Jmat.Real.dawson

Average Error: 29.8 → 0.0

Time: 27.0s

Precision: binary64

Cost: 20872

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := x \cdot \left(x \cdot t_0\right)\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ \mathbf{if}\;x \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;x \cdot \frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + t_0 \cdot \left(\left(x \cdot x\right) \cdot 0.0072644182 + 0.0424060604\right)\right) + t_2 \cdot \left(\left(x \cdot x\right) \cdot 0.0001789971 + 0.0005064034\right)}{\left(\left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.2909738639 + 0.7715471019\right)\right) + t_1 \cdot \left(\left(x \cdot x\right) \cdot 0.0140005442 + 0.0694555761\right)\right) + \left(\left(x \cdot x\right) \cdot t_2\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0003579942 + 0.0008327945\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{x} + \left(\frac{0.2514179000665374}{{x}^{3}} + \left(\frac{0.15298196345929074}{{x}^{5}} + 11.259630434457211 \cdot \frac{1}{{x}^{7}}\right)\right)\\ \end{array} \]

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 (* x (* x (* x x)))) (t_1 (* x (* x t_0))) (t_2 (* (* x x) t_1)))
   (if (<= x -2e+20)
     (/ 0.5 x)
     (if (<= x 200.0)
       (*
        x
        (/
         (+
          (+
           (+ 1.0 (* 0.1049934947 (* x x)))
           (* t_0 (+ (* (* x x) 0.0072644182) 0.0424060604)))
          (* t_2 (+ (* (* x x) 0.0001789971) 0.0005064034)))
         (+
          (+
           (+ 1.0 (* (* x x) (+ (* (* x x) 0.2909738639) 0.7715471019)))
           (* t_1 (+ (* (* x x) 0.0140005442) 0.0694555761)))
          (* (* (* x x) t_2) (+ (* (* x x) 0.0003579942) 0.0008327945)))))
       (+
        (* 0.5 (/ 1.0 x))
        (+
         (/ 0.2514179000665374 (pow x 3.0))
         (+
          (/ 0.15298196345929074 (pow x 5.0))
          (* 11.259630434457211 (/ 1.0 (pow x 7.0))))))))))

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 = x * (x * (x * x));
	double t_1 = x * (x * t_0);
	double t_2 = (x * x) * t_1;
	double tmp;
	if (x <= -2e+20) {
		tmp = 0.5 / x;
	} else if (x <= 200.0) {
		tmp = x * ((((1.0 + (0.1049934947 * (x * x))) + (t_0 * (((x * x) * 0.0072644182) + 0.0424060604))) + (t_2 * (((x * x) * 0.0001789971) + 0.0005064034))) / (((1.0 + ((x * x) * (((x * x) * 0.2909738639) + 0.7715471019))) + (t_1 * (((x * x) * 0.0140005442) + 0.0694555761))) + (((x * x) * t_2) * (((x * x) * 0.0003579942) + 0.0008327945))));
	} else {
		tmp = (0.5 * (1.0 / x)) + ((0.2514179000665374 / pow(x, 3.0)) + ((0.15298196345929074 / pow(x, 5.0)) + (11.259630434457211 * (1.0 / pow(x, 7.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((((1.0d0 + (0.1049934947d0 * (x * x))) + (0.0424060604d0 * ((x * x) * (x * x)))) + (0.0072644182d0 * (((x * x) * (x * x)) * (x * x)))) + (0.0005064034d0 * ((((x * x) * (x * x)) * (x * x)) * (x * x)))) + (0.0001789971d0 * (((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)))) / ((((((1.0d0 + (0.7715471019d0 * (x * x))) + (0.2909738639d0 * ((x * x) * (x * x)))) + (0.0694555761d0 * (((x * x) * (x * x)) * (x * x)))) + (0.0140005442d0 * ((((x * x) * (x * x)) * (x * x)) * (x * x)))) + (0.0008327945d0 * (((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)))) + ((2.0d0 * 0.0001789971d0) * ((((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)) * (x * x))))) * x
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x * (x * (x * x))
    t_1 = x * (x * t_0)
    t_2 = (x * x) * t_1
    if (x <= (-2d+20)) then
        tmp = 0.5d0 / x
    else if (x <= 200.0d0) then
        tmp = x * ((((1.0d0 + (0.1049934947d0 * (x * x))) + (t_0 * (((x * x) * 0.0072644182d0) + 0.0424060604d0))) + (t_2 * (((x * x) * 0.0001789971d0) + 0.0005064034d0))) / (((1.0d0 + ((x * x) * (((x * x) * 0.2909738639d0) + 0.7715471019d0))) + (t_1 * (((x * x) * 0.0140005442d0) + 0.0694555761d0))) + (((x * x) * t_2) * (((x * x) * 0.0003579942d0) + 0.0008327945d0))))
    else
        tmp = (0.5d0 * (1.0d0 / x)) + ((0.2514179000665374d0 / (x ** 3.0d0)) + ((0.15298196345929074d0 / (x ** 5.0d0)) + (11.259630434457211d0 * (1.0d0 / (x ** 7.0d0)))))
    end if
    code = tmp
end function

