Jmat.Real.dawson

Average Error: 29.3 → 0.0

Time: 13.7s

Precision: binary64

Cost: 14276

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := t_0 \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ \mathbf{if}\;x \leq -200:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{x}}{x \cdot x} + \left(\frac{0.5}{x} + \left(\frac{0.15298196345929074}{{x}^{5}} + \frac{11.259630434457211}{{x}^{7}}\right)\right)\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot t_0\right)\right) + \left(0.0005064034 \cdot t_1 + 0.0001789971 \cdot t_2\right)}{\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + \left(t_0 \cdot 0.2909738639 + \left(x \cdot x\right) \cdot \left(t_0 \cdot 0.0694555761\right)\right)\right) + t_1 \cdot 0.0140005442\right) + \left(t_2 \cdot 0.0008327945 + 0.0003579942 \cdot \left(t_0 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{x}}\\ \end{array} \]

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 (* x (* x (* x x)))) (t_1 (* t_0 t_0)) (t_2 (* (* x x) t_1)))
   (if (<= x -200.0)
     (+
      (/ (/ 0.2514179000665374 x) (* x x))
      (+
       (/ 0.5 x)
       (+
        (/ 0.15298196345929074 (pow x 5.0))
        (/ 11.259630434457211 (pow x 7.0)))))
     (if (<= x 500.0)
       (*
        x
        (/
         (+
          (+
           (+ (+ 1.0 (* (* x x) 0.1049934947)) (* 0.0424060604 t_0))
           (* 0.0072644182 (* (* x x) t_0)))
          (+ (* 0.0005064034 t_1) (* 0.0001789971 t_2)))
         (+
          (+
           (+
            (+ 1.0 (* (* x x) 0.7715471019))
            (+ (* t_0 0.2909738639) (* (* x x) (* t_0 0.0694555761))))
           (* t_1 0.0140005442))
          (+ (* t_2 0.0008327945) (* 0.0003579942 (* t_0 t_1))))))
       (/ 1.0 (+ (* x 2.0) (/ -1.0056716002661497 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 = x * (x * (x * x));
	double t_1 = t_0 * t_0;
	double t_2 = (x * x) * t_1;
	double tmp;
	if (x <= -200.0) {
		tmp = ((0.2514179000665374 / x) / (x * x)) + ((0.5 / x) + ((0.15298196345929074 / pow(x, 5.0)) + (11.259630434457211 / pow(x, 7.0))));
	} else if (x <= 500.0) {
		tmp = x * (((((1.0 + ((x * x) * 0.1049934947)) + (0.0424060604 * t_0)) + (0.0072644182 * ((x * x) * t_0))) + ((0.0005064034 * t_1) + (0.0001789971 * t_2))) / ((((1.0 + ((x * x) * 0.7715471019)) + ((t_0 * 0.2909738639) + ((x * x) * (t_0 * 0.0694555761)))) + (t_1 * 0.0140005442)) + ((t_2 * 0.0008327945) + (0.0003579942 * (t_0 * t_1)))));
	} else {
		tmp = 1.0 / ((x * 2.0) + (-1.0056716002661497 / x));
	}
	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 = t_0 * t_0
    t_2 = (x * x) * t_1
    if (x <= (-200.0d0)) then
        tmp = ((0.2514179000665374d0 / x) / (x * x)) + ((0.5d0 / x) + ((0.15298196345929074d0 / (x ** 5.0d0)) + (11.259630434457211d0 / (x ** 7.0d0))))
    else if (x <= 500.0d0) then
        tmp = x * (((((1.0d0 + ((x * x) * 0.1049934947d0)) + (0.0424060604d0 * t_0)) + (0.0072644182d0 * ((x * x) * t_0))) + ((0.0005064034d0 * t_1) + (0.0001789971d0 * t_2))) / ((((1.0d0 + ((x * x) * 0.7715471019d0)) + ((t_0 * 0.2909738639d0) + ((x * x) * (t_0 * 0.0694555761d0)))) + (t_1 * 0.0140005442d0)) + ((t_2 * 0.0008327945d0) + (0.0003579942d0 * (t_0 * t_1)))))
    else
        tmp = 1.0d0 / ((x * 2.0d0) + ((-1.0056716002661497d0) / x))
    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 = t_0 * t_0;
	double t_2 = (x * x) * t_1;
	double tmp;
	if (x <= -200.0) {
		tmp = ((0.2514179000665374 / x) / (x * x)) + ((0.5 / x) + ((0.15298196345929074 / Math.pow(x, 5.0)) + (11.259630434457211 / Math.pow(x, 7.0))));
	} else if (x <= 500.0) {
		tmp = x * (((((1.0 + ((x * x) * 0.1049934947)) + (0.0424060604 * t_0)) + (0.0072644182 * ((x * x) * t_0))) + ((0.0005064034 * t_1) + (0.0001789971 * t_2))) / ((((1.0 + ((x * x) * 0.7715471019)) + ((t_0 * 0.2909738639) + ((x * x) * (t_0 * 0.0694555761)))) + (t_1 * 0.0140005442)) + ((t_2 * 0.0008327945) + (0.0003579942 * (t_0 * t_1)))));
