Jmat.Real.dawson

Average Error: 29.6 → 0.0

Time: 5.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.2514179000665374, {x}^{-3}, \frac{0.5}{x}\right)\\ t_1 := {\left(x \cdot x\right)}^{5}\\ t_2 := {\left(x \cdot x\right)}^{2}\\ t_3 := {\left(x \cdot x\right)}^{3}\\ t_4 := \left(x \cdot x\right) \cdot t_3\\ \mathbf{if}\;x \leq -2000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(\mathsf{fma}\left(0.0424060604, t_2, 0.0072644182 \cdot t_3\right) + \mathsf{fma}\left(0.0005064034, t_4, 0.0001789971 \cdot t_1\right)\right)}{\left(1 + \mathsf{fma}\left(x \cdot x, 0.7715471019, t_2 \cdot 0.2909738639\right)\right) + \left(\mathsf{fma}\left(t_3, 0.0694555761, t_4 \cdot 0.0140005442\right) + \mathsf{fma}\left(t_1, 0.0008327945, 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot t_1\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (fma 0.2514179000665374 (pow x -3.0) (/ 0.5 x)))
        (t_1 (pow (* x x) 5.0))
        (t_2 (pow (* x x) 2.0))
        (t_3 (pow (* x x) 3.0))
        (t_4 (* (* x x) t_3)))
   (if (<= x -2000000000.0)
     t_0
     (if (<= x 10000.0)
       (log1p
        (expm1
         (*
          x
          (/
           (+
            (+ 1.0 (* 0.1049934947 (* x x)))
            (+
             (fma 0.0424060604 t_2 (* 0.0072644182 t_3))
             (fma 0.0005064034 t_4 (* 0.0001789971 t_1))))
           (+
            (+ 1.0 (fma (* x x) 0.7715471019 (* t_2 0.2909738639)))
            (+
             (fma t_3 0.0694555761 (* t_4 0.0140005442))
             (fma t_1 0.0008327945 (* 0.0003579942 (* (* x x) t_1)))))))))
       t_0))))

C

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

↓

double code(double x) {
	double t_0 = fma(0.2514179000665374, pow(x, -3.0), (0.5 / x));
	double t_1 = pow((x * x), 5.0);
	double t_2 = pow((x * x), 2.0);
	double t_3 = pow((x * x), 3.0);
	double t_4 = (x * x) * t_3;
	double tmp;
	if (x <= -2000000000.0) {
		tmp = t_0;
	} else if (x <= 10000.0) {
		tmp = log1p(expm1((x * (((1.0 + (0.1049934947 * (x * x))) + (fma(0.0424060604, t_2, (0.0072644182 * t_3)) + fma(0.0005064034, t_4, (0.0001789971 * t_1)))) / ((1.0 + fma((x * x), 0.7715471019, (t_2 * 0.2909738639))) + (fma(t_3, 0.0694555761, (t_4 * 0.0140005442)) + fma(t_1, 0.0008327945, (0.0003579942 * ((x * x) * t_1)))))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

↓

function code(x)
	t_0 = fma(0.2514179000665374, (x ^ -3.0), Float64(0.5 / x))
	t_1 = Float64(x * x) ^ 5.0
	t_2 = Float64(x * x) ^ 2.0
	t_3 = Float64(x * x) ^ 3.0
	t_4 = Float64(Float64(x * x) * t_3)
	tmp = 0.0
	if (x <= -2000000000.0)
		tmp = t_0;
	elseif (x <= 10000.0)
		tmp = log1p(expm1(Float64(x * Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(fma(0.0424060604, t_2, Float64(0.0072644182 * t_3)) + fma(0.0005064034, t_4, Float64(0.0001789971 * t_1)))) / Float64(Float64(1.0 + fma(Float64(x * x), 0.7715471019, Float64(t_2 * 0.2909738639))) + Float64(fma(t_3, 0.0694555761, Float64(t_4 * 0.0140005442)) + fma(t_1, 0.0008327945, Float64(0.0003579942 * Float64(Float64(x * x) * t_1)))))))));
	else
		tmp = t_0;
	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[(0.2514179000665374 * N[Power[x, -3.0], $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x * x), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(x * x), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(x * x), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(x * x), $MachinePrecision] * t$95$3), $MachinePrecision]}, If[LessEqual[x, -2000000000.0], t$95$0, If[LessEqual[x, 10000.0], N[Log[1 + N[(Exp[N[(x * N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0424060604 * t$95$2 + N[(0.0072644182 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$4 + N[(0.0001789971 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.7715471019 + N[(t$95$2 * 0.2909738639), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 * 0.0694555761 + N[(t$95$4 * 0.0140005442), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * 0.0008327945 + N[(0.0003579942 * N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -2e9 or 1e4 < x

   1. Initial program 59.9

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

   2. Taylor expanded in x around inf 0.0

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

   3. Simplified 0.0

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

   4. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.2514179000665374, {x}^{-3}, \frac{0.5}{x}\right)} \]

   if -2e9 < x < 1e4

   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. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{\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)}{\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)}} \cdot \sqrt[3]{\frac{\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)}{\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 \sqrt[3]{\frac{\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)}{\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. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(\mathsf{fma}\left(0.0424060604, {\left(x \cdot x\right)}^{2}, 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right) + \mathsf{fma}\left(0.0005064034, \left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}, 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)\right)}{\left(1 + \mathsf{fma}\left(x \cdot x, 0.7715471019, {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{3}, 0.0694555761, \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{5}\right)\right)\right)}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2000000000:\\ \;\;\;\;\mathsf{fma}\left(0.2514179000665374, {x}^{-3}, \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 10000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(\mathsf{fma}\left(0.0424060604, {\left(x \cdot x\right)}^{2}, 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right) + \mathsf{fma}\left(0.0005064034, \left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}, 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)\right)}{\left(1 + \mathsf{fma}\left(x \cdot x, 0.7715471019, {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{3}, 0.0694555761, \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{5}\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.2514179000665374, {x}^{-3}, \frac{0.5}{x}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))