Java

public static 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;
}

↓

public static double code(double x) {
	double t_0 = x * (x * (x * x));
	double t_1 = x * (x * t_0);
	double t_2 = (x * x) * t_1;
	double tmp;
	if (x <= -2e+20) {
		tmp = 0.5 / x;
	} else if (x <= 200.0) {
		tmp = x * ((((1.0 + (0.1049934947 * (x * x))) + (t_0 * (((x * x) * 0.0072644182) + 0.0424060604))) + (t_2 * (((x * x) * 0.0001789971) + 0.0005064034))) / (((1.0 + ((x * x) * (((x * x) * 0.2909738639) + 0.7715471019))) + (t_1 * (((x * x) * 0.0140005442) + 0.0694555761))) + (((x * x) * t_2) * (((x * x) * 0.0003579942) + 0.0008327945))));
	} else {
		tmp = (0.5 * (1.0 / x)) + ((0.2514179000665374 / Math.pow(x, 3.0)) + ((0.15298196345929074 / Math.pow(x, 5.0)) + (11.259630434457211 * (1.0 / Math.pow(x, 7.0)))));
	}
	return tmp;
}

Python

def code(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

↓

def code(x):
	t_0 = x * (x * (x * x))
	t_1 = x * (x * t_0)
	t_2 = (x * x) * t_1
	tmp = 0
	if x <= -2e+20:
		tmp = 0.5 / x
	elif x <= 200.0:
		tmp = x * ((((1.0 + (0.1049934947 * (x * x))) + (t_0 * (((x * x) * 0.0072644182) + 0.0424060604))) + (t_2 * (((x * x) * 0.0001789971) + 0.0005064034))) / (((1.0 + ((x * x) * (((x * x) * 0.2909738639) + 0.7715471019))) + (t_1 * (((x * x) * 0.0140005442) + 0.0694555761))) + (((x * x) * t_2) * (((x * x) * 0.0003579942) + 0.0008327945))))
	else:
		tmp = (0.5 * (1.0 / x)) + ((0.2514179000665374 / math.pow(x, 3.0)) + ((0.15298196345929074 / math.pow(x, 5.0)) + (11.259630434457211 * (1.0 / math.pow(x, 7.0)))))
	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 * Float64(x * Float64(x * x)))
	t_1 = Float64(x * Float64(x * t_0))
	t_2 = Float64(Float64(x * x) * t_1)
	tmp = 0.0
	if (x <= -2e+20)
		tmp = Float64(0.5 / x);
	elseif (x <= 200.0)
		tmp = Float64(x * Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(t_0 * Float64(Float64(Float64(x * x) * 0.0072644182) + 0.0424060604))) + Float64(t_2 * Float64(Float64(Float64(x * x) * 0.0001789971) + 0.0005064034))) / Float64(Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * 0.2909738639) + 0.7715471019))) + Float64(t_1 * Float64(Float64(Float64(x * x) * 0.0140005442) + 0.0694555761))) + Float64(Float64(Float64(x * x) * t_2) * Float64(Float64(Float64(x * x) * 0.0003579942) + 0.0008327945)))));
	else
		tmp = Float64(Float64(0.5 * Float64(1.0 / x)) + Float64(Float64(0.2514179000665374 / (x ^ 3.0)) + Float64(Float64(0.15298196345929074 / (x ^ 5.0)) + Float64(11.259630434457211 * Float64(1.0 / (x ^ 7.0))))));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = ((((((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;
end

