Jmat.Real.dawson

Average Accuracy: 54.1% → 100.0%

Time: 8.8s

Precision: binary64

Cost: 77193

Math

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

↓

\[\begin{array}{l} t_0 := {\left(x \cdot x\right)}^{2}\\ t_1 := {\left(x \cdot x\right)}^{3}\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ \mathbf{if}\;x \leq -13200000000 \lor \neg \left(x \leq 47000000\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot t_0 + 0.0072644182 \cdot t_1\right)\right) + \left(0.0005064034 \cdot t_2 + 0.0001789971 \cdot {x}^{10}\right)\right) \cdot \frac{1}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + t_0 \cdot 0.2909738639\right)\right) + \left(t_1 \cdot 0.0694555761 + t_2 \cdot 0.0140005442\right)\right) + \left(\left(t_0 \cdot t_1\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(t_0 \cdot t_2\right)\right)}\right)\\ \end{array} \]

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 (pow (* x x) 2.0)) (t_1 (pow (* x x) 3.0)) (t_2 (* (* x x) t_1)))
   (if (or (<= x -13200000000.0) (not (<= x 47000000.0)))
     (/ 0.5 x)
     (*
      x
      (*
       (+
        (+
         (+ 1.0 (* 0.1049934947 (* x x)))
         (+ (* 0.0424060604 t_0) (* 0.0072644182 t_1)))
        (+ (* 0.0005064034 t_2) (* 0.0001789971 (pow x 10.0))))
       (/
        1.0
        (+
         (+
          (+ 1.0 (+ (* (* x x) 0.7715471019) (* t_0 0.2909738639)))
          (+ (* t_1 0.0694555761) (* t_2 0.0140005442)))
         (+ (* (* t_0 t_1) 0.0008327945) (* 0.0003579942 (* t_0 t_2))))))))))

C

double code(double x) {
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * ((x * x) * (x * x)))) + (0.0072644182 * (((x * x) * (x * x)) * (x * x)))) + (0.0005064034 * ((((x * x) * (x * x)) * (x * x)) * (x * x)))) + (0.0001789971 * (((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)))) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * ((x * x) * (x * x)))) + (0.0694555761 * (((x * x) * (x * x)) * (x * x)))) + (0.0140005442 * ((((x * x) * (x * x)) * (x * x)) * (x * x)))) + (0.0008327945 * (((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)))) + ((2.0 * 0.0001789971) * ((((((x * x) * (x * x)) * (x * x)) * (x * x)) * (x * x)) * (x * x))))) * x;
}

