cdot \left(x \cdot {\left(x \cdot x\right)}^{3}\right), 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)\right)}{\left(1 + \mathsf{fma}\left(x \cdot x, 0.7715471019, {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{3}, 0.0694555761, \left(x \cdot x\right) \cdot \left({\left(x \cdot x\right)}^{3} \cdot 0.0140005442\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{5}\right)\right)\right)} \]

      associate-*r/ [=>] 0.04

      \[ \color{blue}{\frac{1 \cdot \left(x \cdot \left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \mathsf{fma}\left(0.0424060604, {\left(x \cdot x\right)}^{2}, 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + \mathsf{fma}\left(0.0005064034, x \cdot \left(x \cdot {\left(x \cdot x\right)}^{3}\right), 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)\right)\right)}{\left(1 + \mathsf{fma}\left(x \cdot x, 0.7715471019, {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{3}, 0.0694555761, \left(x \cdot x\right) \cdot \left({\left(x \cdot x\right)}^{3} \cdot 0.0140005442\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{5}\right)\right)\right)}} \]

   5. Applied egg-rr 0.04

      \[\leadsto \color{blue}{1 \cdot \frac{x \cdot \left(1 + \left(\left(0.1049934947 \cdot x\right) \cdot x + \left(\mathsf{fma}\left(0.0424060604, x \cdot {x}^{3}, 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right) + \mathsf{fma}\left(0.0005064034, \left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}, 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)\right)\right)\right)}{1 + \left(\mathsf{fma}\left(x \cdot x, 0.7715471019, x \cdot \left({x}^{3} \cdot 0.2909738639\right)\right) + \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{3}, 0.0694555761, \left(x \cdot x\right) \cdot \left({\left(x \cdot x\right)}^{3} \cdot 0.0140005442\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, {\left(x \cdot x\right)}^{5} \cdot \left(\left(0.0003579942 \cdot x\right) \cdot x\right)\right)\right)\right)}} \]

   6. Simplified 0.04

      \[\leadsto \color{blue}{\frac{x}{\frac{1 + \left(\mathsf{fma}\left({x}^{6}, 0.0694555761, {\left(x \cdot x\right)}^{4} \cdot 0.0140005442\right) + \left({\left(x \cdot x\right)}^{5} \cdot \left(0.0008327945 + x \cdot \left(x \cdot 0.0003579942\right)\right) + \mathsf{fma}\left(x \cdot x, 0.7715471019, 0.2909738639 \cdot {x}^{4}\right)\right)\right)}{1 + \left(\mathsf{fma}\left(0.0005064034, {\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right) + \mathsf{fma}\left(x, x \cdot 0.1049934947, \mathsf{fma}\left(0.0424060604, {x}^{4}, 0.0072644182 \cdot {x}^{6}\right)\right)\right)}}} \]
      Proof

      [Start] 0.04	\[ 1 \cdot \frac{x \cdot \left(1 + \left(\left(0.1049934947 \cdot x\right) \cdot x + \left(\mathsf{fma}\left(0.0424060604, x \cdot {x}^{3}, 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right) + \mathsf{fma}\left(0.0005064034, \left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}, 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)\right)\right)\right)}{1 + \left(\mathsf{fma}\left(x \cdot x, 0.7715471019, x \cdot \left({x}^{3} \cdot 0.2909738639\right)\right) + \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{3}, 0.0694555761, \left(x \cdot x\right) \cdot \left({\left(x \cdot x\right)}^{3} \cdot 0.0140005442\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, {\left(x \cdot x\right)}^{5} \cdot \left(\left(0.0003579942 \cdot x\right) \cdot x\right)\right)\right)\right)} \]
      *-commutative [=>] 0.04	\[ \color{blue}{\frac{x \cdot \left(1 + \left(\left(0.1049934947 \cdot x\right) \cdot x + \left(\mathsf{fma}\left(0.0424060604, x \cdot {x}^{3}, 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right) + \mathsf{fma}\left(0.0005064034, \left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}, 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)\right)\right)\right)}{1 + \left(\mathsf{fma}\left(x \cdot x, 0.7715471019, x \cdot \left({x}^{3} \cdot 0.2909738639\right)\right) + \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{3}, 0.0694555761, \left(x \cdot x\right) \cdot \left({\left(x \cdot x\right)}^{3} \cdot 0.0140005442\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, {\left(x \cdot x\right)}^{5} \cdot \left(\left(0.0003579942 \cdot x\right) \cdot x\right)\right)\right)\right)} \cdot 1} \]
      associate-/l* [=>] 0.04	\[ \color{blue}{\frac{x}{\frac{1 + \left(\mathsf{fma}\left(x \cdot x, 0.7715471019, x \cdot \left({x}^{3} \cdot 0.2909738639\right)\right) + \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{3}, 0.0694555761, \left(x \cdot x\right) \cdot \left({\left(x \cdot x\right)}^{3} \cdot 0.0140005442\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, {\left(x \cdot x\right)}^{5} \cdot \left(\left(0.0003579942 \cdot x\right) \cdot x\right)\right)\right)\right)}{1 + \left(\left(0.1049934947 \cdot x\right) \cdot x + \left(\mathsf{fma}\left(0.0424060604, x \cdot {x}^{3}, 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right) + \mathsf{fma}\left(0.0005064034, \left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}, 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)\right)\right)}}} \cdot 1 \]