	} else {
		tmp = 1.0 / ((x * 2.0) + (-1.0056716002661497 / x));
	}
	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 = t_0 * t_0
	t_2 = (x * x) * t_1
	tmp = 0
	if x <= -200.0:
		tmp = ((0.2514179000665374 / x) / (x * x)) + ((0.5 / x) + ((0.15298196345929074 / math.pow(x, 5.0)) + (11.259630434457211 / math.pow(x, 7.0))))
	elif x <= 500.0:
		tmp = x * (((((1.0 + ((x * x) * 0.1049934947)) + (0.0424060604 * t_0)) + (0.0072644182 * ((x * x) * t_0))) + ((0.0005064034 * t_1) + (0.0001789971 * t_2))) / ((((1.0 + ((x * x) * 0.7715471019)) + ((t_0 * 0.2909738639) + ((x * x) * (t_0 * 0.0694555761)))) + (t_1 * 0.0140005442)) + ((t_2 * 0.0008327945) + (0.0003579942 * (t_0 * t_1)))))
	else:
		tmp = 1.0 / ((x * 2.0) + (-1.0056716002661497 / 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 * Float64(x * Float64(x * x)))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(Float64(x * x) * t_1)
	tmp = 0.0
	if (x <= -200.0)
		tmp = Float64(Float64(Float64(0.2514179000665374 / x) / Float64(x * x)) + Float64(Float64(0.5 / x) + Float64(Float64(0.15298196345929074 / (x ^ 5.0)) + Float64(11.259630434457211 / (x ^ 7.0)))));
	elseif (x <= 500.0)
		tmp = Float64(x * Float64(Float64(Float64(Float64(Float64(1.0 + Float64(Float64(x * x) * 0.1049934947)) + Float64(0.0424060604 * t_0)) + Float64(0.0072644182 * Float64(Float64(x * x) * t_0))) + Float64(Float64(0.0005064034 * t_1) + Float64(0.0001789971 * t_2))) / Float64(Float64(Float64(Float64(1.0 + Float64(Float64(x * x) * 0.7715471019)) + Float64(Float64(t_0 * 0.2909738639) + Float64(Float64(x * x) * Float64(t_0 * 0.0694555761)))) + Float64(t_1 * 0.0140005442)) + Float64(Float64(t_2 * 0.0008327945) + Float64(0.0003579942 * Float64(t_0 * t_1))))));
	else
		tmp = Float64(1.0 / Float64(Float64(x * 2.0) + Float64(-1.0056716002661497 / x)));
	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 = t_0 * t_0;
	t_2 = (x * x) * t_1;
	tmp = 0.0;
	if (x <= -200.0)
		tmp = ((0.2514179000665374 / x) / (x * x)) + ((0.5 / x) + ((0.15298196345929074 / (x ^ 5.0)) + (11.259630434457211 / (x ^ 7.0))));
	elseif (x <= 500.0)
		tmp = x * (((((1.0 + ((x * x) * 0.1049934947)) + (0.0424060604 * t_0)) + (0.0072644182 * ((x * x) * t_0))) + ((0.0005064034 * t_1) + (0.0001789971 * t_2))) / ((((1.0 + ((x * x) * 0.7715471019)) + ((t_0 * 0.2909738639) + ((x * x) * (t_0 * 0.0694555761)))) + (t_1 * 0.0140005442)) + ((t_2 * 0.0008327945) + (0.0003579942 * (t_0 * t_1)))));
	else
		tmp = 1.0 / ((x * 2.0) + (-1.0056716002661497 / x));
	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[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[x, -200.0], N[(N[(N[(0.2514179000665374 / x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / x), $MachinePrecision] + N[(N[(0.15298196345929074 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(11.259630434457211 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 500.0], N[(x * N[(N[(N[(N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.1049934947), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0005064034 * t$95$1), $MachinePrecision] + N[(0.0001789971 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.7715471019), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * 0.2909738639), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * 0.0694555761), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * 0.0140005442), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 * 0.0008327945), $MachinePrecision] + N[(0.0003579942 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * 2.0), $MachinePrecision] + N[(-1.0056716002661497 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -200

