Jmat.Real.dawson

Average Error: 29.3 → 0.4

Time: 3.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := {\left(x \cdot x\right)}^{3}\\ t_1 := {\left(x \cdot x\right)}^{2}\\ t_2 := t_1 \cdot t_0\\ t_3 := \frac{0.2514179000665374}{{x}^{3}} + \frac{0.5}{x}\\ t_4 := \left(x \cdot x\right) \cdot t_0\\ \mathbf{if}\;x \leq -3.1556112308066137 \cdot 10^{+32}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 0.516789691633757:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + 0.0424060604 \cdot t_1\right)\right) + 0.0072644182 \cdot t_0\right) + 0.0005064034 \cdot t_4\right) + 0.0001789971 \cdot t_2}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + t_1 \cdot 0.2909738639\right)\right) + \left(t_0 \cdot 0.0694555761 + t_4 \cdot 0.0140005442\right)\right) + \left(t_2 \cdot 0.0008327945 + 0.0003579942 \cdot \left(t_1 \cdot t_4\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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) 3.0))
        (t_1 (pow (* x x) 2.0))
        (t_2 (* t_1 t_0))
        (t_3 (+ (/ 0.2514179000665374 (pow x 3.0)) (/ 0.5 x)))
        (t_4 (* (* x x) t_0)))
   (if (<= x -3.1556112308066137e+32)
     t_3
     (if (<= x 0.516789691633757)
       (log1p
        (expm1
         (*
          x
          (/
           (+
            (+
             (+
              (+ 1.0 (+ (* 0.1049934947 (* x x)) (* 0.0424060604 t_1)))
              (* 0.0072644182 t_0))
             (* 0.0005064034 t_4))
            (* 0.0001789971 t_2))
           (+
            (+
             (+ 1.0 (+ (* (* x x) 0.7715471019) (* t_1 0.2909738639)))
             (+ (* t_0 0.0694555761) (* t_4 0.0140005442)))
            (+ (* t_2 0.0008327945) (* 0.0003579942 (* t_1 t_4))))))))
       t_3))))

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), 3.0);
	double t_1 = pow((x * x), 2.0);
	double t_2 = t_1 * t_0;
	double t_3 = (0.2514179000665374 / pow(x, 3.0)) + (0.5 / x);
	double t_4 = (x * x) * t_0;
	double tmp;
	if (x <= -3.1556112308066137e+32) {
		tmp = t_3;
	} else if (x <= 0.516789691633757) {
		tmp = log1p(expm1((x * (((((1.0 + ((0.1049934947 * (x * x)) + (0.0424060604 * t_1))) + (0.0072644182 * t_0)) + (0.0005064034 * t_4)) + (0.0001789971 * t_2)) / (((1.0 + (((x * x) * 0.7715471019) + (t_1 * 0.2909738639))) + ((t_0 * 0.0694555761) + (t_4 * 0.0140005442))) + ((t_2 * 0.0008327945) + (0.0003579942 * (t_1 * t_4))))))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