   if 5e7 < x

   1. Initial program 94.61

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

   2. Simplified 94.62

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

      [Start] 94.61

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

      *-commutative [=>] 94.61

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

   3. Taylor expanded in x around inf 0

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.02

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7000:\\ \;\;\;\;\frac{1}{x \cdot x} \cdot \frac{0.2514179000665374}{x} + \frac{0.5}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;\frac{x}{\frac{1 + \left(\mathsf{fma}\left({x}^{6}, 0.0694555761, {\left(x \cdot x\right)}^{4} \cdot 0.0140005442\right) + \left({\left(x \cdot x\right)}^{5} \cdot \left(0.0008327945 + x \cdot \left(x \cdot 0.0003579942\right)\right) + \mathsf{fma}\left(x \cdot x, 0.7715471019, 0.2909738639 \cdot {x}^{4}\right)\right)\right)}{1 + \left(\mathsf{fma}\left(0.0005064034, {\left(x \cdot x\right)}^{4}, {\left(x \cdot x\right)}^{5} \cdot 0.0001789971\right) + \mathsf{fma}\left(x, x \cdot 0.1049934947, \mathsf{fma}\left(0.0424060604, {x}^{4}, {x}^{6} \cdot 0.0072644182\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.03%
Cost	77705

\[\begin{array}{l} t_0 := {\left(x \cdot x\right)}^{2}\\ t_1 := {\left(x \cdot x\right)}^{3}\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ t_3 := t_0 \cdot t_1\\ \mathbf{if}\;x \leq -4 \cdot 10^{+14} \lor \neg \left(x \leq 50000000\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + \left(0.0424060604 \cdot t_0 + 0.0072644182 \cdot t_1\right)\right) + \left(0.0005064034 \cdot t_2 + 0.0001789971 \cdot t_3\right)\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + 0.2909738639 \cdot t_0\right)\right) + \left(0.0694555761 \cdot t_1 + 0.0140005442 \cdot t_2\right)\right) + \left(0.0008327945 \cdot t_3 + 0.0003579942 \cdot \left(t_2 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \]

Alternative 2
Error	0.02%
Cost	11208

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := t_0 \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ \mathbf{if}\;x \leq -10000:\\ \;\;\;\;\frac{1}{x \cdot x} \cdot \frac{0.2514179000665374}{x} + \frac{0.5}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot t_0\right)\right) + \left(0.0005064034 \cdot t_1 + 0.0001789971 \cdot t_2\right)}{\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + \left(0.2909738639 \cdot t_0 + \left(x \cdot x\right) \cdot \left(0.0694555761 \cdot t_0\right)\right)\right) + 0.0140005442 \cdot t_1\right) + \left(0.0008327945 \cdot t_2 + 0.0003579942 \cdot \left(t_0 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 3
Error	0.5%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{1}{x \cdot x} \cdot \frac{0.2514179000665374}{x} + \frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + \left(x \cdot x\right) \cdot 0.6665536072}\\ \end{array} \]

Alternative 4
Error	0.73%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -0.8 \lor \neg \left(x \leq 0.8\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.6665536072\right)\\ \end{array} \]

Alternative 5
Error	0.73%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;x + x \cdot \left(\left(x \cdot x\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 6
Error	0.73%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;\frac{x}{1 + \left(x \cdot x\right) \cdot 0.6665536072}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 7
Error	1.02%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -0.72:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 8
Error	48.16%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023115 
(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))