   1. Initial program 58.1

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

   2. Simplified 58.2

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

      [Start] 58.1

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

      associate-*l/ [=>] 58.2

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

   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

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

      [Start] 0.0	\[ 0.2514179000665374 \cdot \frac{1}{{x}^{3}} + \left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + \left(11.259630434457211 \cdot \frac{1}{{x}^{7}} + 0.5 \cdot \frac{1}{x}\right)\right) \]
      associate-*r/ [=>] 0.0	\[ \color{blue}{\frac{0.2514179000665374 \cdot 1}{{x}^{3}}} + \left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + \left(11.259630434457211 \cdot \frac{1}{{x}^{7}} + 0.5 \cdot \frac{1}{x}\right)\right) \]
      metadata-eval [=>] 0.0	\[ \frac{\color{blue}{0.2514179000665374}}{{x}^{3}} + \left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + \left(11.259630434457211 \cdot \frac{1}{{x}^{7}} + 0.5 \cdot \frac{1}{x}\right)\right) \]
      associate-+r+ [=>] 0.0	\[ \frac{0.2514179000665374}{{x}^{3}} + \color{blue}{\left(\left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + 11.259630434457211 \cdot \frac{1}{{x}^{7}}\right) + 0.5 \cdot \frac{1}{x}\right)} \]
      +-commutative [=>] 0.0	\[ \frac{0.2514179000665374}{{x}^{3}} + \color{blue}{\left(0.5 \cdot \frac{1}{x} + \left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + 11.259630434457211 \cdot \frac{1}{{x}^{7}}\right)\right)} \]
      associate-*r/ [=>] 0.0	\[ \frac{0.2514179000665374}{{x}^{3}} + \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + \left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + 11.259630434457211 \cdot \frac{1}{{x}^{7}}\right)\right) \]
      metadata-eval [=>] 0.0	\[ \frac{0.2514179000665374}{{x}^{3}} + \left(\frac{\color{blue}{0.5}}{x} + \left(0.15298196345929074 \cdot \frac{1}{{x}^{5}} + 11.259630434457211 \cdot \frac{1}{{x}^{7}}\right)\right) \]
      associate-*r/ [=>] 0.0	\[ \frac{0.2514179000665374}{{x}^{3}} + \left(\frac{0.5}{x} + \left(\color{blue}{\frac{0.15298196345929074 \cdot 1}{{x}^{5}}} + 11.259630434457211 \cdot \frac{1}{{x}^{7}}\right)\right) \]
      metadata-eval [=>] 0.0	\[ \frac{0.2514179000665374}{{x}^{3}} + \left(\frac{0.5}{x} + \left(\frac{\color{blue}{0.15298196345929074}}{{x}^{5}} + 11.259630434457211 \cdot \frac{1}{{x}^{7}}\right)\right) \]
      associate-*r/ [=>] 0.0	\[ \frac{0.2514179000665374}{{x}^{3}} + \left(\frac{0.5}{x} + \left(\frac{0.15298196345929074}{{x}^{5}} + \color{blue}{\frac{11.259630434457211 \cdot 1}{{x}^{7}}}\right)\right) \]
      metadata-eval [=>] 0.0	\[ \frac{0.2514179000665374}{{x}^{3}} + \left(\frac{0.5}{x} + \left(\frac{0.15298196345929074}{{x}^{5}} + \frac{\color{blue}{11.259630434457211}}{{x}^{7}}\right)\right) \]