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), 3.0);
	double t_1 = Math.pow((x * x), 2.0);
	double t_2 = t_1 * t_0;
	double t_3 = (0.2514179000665374 / Math.pow(x, 3.0)) + (0.5 / x);
	double t_4 = (x * x) * t_0;
	double tmp;
	if (x <= -3.1556112308066137e+32) {
		tmp = t_3;
	} else if (x <= 0.516789691633757) {
		tmp = Math.log1p(Math.expm1((x * (((((1.0 + ((0.1049934947 * (x * x)) + (0.0424060604 * t_1))) + (0.0072644182 * t_0)) + (0.0005064034 * t_4)) + (0.0001789971 * t_2)) / (((1.0 + (((x * x) * 0.7715471019) + (t_1 * 0.2909738639))) + ((t_0 * 0.0694555761) + (t_4 * 0.0140005442))) + ((t_2 * 0.0008327945) + (0.0003579942 * (t_1 * t_4))))))));
	} else {
		tmp = t_3;
	}
	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), 3.0)
	t_1 = math.pow((x * x), 2.0)
	t_2 = t_1 * t_0
	t_3 = (0.2514179000665374 / math.pow(x, 3.0)) + (0.5 / x)
	t_4 = (x * x) * t_0
	tmp = 0
	if x <= -3.1556112308066137e+32:
		tmp = t_3
	elif x <= 0.516789691633757:
		tmp = math.log1p(math.expm1((x * (((((1.0 + ((0.1049934947 * (x * x)) + (0.0424060604 * t_1))) + (0.0072644182 * t_0)) + (0.0005064034 * t_4)) + (0.0001789971 * t_2)) / (((1.0 + (((x * x) * 0.7715471019) + (t_1 * 0.2909738639))) + ((t_0 * 0.0694555761) + (t_4 * 0.0140005442))) + ((t_2 * 0.0008327945) + (0.0003579942 * (t_1 * t_4))))))))
	else:
		tmp = t_3
	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) ^ 3.0
	t_1 = Float64(x * x) ^ 2.0
	t_2 = Float64(t_1 * t_0)
	t_3 = Float64(Float64(0.2514179000665374 / (x ^ 3.0)) + Float64(0.5 / x))
	t_4 = Float64(Float64(x * x) * t_0)
	tmp = 0.0
	if (x <= -3.1556112308066137e+32)
		tmp = t_3;
	elseif (x <= 0.516789691633757)
		tmp = log1p(expm1(Float64(x * Float64(Float64(Float64(Float64(Float64(1.0 + Float64(Float64(0.1049934947 * Float64(x * x)) + Float64(0.0424060604 * t_1))) + Float64(0.0072644182 * t_0)) + Float64(0.0005064034 * t_4)) + Float64(0.0001789971 * t_2)) / Float64(Float64(Float64(1.0 + Float64(Float64(Float64(x * x) * 0.7715471019) + Float64(t_1 * 0.2909738639))) + Float64(Float64(t_0 * 0.0694555761) + Float64(t_4 * 0.0140005442))) + Float64(Float64(t_2 * 0.0008327945) + Float64(0.0003579942 * Float64(t_1 * t_4))))))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(1.0 + N[(0.7715471019 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0694555761 * N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[Power[N[(x * x), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x * x), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.2514179000665374 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[x, -3.1556112308066137e+32], t$95$3, If[LessEqual[x, 0.516789691633757], N[Log[1 + N[(Exp[N[(x * N[(N[(N[(N[(N[(1.0 + N[(N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * 0.7715471019), $MachinePrecision] + N[(t$95$1 * 0.2909738639), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * 0.0694555761), $MachinePrecision] + N[(t$95$4 * 0.0140005442), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 * 0.0008327945), $MachinePrecision] + N[(0.0003579942 * N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.15561123080661368e32 or 0.516789691633756987 < x

   1. Initial program 61.7

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

   2. Taylor expanded in x around inf 0.2

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

   3. Simplified 0.2

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

   if -3.15561123080661368e32 < x < 0.516789691633756987

   1. Initial program 0.6

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

   2. Applied egg-rr 0.6

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + 0.0424060604 \cdot {\left(x \cdot x\right)}^{2}\right)\right) + 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right)\right) + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left({\left(x \cdot x\right)}^{3} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot {\left(x \cdot x\right)}^{2}\right)\right)}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1556112308066137 \cdot 10^{+32}:\\ \;\;\;\;\frac{0.2514179000665374}{{x}^{3}} + \frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.516789691633757:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(\left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + 0.0424060604 \cdot {\left(x \cdot x\right)}^{2}\right)\right) + 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right)\right) + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{2} \cdot {\left(x \cdot x\right)}^{3}\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left({\left(x \cdot x\right)}^{3} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{2} \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left({\left(x \cdot x\right)}^{2} \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2514179000665374}{{x}^{3}} + \frac{0.5}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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))