↓

function tmp_2 = code(x)
	t_0 = x * (x * (x * x));
	t_1 = x * (x * t_0);
	t_2 = (x * x) * t_1;
	tmp = 0.0;
	if (x <= -2e+20)
		tmp = 0.5 / x;
	elseif (x <= 200.0)
		tmp = x * ((((1.0 + (0.1049934947 * (x * x))) + (t_0 * (((x * x) * 0.0072644182) + 0.0424060604))) + (t_2 * (((x * x) * 0.0001789971) + 0.0005064034))) / (((1.0 + ((x * x) * (((x * x) * 0.2909738639) + 0.7715471019))) + (t_1 * (((x * x) * 0.0140005442) + 0.0694555761))) + (((x * x) * t_2) * (((x * x) * 0.0003579942) + 0.0008327945))));
	else
		tmp = (0.5 * (1.0 / x)) + ((0.2514179000665374 / (x ^ 3.0)) + ((0.15298196345929074 / (x ^ 5.0)) + (11.259630434457211 * (1.0 / (x ^ 7.0)))));
	end
	tmp_2 = 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[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[x, -2e+20], N[(0.5 / x), $MachinePrecision], If[LessEqual[x, 200.0], N[(x * N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[(N[(x * x), $MachinePrecision] * 0.0072644182), $MachinePrecision] + 0.0424060604), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(N[(N[(x * x), $MachinePrecision] * 0.0001789971), $MachinePrecision] + 0.0005064034), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.2909738639), $MachinePrecision] + 0.7715471019), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(N[(N[(x * x), $MachinePrecision] * 0.0140005442), $MachinePrecision] + 0.0694555761), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.0003579942), $MachinePrecision] + 0.0008327945), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.2514179000665374 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.15298196345929074 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(11.259630434457211 * N[(1.0 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e20

   1. Initial program 63.0

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

   2. Simplified 63.0

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

      [Start] 63.0

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

      rational.json-simplify-2 [=>] 63.0

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

   3. Taylor expanded in x around inf 0

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

   if -2e20 < x < 200

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0

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

      rational.json-simplify-2 [=>] 0.0

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

   if 200 < x

   1. Initial program 59.4

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

   2. Simplified 59.4

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

      [Start] 59.4

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

      rational.json-simplify-2 [=>] 59.4

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

   3. Taylor expanded in x around inf 0.0

      \[\leadsto \color{blue}{0.2514179000665374 \cdot \frac{1}{{x}^{3}} + \left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + \left(11.259630434457211 \cdot \frac{1}{{x}^{7}} + 0.5 \cdot \frac{1}{x}\right)\right)} \]

   4. Simplified 0.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{1}{x} + \left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + \left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + 11.259630434457211 \cdot \frac{1}{{x}^{7}}\right)\right)} \]
      Proof

      [Start] 0.0	\[ 0.2514179000665374 \cdot \frac{1}{{x}^{3}} + \left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + \left(11.259630434457211 \cdot \frac{1}{{x}^{7}} + 0.5 \cdot \frac{1}{x}\right)\right) \]
      rational.json-simplify-41 [<=] 0.0	\[ 0.2514179000665374 \cdot \frac{1}{{x}^{3}} + \color{blue}{\left(0.5 \cdot \frac{1}{x} + \left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + 11.259630434457211 \cdot \frac{1}{{x}^{7}}\right)\right)} \]
      rational.json-simplify-1 [=>] 0.0	\[ 0.2514179000665374 \cdot \frac{1}{{x}^{3}} + \color{blue}{\left(\left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + 11.259630434457211 \cdot \frac{1}{{x}^{7}}\right) + 0.5 \cdot \frac{1}{x}\right)} \]
      rational.json-simplify-41 [<=] 0.0	\[ \color{blue}{0.5 \cdot \frac{1}{x} + \left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + \left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + 11.259630434457211 \cdot \frac{1}{{x}^{7}}\right)\right)} \]

   5. Taylor expanded in x around 0 0.0

      \[\leadsto 0.5 \cdot \frac{1}{x} + \left(\color{blue}{\frac{0.2514179000665374}{{x}^{3}}} + \left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + 11.259630434457211 \cdot \frac{1}{{x}^{7}}\right)\right) \]