↓

double code(double x) {
	double t_0 = pow((x * x), 2.0);
	double t_1 = pow((x * x), 3.0);
	double t_2 = (x * x) * t_1;
	double tmp;
	if ((x <= -13200000000.0) || !(x <= 47000000.0)) {
		tmp = 0.5 / x;
	} else {
		tmp = x * ((((1.0 + (0.1049934947 * (x * x))) + ((0.0424060604 * t_0) + (0.0072644182 * t_1))) + ((0.0005064034 * t_2) + (0.0001789971 * pow(x, 10.0)))) * (1.0 / (((1.0 + (((x * x) * 0.7715471019) + (t_0 * 0.2909738639))) + ((t_1 * 0.0694555761) + (t_2 * 0.0140005442))) + (((t_0 * t_1) * 0.0008327945) + (0.0003579942 * (t_0 * t_2))))));
	}
	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) ** 2.0d0
    t_1 = (x * x) ** 3.0d0
    t_2 = (x * x) * t_1
    if ((x <= (-13200000000.0d0)) .or. (.not. (x <= 47000000.0d0))) then
        tmp = 0.5d0 / x
    else
        tmp = x * ((((1.0d0 + (0.1049934947d0 * (x * x))) + ((0.0424060604d0 * t_0) + (0.0072644182d0 * t_1))) + ((0.0005064034d0 * t_2) + (0.0001789971d0 * (x ** 10.0d0)))) * (1.0d0 / (((1.0d0 + (((x * x) * 0.7715471019d0) + (t_0 * 0.2909738639d0))) + ((t_1 * 0.0694555761d0) + (t_2 * 0.0140005442d0))) + (((t_0 * t_1) * 0.0008327945d0) + (0.0003579942d0 * (t_0 * t_2))))))
    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 = Math.pow((x * x), 2.0);
	double t_1 = Math.pow((x * x), 3.0);
	double t_2 = (x * x) * t_1;
	double tmp;
	if ((x <= -13200000000.0) || !(x <= 47000000.0)) {
		tmp = 0.5 / x;
	} else {
		tmp = x * ((((1.0 + (0.1049934947 * (x * x))) + ((0.0424060604 * t_0) + (0.0072644182 * t_1))) + ((0.0005064034 * t_2) + (0.0001789971 * Math.pow(x, 10.0)))) * (1.0 / (((1.0 + (((x * x) * 0.7715471019) + (t_0 * 0.2909738639))) + ((t_1 * 0.0694555761) + (t_2 * 0.0140005442))) + (((t_0 * t_1) * 0.0008327945) + (0.0003579942 * (t_0 * t_2))))));
	}
	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 = math.pow((x * x), 2.0)
	t_1 = math.pow((x * x), 3.0)
	t_2 = (x * x) * t_1
	tmp = 0
	if (x <= -13200000000.0) or not (x <= 47000000.0):
		tmp = 0.5 / x
	else:
		tmp = x * ((((1.0 + (0.1049934947 * (x * x))) + ((0.0424060604 * t_0) + (0.0072644182 * t_1))) + ((0.0005064034 * t_2) + (0.0001789971 * math.pow(x, 10.0)))) * (1.0 / (((1.0 + (((x * x) * 0.7715471019) + (t_0 * 0.2909738639))) + ((t_1 * 0.0694555761) + (t_2 * 0.0140005442))) + (((t_0 * t_1) * 0.0008327945) + (0.0003579942 * (t_0 * t_2))))))
	return tmp

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(0.0424060604 * Float64(Float64(x * x) * Float64(x * x)))) + Float64(0.0072644182 * Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0005064034 * Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0001789971 * Float64(Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) / Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.7715471019 * Float64(x * x))) + Float64(0.2909738639 * Float64(Float64(x * x) * Float64(x * x)))) + Float64(0.0694555761 * Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0140005442 * Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) + Float64(0.0008327945 * Float64(Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)))) + Float64(Float64(2.0 * 0.0001789971) * Float64(Float64(Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x)) * Float64(x * x))))) * x)
end

↓

function code(x)
	t_0 = Float64(x * x) ^ 2.0
	t_1 = Float64(x * x) ^ 3.0
	t_2 = Float64(Float64(x * x) * t_1)
	tmp = 0.0
	if ((x <= -13200000000.0) || !(x <= 47000000.0))
		tmp = Float64(0.5 / x);
	else
		tmp = Float64(x * Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(Float64(0.0424060604 * t_0) + Float64(0.0072644182 * t_1))) + Float64(Float64(0.0005064034 * t_2) + Float64(0.0001789971 * (x ^ 10.0)))) * Float64(1.0 / Float64(Float64(Float64(1.0 + Float64(Float64(Float64(x * x) * 0.7715471019) + Float64(t_0 * 0.2909738639))) + Float64(Float64(t_1 * 0.0694555761) + Float64(t_2 * 0.0140005442))) + Float64(Float64(Float64(t_0 * t_1) * 0.0008327945) + Float64(0.0003579942 * Float64(t_0 * t_2)))))));
	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) ^ 2.0;
	t_1 = (x * x) ^ 3.0;
	t_2 = (x * x) * t_1;
	tmp = 0.0;
	if ((x <= -13200000000.0) || ~((x <= 47000000.0)))
		tmp = 0.5 / x;
	else
		tmp = x * ((((1.0 + (0.1049934947 * (x * x))) + ((0.0424060604 * t_0) + (0.0072644182 * t_1))) + ((0.0005064034 * t_2) + (0.0001789971 * (x ^ 10.0)))) * (1.0 / (((1.0 + (((x * x) * 0.7715471019) + (t_0 * 0.2909738639))) + ((t_1 * 0.0694555761) + (t_2 * 0.0140005442))) + (((t_0 * t_1) * 0.0008327945) + (0.0003579942 * (t_0 * t_2))))));
	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[Power[N[(x * x), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x * x), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[Or[LessEqual[x, -13200000000.0], N[Not[LessEqual[x, 47000000.0]], $MachinePrecision]], N[(0.5 / x), $MachinePrecision], N[(x * N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0424060604 * t$95$0), $MachinePrecision] + N[(0.0072644182 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0005064034 * t$95$2), $MachinePrecision] + N[(0.0001789971 * N[Power[x, 10.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * 0.7715471019), $MachinePrecision] + N[(t$95$0 * 0.2909738639), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * 0.0694555761), $MachinePrecision] + N[(t$95$2 * 0.0140005442), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * 0.0008327945), $MachinePrecision] + N[(0.0003579942 * N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.32e10 or 4.7e7 < x