   5. Applied egg-rr 0.0

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

   6. Applied egg-rr 0.0

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

   if -200 < x < 500

   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

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

      [Start] 0.0

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

      *-commutative [=>] 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 500 < 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

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

      [Start] 59.4

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

      *-commutative [=>] 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

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

   4. Simplified 0.0

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

      [Start] 0.0	\[ 0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x} \]
      associate-*r/ [=>] 0.0	\[ \color{blue}{\frac{0.2514179000665374 \cdot 1}{{x}^{3}}} + 0.5 \cdot \frac{1}{x} \]
      metadata-eval [=>] 0.0	\[ \frac{\color{blue}{0.2514179000665374}}{{x}^{3}} + 0.5 \cdot \frac{1}{x} \]
      associate-*r/ [=>] 0.0	\[ \frac{0.2514179000665374}{{x}^{3}} + \color{blue}{\frac{0.5 \cdot 1}{x}} \]
      metadata-eval [=>] 0.0	\[ \frac{0.2514179000665374}{{x}^{3}} + \frac{\color{blue}{0.5}}{x} \]

   5. Applied egg-rr 0.0

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

   6. Applied egg-rr 0.1

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

   7. Taylor expanded in x around inf 0.1

      \[\leadsto \frac{1}{\color{blue}{2 \cdot x - 1.0056716002661497 \cdot \frac{1}{x}}} \]

   8. Simplified 0.1

      \[\leadsto \frac{1}{\color{blue}{x \cdot 2 - \frac{1.0056716002661497}{x}}} \]
      Proof

      [Start] 0.1	\[ \frac{1}{2 \cdot x - 1.0056716002661497 \cdot \frac{1}{x}} \]
      *-commutative [<=] 0.1	\[ \frac{1}{\color{blue}{x \cdot 2} - 1.0056716002661497 \cdot \frac{1}{x}} \]
      associate-*r/ [=>] 0.1	\[ \frac{1}{x \cdot 2 - \color{blue}{\frac{1.0056716002661497 \cdot 1}{x}}} \]
      metadata-eval [=>] 0.1	\[ \frac{1}{x \cdot 2 - \frac{\color{blue}{1.0056716002661497}}{x}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{x}}{x \cdot x} + \left(\frac{0.5}{x} + \left(\frac{0.15298196345929074}{{x}^{5}} + \frac{11.259630434457211}{{x}^{7}}\right)\right)\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\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(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0694555761\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(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\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)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	11209

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := t_0 \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ \mathbf{if}\;x \leq -5000 \lor \neg \left(x \leq 500\right):\\ \;\;\;\;\frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot t_0\right)\right) + \left(0.0005064034 \cdot t_1 + 0.0001789971 \cdot t_2\right)}{\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + \left(t_0 \cdot 0.2909738639 + \left(x \cdot x\right) \cdot \left(t_0 \cdot 0.0694555761\right)\right)\right) + t_1 \cdot 0.0140005442\right) + \left(t_2 \cdot 0.0008327945 + 0.0003579942 \cdot \left(t_0 \cdot t_1\right)\right)}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	1225

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.265709700396151 + -0.6665536072\right)\right)\\ \end{array} \]

Alternative 3
Error	0.3
Cost	1225

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.265709700396151 + -0.6665536072\right)\right)\\ \end{array} \]

Alternative 4
Error	0.5
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	0.4
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(\left(x \cdot x\right) \cdot -0.6665536072\right)\\ \end{array} \]

Alternative 6
Error	0.5
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 7
Error	0.7
Cost	456

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

Alternative 8
Error	31.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023039 
(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))