   6. Taylor expanded in x around 0 0.0

      \[\leadsto 0.5 \cdot \frac{1}{x} + \left(\frac{0.2514179000665374}{{x}^{3}} + \left(\color{blue}{\frac{0.15298196345929074}{{x}^{5}}} + 11.259630434457211 \cdot \frac{1}{{x}^{7}}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;x \cdot \frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0072644182 + 0.0424060604\right)\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0001789971 + 0.0005064034\right)}{\left(\left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.2909738639 + 0.7715471019\right)\right) + \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0140005442 + 0.0694555761\right)\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0003579942 + 0.0008327945\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{x} + \left(\frac{0.2514179000665374}{{x}^{3}} + \left(\frac{0.15298196345929074}{{x}^{5}} + 11.259630434457211 \cdot \frac{1}{{x}^{7}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	7368

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := x \cdot \left(x \cdot t_0\right)\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ \mathbf{if}\;x \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;x \cdot \frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + t_0 \cdot \left(\left(x \cdot x\right) \cdot 0.0072644182 + 0.0424060604\right)\right) + t_2 \cdot \left(\left(x \cdot x\right) \cdot 0.0001789971 + 0.0005064034\right)}{\left(\left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.2909738639 + 0.7715471019\right)\right) + t_1 \cdot \left(\left(x \cdot x\right) \cdot 0.0140005442 + 0.0694555761\right)\right) + \left(\left(x \cdot x\right) \cdot t_2\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0003579942 + 0.0008327945\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 2
Error	0.0
Cost	4808

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;x \cdot \frac{1 + x \cdot \left(x \cdot \left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + x \cdot \left(x \cdot \left(0.0072644182 + \left(x \cdot x\right) \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)\right)\right)\right)}{1 + x \cdot \left(x \cdot \left(0.7715471019 + \left(x \cdot x\right) \cdot \left(0.2909738639 + \left(x \cdot x\right) \cdot \left(0.0694555761 + \left(x \cdot x\right) \cdot \left(0.0140005442 + x \cdot \left(x \cdot \left(0.0008327945 + x \cdot \left(x \cdot 0.0003579942\right)\right)\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 3
Error	0.0
Cost	4808

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;x \cdot \frac{1 + x \cdot \left(x \cdot \left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + x \cdot \left(x \cdot \left(0.0072644182 + \left(x \cdot x\right) \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(0.7715471019 + x \cdot \left(x \cdot \left(0.2909738639 + \left(x \cdot x\right) \cdot \left(0.0694555761 + x \cdot \left(x \cdot \left(0.0140005442 + x \cdot \left(x \cdot \left(0.0008327945 + x \cdot \left(x \cdot 0.0003579942\right)\right)\right)\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	4424

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 2.3:\\ \;\;\;\;x \cdot \frac{1 + x \cdot \left(x \cdot \left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + x \cdot \left(x \cdot \left(0.0072644182 + \left(x \cdot x\right) \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)\right)\right)\right)}{1 + x \cdot \left(x \cdot \left(0.7715471019 + \left(x \cdot x\right) \cdot \left(0.2909738639 + \left(x \cdot x\right) \cdot \left(0.0694555761 + \left(x \cdot x\right) \cdot \left(0.0140005442 + x \cdot \left(x \cdot 0.0008327945\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 5
Error	0.3
Cost	3656

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 2.4:\\ \;\;\;\;x \cdot \frac{1 + \left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + \left(0.0072644182 + x \cdot \left(x \cdot 0.0005064034\right)\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)}{1 + \left(0.7715471019 + x \cdot \left(x \cdot \left(0.2909738639 + \left(x \cdot x\right) \cdot \left(0.0694555761 + 0.0140005442 \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 6
Error	0.4
Cost	3272

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;x \cdot \frac{1 + x \cdot \left(x \cdot \left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + x \cdot \left(x \cdot \left(0.0072644182 + \left(x \cdot x\right) \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(0.7715471019 + x \cdot \left(x \cdot 0.2909738639\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 7
Error	0.4
Cost	1736

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 3.8:\\ \;\;\;\;x \cdot \frac{1 + x \cdot \left(x \cdot 0.1049934947\right)}{1 + \left(x \cdot x\right) \cdot \left(0.7715471019 + x \cdot \left(x \cdot 0.2909738639\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 8
Error	0.5
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 9
Error	31.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023077 
(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))