   1. Initial program 5.1%

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

   2. Simplified 5.1%

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

      [Start] 5.1

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

      *-commutative [=>] 5.1

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

   3. Taylor expanded in x around inf 100.0%

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

   if -1.32e10 < x < 4.7e7

   1. Initial program 99.9%

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

   2. Applied egg-rr 99.9%

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

      [Start] 99.9

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

      div-inv [=>] 99.9

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

   3. Taylor expanded in x around 0 99.9%

      \[\leadsto \left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot {\left(x \cdot x\right)}^{2} + 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) + \color{blue}{0.0001789971 \cdot {x}^{10}}\right)\right) \cdot \frac{1}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left({\left(x \cdot x\right)}^{3} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot {\left(x \cdot x\right)}^{2}\right)\right)}\right) \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -13200000000 \lor \neg \left(x \leq 47000000\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot {\left(x \cdot x\right)}^{2} + 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) + 0.0001789971 \cdot {x}^{10}\right)\right) \cdot \frac{1}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left({\left(x \cdot x\right)}^{3} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{2} \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left({\left(x \cdot x\right)}^{2} \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right)\right)\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	11209

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := t_0 \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ \mathbf{if}\;x \leq -2000000000 \lor \neg \left(x \leq 48000000\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot t_0\right)\right) + \left(0.0005064034 \cdot t_1 + 0.0001789971 \cdot t_2\right)}{\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + \left(0.2909738639 \cdot t_0 + \left(x \cdot x\right) \cdot \left(0.0694555761 \cdot t_0\right)\right)\right) + 0.0140005442 \cdot t_1\right) + \left(0.0008327945 \cdot t_2 + 0.0003579942 \cdot \left(t_0 \cdot t_1\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	11209

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ t_3 := \left(x \cdot x\right) \cdot t_2\\ \mathbf{if}\;x \leq -2000000000 \lor \neg \left(x \leq 48000000\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot t_1\right) + 0.0005064034 \cdot t_2\right) + 0.0001789971 \cdot t_3}{\left(\left(\left(\left(0.2909738639 \cdot t_0 + \left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right)\right) + 0.0694555761 \cdot t_1\right) + 0.0140005442 \cdot t_2\right) + 0.0008327945 \cdot t_3\right) + 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot t_3\right)}\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -0.95 \lor \neg \left(x \leq 0.95\right):\\ \;\;\;\;\frac{0.5}{x} + \frac{\frac{0.2514179000665374}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.6665536072\right)\\ \end{array} \]

Alternative 4
Accuracy	99.2%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -0.78:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.78:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 5
Accuracy	98.9%
Cost	456

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

Alternative 6
Accuracy	51.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023133 
(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))